P Hardy

Dickes

S Dolbeault
email: sylvie.dolbeault@curie.net

S Cayrou

A Bre ´ Dart

A L Viala

B Desclaux

P Saltel

A Gauvain-Piquard

P Dickes

The effectiveness of a psycho-educational group after early-stage breast cancer treatment: results of a randomized French study

Keywords: group therapy, psycho-education, randomized controlled trial, breast cancer, oncology

The effectiveness of a psycho-educational group after

early-stage breast cancer treatment: results of a randomized French study

Introduction

Women facing a diagnosis of breast cancer frequently develop psychopathological disturbances during medical treatment [START_REF] Kissane | Psychiatric disorder in women with early stage and advanced breast cancer: a comparative analysis[END_REF] or during follow-up [START_REF] Burgess | Depression and anxiety in women with early breast cancer: five year observational cohort study[END_REF], particularly anxiety, depression or both.

After treatment, women are concerned by the threat of death, the risk of recurrence, their uncertainty for the future, physical and treatment related problems, fatigue, sexuality, body image, relationships and occupational difficulties [START_REF] Stanton | Promoting adjustment after PEG after breast cancer treatment treatment for cancer[END_REF][START_REF] Kornblith | Psychosocial and sexual functioning of survivors of breast cancer[END_REF][START_REF] Kissane | Cognitiveexistential group psychotherapy for women with primary breast cancer: a randomized controlled trial[END_REF][START_REF] Shimozuma | Quality of life in the first year after breast cancer surgery: rehabilitation needs and patterns of recovery[END_REF]. In this remission phase also called re-entry transition [START_REF] Stanton | Outcomes from the Moving Beyond Cancer psychoeducational, randomized, controlled trial with breast cancer patients[END_REF], they are confronted with many individual challenges in reverting from 'cancer patient' back to 'person' and they have to cope with a new experience of the disease in a complex rehabilitation process [START_REF] Carver | Enhancing adaptation during treatment and the role of individual differences[END_REF][START_REF] Ronson | Psychosocial rehabilitation of cancer patients after curative therapy[END_REF].

All these concerns can create psychosocial adjustment disturbances, including anxiety, depression, other affects or coping disabilities. Symptoms of anxiety can arise from follow-up medical examinations, as can post-traumatic symptom disorders when dealing with physical and mental changes, feelings of abandonment by professionals or isolation from the partner and family [START_REF] Stanton | Outcomes from the Moving Beyond Cancer psychoeducational, randomized, controlled trial with breast cancer patients[END_REF].

Because of the high prevalence of psychological or psychiatric problems in breast cancer patients, a wide range of psychosocial interventions have been developed, which are diverse in type, target population, format and outcome [START_REF] Newell | Systematic review of psychological therapies for cancer patients: overview and recommendations for future research[END_REF][START_REF] Fawzy | Psychosocial intervention for patients with cancer: what works and what doesn't[END_REF][START_REF] Vos | Effects of delayed psychosocial interventions versus early psychosocial interventions for women with early stage breast cancer[END_REF][START_REF] Meyer | Effects of psychosocial interventions with adult cancer patients: a meta-analysis of randomized experiments[END_REF]. The meta-analysis by Sheard and Maguire [START_REF] Sheard | The effect of psychological interventions on anxiety and depression in cancer patients: results of two meta-analyses[END_REF] exploring anxiety and depression, reported a positive impact of psychotherapeutic interventions, with group interventions proving to be superior.

A number of studies have assessed the efficacy of cognitive behavioural techniques (CBT) or psychoeducational methods inspired from CBT in cancer patients, and a majority of these studies found significant positive outcomes for anxiety, depression, fatigue and general cancer distress [START_REF] Osborn | Psychosocial interventions for depression, anxiety, and quality of life in cancer survivors: meta-analyses[END_REF][START_REF] Savard | Randomized clinical trial on cognitive therapy for depression in women with metastatic breast cancer: psychological and immunological effects[END_REF]; among them, a recent meta-analysis [START_REF] Tatrow | Cognitive behavioral therapy techniques for distress and pain in breast cancer patients: a meta-analysis[END_REF] concerning breast cancer confirmed the efficacy of CBT both on distress and pain, and confirmed effect sizes found for distress in previously published meta-analyses [START_REF] Sheard | The effect of psychological interventions on anxiety and depression in cancer patients: results of two meta-analyses[END_REF][START_REF] Osborn | Psychosocial interventions for depression, anxiety, and quality of life in cancer survivors: meta-analyses[END_REF][START_REF] Rehse | Effects of psychosocial interventions on quality of life in adult cancer patients: meta analysis of 37 published controlled outcome studies[END_REF].

To our knowledge, most studies assessing the effectiveness of psychosocial interventions in earlystage breast cancer have been performed for diagnosis or initial treatment rather than during the remission phase [START_REF] Stanton | Promoting adjustment after PEG after breast cancer treatment treatment for cancer[END_REF][START_REF] Vos | Effects of delayed psychosocial interventions versus early psychosocial interventions for women with early stage breast cancer[END_REF][START_REF] Bloom | Risk and timing of counseling and support interventions for younger women with breast cancer[END_REF][START_REF] Scheier | Interventions to enhance physical and psychological functioning among younger women who are ending nonhormonal adjuvant treatment for early-stage breast cancer[END_REF].

With some exceptions [START_REF] Fukui | A psychosocial group intervention for Japanese women with primary breast carcinoma: a randomized controlled trial[END_REF][START_REF] Okamura | Psychoeducational intervention for patients with primary breast cancer and patient satisfaction with information: an exploratory analysis[END_REF], most trials have been performed in English-speaking cultures, preventing extrapolation of results to the French context. Indeed, psycho-oncology in France has been developed mainly by the professionals trained in the use of psychodynamic techniques, involving individual patient follow-up using psychoanalytic methods, with a reluctance for consultation-liaison techniques.

Very few psycho-oncologists are trained in CBT, brief therapy techniques and concepts of patient education. French patients are not familiar with group approaches, whether they concern psychology, psychiatry or oncology.

Given our cultural context, which is quite different from Anglo-American cultures and those of many other European countries, the challenge was to explore the feasibility of new psychooncology approaches in France [START_REF] Dolbeault | Psychoeducational group: a model of education for cancer patients[END_REF], using both group formats and therapeutic tools from the psycho-educational field.

Methods

Purpose

This randomized trial aimed to assess the effectiveness of a psycho-educational group (PEG) intervention on anxiety, mood, mental adjustment and quality of life in early-stage breast cancer. As our first objective, we hypothesized a significant reduction in anxiety-a possible result of posttreatment patients' feelings-in the PEG compared with the control group (CG). Our secondary objectives included an improvement of patients' adjustment strategies and quality of life profiles as a result of the group intervention.

Design and sample

After receiving permission from the Institutional Review Board of Clinical Research and National Ethics Committee, this randomized pre-post intervention study was conducted in three French cancer centres. Eligible women (1) were 18 or older, (2) had completed primary breast cancer treatment (radiation alone or combined chemoradiotherapy 15 days to 1 year before testing), (3) had no recurrence or metastases, (4) had working knowledge of French, (5) had no psychiatric diagnosis such as severe cognitive disorders, mood disorders (ongoing or recent history of depression requiring hospitalization) or serious personality disorders.

Approximately 1000 letters were sent out, and an estimated 20% of those patients contacted answered positively. Randomization was performed to distribute the eligible patients after an interview to clarify their motivations. The final sample consisted of 203 early-stage breast cancer patients (Paris, 107; Toulouse, 51; Lyon, 45) randomized between the 'immediate' treated group (TG: n 5 102) and the 'deferred' control group (CG: n 5 101). Randomization by sealed letter was performed at each site, with a readjustment of the number of subjects in each group after every eighth subject. After randomization, patients who dropped out were not replaced. Patients who missed four group sessions were excluded from the analyses.

Assessments were performed three times on all participants: 1 week before starting the immediate intervention (E1), the second after completion of the intervention (E2) and the third, one week before the beginning of the deferred intervention (E3). Assessment at E1 was performed in the hospital and E2 and E3 were done at home (Figure 1). approach, all PEG exercises were initiations combined with general medical information and peer exchanges on defined themes (causes and significance of cancer, impact of treatments on body image, managing uncertainty, improving communication with loved ones, etc.). The programme included 8 weekly 2 h sessions. The groups were composed of 8-12 participants led by 2 therapists, either psychologists or psychiatrists, trained in group therapy and BCT [START_REF] Dolbeault | Psychoeducational group: a model of education for cancer patients[END_REF].

The intervention was free of charge for patients, therapists being paid by hospital and research funding. The detailed content of this PEG has been described elsewhere [START_REF] Cayrou | Groupes psycho-e´ ducationnels pour femmes atteintes de cancer du sein en France: the` mes et techniques[END_REF] and a manual (2001) can be obtained by written request from the first author. Monthly meetings were organised with leaders, co-leaders and investigators to ensure the coherence of intervention delivery.

Measurements

Socio-demographic and medical data, disease stage, time since diagnosis and treatments received -either medical (surgery, chemotherapy, radiotherapy) or psychological (individual support, psychotropic drugs) -were recorded (Table 1). For missing data, we applied the method of data allocation recommended by Jo¨reskog & So¨rbom [START_REF] Jo¨reskog | Prelis 2. User's Reference Guide[END_REF] using PRELIS software. Statistical analyses were performed using SPSS and LISREL software. Statistical significance was set at p = 0.05.

PEG after breast cancer treatment

The main objective of the statistical analyses was to compare the course of the two groups between E1 and E3, including E2, on STAI state and trait anxiety scales, after confirming their comparability at E1 on all socio-demographic and medical variables as well as psychosocial evaluations.

Quantitative socio-demographic data were ana- Patients completed validated French versions of the following self-report questionnaires.

Anxiety was measured using the State-Trait Anxiety Inventory (STAI, whose scores range from 20 to 80 with a higher score reflecting a higher level of anxiety [START_REF] Spielberger | Manual for the State-Trait Anxiety Inventory (Forme Y)[END_REF][START_REF] Schweitzer | Manuel pour l'Inventaire d'Anxie´te´ Trait-Etat (Forme Y)[END_REF]). Psychological adjustment was assessed with the Profile of Mood States (POMS) [START_REF] Mcnair | EdITS Manual for the Profile of Mood States. Educational and Industrial Testing Service[END_REF][START_REF] Cayrou | Validation de la traduction franc -aise du POMS (Profile Of Mood States)[END_REF][START_REF] Cayrou | Version franc -aise du profile of mood states[END_REF], comprising different scales: anxiety (with scores ranging from 0 to 36), anger (0 to 48), confusion (0 to 28), depression (0 to 60), fatigue (0 to 28), vigor (0 to 32), interpersonal relationships (0 to 28). It includes a global score (from -32 to 200). A higher score indicates an increasing intensity of the mood state, whether it be negative affects (e.g.: anxiety, anger) or positive affects (e.g.: vigor, relationships). The Mental Adjustment to Cancer Scale (MAC) [START_REF] Watson | Mental Adjustment to Cancer Scale: User's Manual[END_REF][START_REF] Cayrou | Validation d'une version franc -aise de la MAC (Mental Adjustment to Cancer) scale[END_REF][START_REF] Cayrou | The Mental Adjustment to Cancer (MAC): French replication and assessment of positive and negative adjustment dimensions[END_REF] provides 5 subscales measuring fighting spirit (13-52), helplessnesshopelessness , anxious preoccupation , avoidance [START_REF] Kissane | Cognitiveexistential group psychotherapy for women with primary breast cancer: a randomized controlled trial[END_REF][START_REF] Shimozuma | Quality of life in the first year after breast cancer surgery: rehabilitation needs and patterns of recovery[END_REF][START_REF] Stanton | Outcomes from the Moving Beyond Cancer psychoeducational, randomized, controlled trial with breast cancer patients[END_REF][START_REF] Carver | Enhancing adaptation during treatment and the role of individual differences[END_REF][START_REF] Ronson | Psychosocial rehabilitation of cancer patients after curative therapy[END_REF][START_REF] Newell | Systematic review of psychological therapies for cancer patients: overview and recommendations for future research[END_REF][START_REF] Fawzy | Psychosocial intervention for patients with cancer: what works and what doesn't[END_REF][START_REF] Vos | Effects of delayed psychosocial interventions versus early psychosocial interventions for women with early stage breast cancer[END_REF][START_REF] Meyer | Effects of psychosocial interventions with adult cancer patients: a meta-analysis of randomized experiments[END_REF][START_REF] Sheard | The effect of psychological interventions on anxiety and depression in cancer patients: results of two meta-analyses[END_REF][START_REF] Osborn | Psychosocial interventions for depression, anxiety, and quality of life in cancer survivors: meta-analyses[END_REF][START_REF] Savard | Randomized clinical trial on cognitive therapy for depression in women with metastatic breast cancer: psychological and immunological effects[END_REF][START_REF] Tatrow | Cognitive behavioral therapy techniques for distress and pain in breast cancer patients: a meta-analysis[END_REF][START_REF] Rehse | Effects of psychosocial interventions on quality of life in adult cancer patients: meta analysis of 37 published controlled outcome studies[END_REF][START_REF] Bloom | Risk and timing of counseling and support interventions for younger women with breast cancer[END_REF][START_REF] Scheier | Interventions to enhance physical and psychological functioning among younger women who are ending nonhormonal adjuvant treatment for early-stage breast cancer[END_REF] and denial [START_REF] Stanton | Promoting adjustment after PEG after breast cancer treatment treatment for cancer[END_REF][START_REF] Kornblith | Psychosocial and sexual functioning of survivors of breast cancer[END_REF][START_REF] Kissane | Cognitiveexistential group psychotherapy for women with primary breast cancer: a randomized controlled trial[END_REF][START_REF] Shimozuma | Quality of life in the first year after breast cancer surgery: rehabilitation needs and patterns of recovery[END_REF][START_REF] Stanton | Outcomes from the Moving Beyond Cancer psychoeducational, randomized, controlled trial with breast cancer patients[END_REF][START_REF] Carver | Enhancing adaptation during treatment and the role of individual differences[END_REF][START_REF] Ronson | Psychosocial rehabilitation of cancer patients after curative therapy[END_REF][START_REF] Newell | Systematic review of psychological therapies for cancer patients: overview and recommendations for future research[END_REF][START_REF] Fawzy | Psychosocial intervention for patients with cancer: what works and what doesn't[END_REF][START_REF] Vos | Effects of delayed psychosocial interventions versus early psychosocial interventions for women with early stage breast cancer[END_REF]. High-er scores indicate a higher level of the mental adjustment score.

Patients' quality of life profiles were evaluated by the EORTC core quality of life questionnaire (EORTC QLQ-C30) [START_REF]EORTC Quality Of Life Study Group[END_REF][START_REF] Aaronson | The European organization for research of treatment of cancer QLQ-C30. Quality of life instrument use in international clinical trials in oncology[END_REF] and breast cancer module (EORTC QLQ-BR23). Items are scored from 1 to 4, except the global health status items, which range from 1 to 7. A high score for functional or global health status scales represents a healthy level of functioning; a high score for a symptom scale/item indicates a high level of symptoms or problems [START_REF] Sprangers | The European organization for research and treatment of cancer breast cancer-specific quality-of-life questionnaire module: first results from a three-country field study[END_REF]. advantage of providing t-test values for each variable, and a general value (Wilks' lambda) that takes all the variables entered into the model and their correlations into account.

We compared the course of the two groups between E1 and E3 using the general linear models with repeated measurements, which take all three measures into account, to distinguish between intra-subject effects (time effect and time-by-group interaction) and inter-subject effects. Statistical analyses also controlled for confounding variables, partly responsible for the variance of the scores of the variables measured.

The effects of confounding variables were tested in a preliminary analysis. First, they were entered as fixed criteria, without the group variable, in a repeated measures model for each dependent variable. If a significant effect was observed, then the variables were considered as covariates and remained in the complete model when the group differences were tested. If no significant effects were observed, only the groups were considered as independent variables, without covariates.

Results

Sample characteristics

Of the 203 women recruited for the study, 33 (16.3%) dropped out: 14 from the TG and 19 from the CG. The main reasons were: disappointment with the group (n = 7), disappointment with the randomized allocation (n = 6), work problems (n = 4), transportation problems (n = 3), family problems (n = 3), other non-cancer health problems PEG after breast cancer treatment (n = 4), cancer recurrence (n = 1), other reasons (n = =).

Most patients attended all sessions. Absences were rare and were always motivated by material or health problems and were equivalent in both the TG and CG.

Patients who dropped out were not significantly different from PEG-participating patients as concerns their STAI, POMS and MAC scores at E1, but their socio-demographic characteristics were different. They were more frequently from the working class or were unemployed, compared with full-length participants who were more frequently from middle and high-level executive categories. They had also needed more extensive surgery (mastectomy with lymph node dissection versus tumorectomy) than did the other participants.

The overall missing data rate in questionnaires ranged from 0.02 to 6.29% (the sum of missing data divided by the product of the number of subjects and questionnaire items) depending on the questionnaire and assessment time. For up to 10% of the missing data, the described procedures were used to impute values, questionnaires being excluded afterwards.

Comparisons of baseline characteristics between groups

The two groups were comparable for all variables except age (p = 0.03) and the presence of chemotherapy (p = 0.02). The CG was younger (mean age: 51.60 versus 54.47) and had more frequently received chemotherapy (57 versus 43%). No significant difference was found for scores on the STAI, POMS, MAC, EORTC QLQ-C30 and QLQ-BR23 scales, except the EORTC QLQ-C30 'global health state' scale (p = 0.01), where the CG reported better health status (data not shown).

Age (50 or under versus over 50) influenced the time-course of the POMS anxiety, anger, fatigue and global scores, the EORTC QLQ-C30 or QLQ-BR23 emotional and role functioning, fatigue, body image, sexuality, adverse effects and symptom scales. The presence of chemotherapy influenced the course of the EORTC QLQ-C30 pain, appetite, constipation and QLQ-BR23 breast symptom scores.

Comparisons of change in scores within and between treatment groups Some patients were removed from the statistical multivariate analysis because they did not complete the three evaluations, a fact which explains the reduction in number from 102 to 81 in the TG and from 101 to 87 in the CG.

As described in Table 3, in both groups changes were observed over time in the STAI trait and state anxiety scores, the POMS anxiety, anger, confusion, depression and global scores, the MAC helplessnes-shopelessness and anxious preoccupations scores, in the EORTC QLQ-C30 scores for physical, emotional, cognitive and social functioning, dyspnoea, sleep and financial difficulties, and in the QLQ-BR23 body image, future prospects and breast symptom scores.

Controlling for a time effect, significant group/ time interactions indicate a positive effect of the intervention on anxiety, our primary outcome measure. This was evidenced for the STAI state and trait anxiety scales, explaining 6% and 4% of the variance in the STAI-state and STAI-trait anxiety scores, respectively (Table 3).

We also found positive results on other variables identified as secondary outcome measures. A greater reduction of negative affects and improvement in positive affects and in quality of life functional or symptom scales were observed in the TG compared with the CG. This concerned the POMS anxiety and global scores (8% of the variance explained by the model including the time/group interaction term), scores for fatigue (7%), anger (5%), interpersonal relationships (4%), vigor (3%) and depression (2%) and the EORTC QLQ-C30 scores, emotional functioning (9%), role functioning (3%), global health status (3%) and fatigue (3%).

In contrast, no effect of the PEG was evidenced on the MAC scale or on the POMS confusion scores.

The study aimed to test the effect of PEG interventions on psychosocial adjustment in posttreatment early-stage breast cancer patients, in a multi-site French study. Controlling for age and past chemotherapy, a PEG intervention effect was clearly visible, as measured 1-month post-intervention, regardless of the passage of time, on patients' levels of anxiety and on mood states (except confusion), emotional and role functioning, fatigue and overall health status. However, no effect of the PEG was observed on mental adjustment, i.e. measures of emotional, cognitive or behavioural responses to the breast cancer experience in these patients in remission. Lower levels of helplessness-hopelessness or anxious preoccu pation were evidenced in TG and CG alike, reflecting spontaneous psychological improvement over time.

Our positive results are in line with data from other studies in different cultural contexts on the efficacy of group psychological therapy on mood disturbance, anxiety or depression in early-stage breast cancer patients [START_REF] Kissane | Cognitiveexistential group psychotherapy for women with primary breast cancer: a randomized controlled trial[END_REF][START_REF] Fukui | A psychosocial group intervention for Japanese women with primary breast carcinoma: a randomized controlled trial[END_REF][START_REF] Spiegel | Group psychotherapy for recently diagnosed breast cancer patients: a multicenter feasibility study[END_REF], as well as on quality of life aspects, probably as a result of positive variations in emotional states [START_REF] Shimozuma | Quality of life in the first year after breast cancer surgery: rehabilitation needs and patterns of recovery[END_REF][START_REF] Jacobsen | Efficacy and costs of two forms of stress management training for cancer patients undergoing chemotherapy[END_REF].

As noted in this study, Kissane et al. [START_REF] Kissane | Cognitiveexistential group psychotherapy for women with primary breast cancer: a randomized controlled trial[END_REF] did not find any effect of their intervention on mental adjustment scores. Conversely, using a similar intervention, Fukui et al. [START_REF] Fukui | A psychosocial group intervention for Japanese women with primary breast carcinoma: a randomized controlled trial[END_REF] found higher scores for fighting spirit among treated patients, immediately or 6 months after an educational intervention, and Helgeson et al. [START_REF] Helgeson | Long-term effects of educational and peer discussion group interventions on adjustment to breast cancer[END_REF] highlighted better adjustment in post-diagnosis breast cancer patients participating in educational intervention compared with peer discussion groups.

Several hypotheses could explain our results; that is, the PEG effect observed on emotional states but not on mental adjustment.

The MAC scale has demonstrated its clinical sensitivity [START_REF] Moorey | A comparison of adjuvant psychological therapy and supportive counselling in patients with cancer[END_REF]. The lack of PEG effect on mental adjustment may be related to the psychotherapy technique used. Although our PEG was intended to teach targeted coping skills through techniques inspired from BCT, it was mixed with supportiveexpressive moments in each session, which may have emphasized the experiential facet of the therapy to the detriment of the educational part. The PEG respected the reluctance of some of the groups to directly confront subjects considered to be taboo-some authors [START_REF] Classen | Brief Supportive-expressive Group Therapy for Women with Primary Breast Cancer: a Treatment Manual[END_REF][START_REF] Fawzy | Patient Manual: Intervention for Cancer Patients[END_REF] think that this could be more effective in the long term. Indeed, groups may alleviate general anxiety by enabling patients to feel less isolated, legitimise personal feelings and find reassurance through peer exchanges, while maintaining a general avoidance attitude.

BCT-inspired techniques might need more than eight sessions to demonstrate improvement in adjustment strategies. Increases in anxiety for some patients while facing their experience during therapy could be a sign of mental adjustment, which could be beneficial in the long run. Further research is needed to pinpoint the specific mechanisms by which this type of intervention produces relief from anxiety or negative mood [START_REF] Bre´ Dart | Letter to the Editor: Re: systematic review of psychological therapies for cancer patients: overview and recommendantions for future[END_REF].

Our study presents certain methodological limitations.

The early implementation of the 'deferred' group therapy (for ethical reasons) prevented comparative assessment of long-term effects of the intervention.

Our results are based solely on subjective measures of psychological status, so there is a possible psychosocial bias (e.g. Hawthorne effect [START_REF] Adair | The Hawthorne effect: a reconsideration of the methodological artifact[END_REF]). Objective measures such as return to work might be an important psychosocial outcome in breast cancer survivors [START_REF] Stanton | Psychosocial concerns and interventions for cancer survivors[END_REF].

There is no comparison of participants to nonparticipants.

The lack of complete data for one-fifth of the patients, who did not complete the questionnaires at all three evaluation times, weakens our results.

The sample was too small to evaluate differences between participating therapists (despite many efforts to ensure the consistency of intervention delivery), or to assess the specific impact of the PEG on sub-groups of patients defined by psychological characteristics.

However, despite these weaknesses, the PEGs clearly had an effect on our patients' anxiety and some aspects of mood and QOL.

Adding tools from group dynamics analysis could be useful in our PEGs so as to better understand the therapeutic process, including mechanisms of change.

Choosing anxiety as a primary outcome measure can be debated. Owen et al. [START_REF] Owen | Psychosocial interventions for cancer: review and analysis using a three-tiered outcomes model[END_REF] suggested variables reflecting patients' well-being as the most appropriate outcome measures to assess the effectiveness of psychological therapy in cancer patients.

Our PEG intervention recruited patients without criteria of significant clinical distress but, as recently suggested by various authors [START_REF] Vos | Psychosocial intervention for women with primary, non-metastatic breast cancer: a comparison between participants and non-participants[END_REF][START_REF] Boesen | Psychoeducational intervention for patients with cutaneous malignant melanoma: a replication study[END_REF][START_REF] Classen | Supportive-expressive group therapy for primary breast cancer patients: a randomized prospective multicenter trial[END_REF], short-term interventions could offer notable benefit to clinically distressed patients.

The interest of such interventions should now be tested in specific populations, bringing other sociodemographic variables into account like income or cultural origin (given the diversity of the French population), to determine which patients could benefit most [START_REF] Cordova | Mood disturbance in community cancer support groups. The role of emotional suppression and fighting spirit[END_REF][START_REF] Scheier | Moderators of interventions designed to enhance physical and psychological functioning among younger women with early-stage breast cancer[END_REF].

Conclusion

This experience proves that structured PEGs for French patients with early-stage breast cancer in remission have a positive effect on emotional states, especially in reducing anxiety levels. Although improvement in these outcomes was observed over time in both TG and CG , the PEG appeared to act as an accelerator of these effects. Further work should seek to identify and integrate breast cancer patients who particularly need this psychological approach, and to develop and test these psychotherapeutic techniques in the French cancer treatment milieu. PEGs represent an excellent complement or an alternative to the individual psycho-oncological treatment techniques, which are widely offered in France, and should now be tested in other types of cancer patients and in other stages of the disease.

Table 1 .

 1 Socio-demographic and clinical variables by randomization arm (significant data only)

	Intervention
	On the basis of previous studies with group
	interventions for women with breast cancer [24-
	28], a psychoeducationally structured model based
	on CBT principles was developed. Patients were
	taught to routinely use thought records, to practise
	problem-solving and cognitive restructur-ing, to
	communicate better with caregivers and health
	professionals through role-play, and to practice
	relaxation. Unlike	a comprehensive CBT

Table 2 .

 2 Variables influenced by age and chemotherapyThe EORTC core was administered only at E1 and E3 to reduce evaluation fatigue.

			Inter-subject	Intra-subject
		Confounding	effects a	effects b
		Variables	P<0.05	P<0.05	Statistical analyses
	Anx	Age		0.006
	Ang	Age		0.006
	Fat	Age	0.012	0.048
	ScG	Age		0.005
	RF	Age	0.039
	EF	Age	0.033
	FA: fatigue	Age	0.004
	NV: nausea	Age*cte	0.034
	PA: pain	Age	0.031
		Cte	0.010
	AP: appetite	Cte	0.016
	CO: constipation	Cte		0.012
		Cte*age		0.036
	FI: financial difficulties Age*cte		0.050
	BRBI: body image	Age		0.020

Table 3 .

 3 Comparison of change scores between randomization arms (Group: n = 81; Control: n = 87)

			E1 Mean (SD)	E2 Mean (SD)	E3 Mean (SD)		p	Eta 2
	STAI Y-A	Group	46.27 (13.87)	40.60 (11.43)	39.75 (10.55)	Intra-subject	0.001	0.06
		Control	43.85 (12.16)	45.77 (12.68)	43.85 (10.94)	Inter-subject	0.140	0.01
						Time x group	0.001	0.06
	STAI Y-B	Group	47.53 (10.42)	43.22 (10.02)	(10.04)	Intra-subject	0.000	0.17
		Control	47.48 (9.31)	47.94 (10.46)	45.46 (10.17)	Inter-subject	0.064	0.02
						Time x group	0.010	0.04
	POMS Anxiety a	Group	15.12 (8.11)	10.58 (6.89)	9.90 (6.44)	Intra-subject	0.001	0.07
		Control	14.69 (7.63)	14.71 (8.35)	12.78 (7.45)	Inter-subject	0.059	0.02
						Time x group	0.000	0.08
	POMS Anger a	Group	13.83 (8.71)	9.74 (7.43)	9.49 (6.81)	Intra-subject	0.000	0.07
		Control	13.29 (8.64)	14.28 (9.11)	11.44 (7.91)	Inter-subject	0.124	0.01
						Time x group	0.005	0.05
	POMS Confusion	Group	8.78 (4.85)	7.12 (4.73)	6.91 (4.74)	Intra-subject	0.000	0.11
		Control	9.84 (5.70)	10.11 (6.24)	8.90 (5.83)	Inter-subject	0.008	0.04
						Time x group	0.149	0.01
	POMS Depression	Group	12.38 (11.45)	8.90 (10.21)	7.86 (8.61)	Intra-subject	0.000	0.14
		Control	13.46 (11.95)	13.54 (12.56)	11.40 (10.78)	Inter-subject	0.047	0.02
						Time x group	0.034	0.02
	POMS Fatigue a	Group	10.01 (7.38)	7.74 (6.89)	6.86 (5.58)	Intra-subject	0.069	0.02
		Control	8.78 (6.85)	10.23 (7.08)	8.87 (6.84)	Inter-subject	0.370	0.01
						Time x group	0.000	0.07
	POMS Vigor	Group	15.02 (5.29)	16.22 (6.28)	16.31 (5.96)	Intra-subject	0.109	0.02
		Control	15.21 (6.28)	14.51 (6.48)	15.00 (5.52)	Inter-subject	0.253	0.01
						Time x group	0.027	0.03
	POMS Interpersonal	Group	17.64 (4.17)	18.91 (3.42)	18.80 (3.49)	Intra-subject	0.166	0.01
	Relationships	Control	17.79 (3.74)	17.52 (3.85)	17.41 (3.67)	Inter-subject	0.072	0.02
		Group				Time x group	0.007	0.04
	POMS	Group	45.10 (36.70)	27.86 (35.24)	24.73 (30.91)	Intra-subject	0.001	0.06
	Global score a
		Control	44.85 (37.83)	48.37 (41.77)	38.39 (35.57)	Inter-subject	0.040	0.03
						Time x group	0.000	0.08
	MAC Hopelessness/	Group	19.67 (5.96)	18.65 (5.25)	17.80 (5.10)	Intra-subject	0.000	0.12
	Helplessness
		Control	19.26 (6.12)	19.23 (5.97)	18.46 (5.90)	Inter-subject	0.737	0.00
						Time x group	0.063	0.02
	MAC	Group	27.58 (8.00)	24.99 (6.54)	24.22 (6.67)	Intra-subject	0.000	0.29
	Anxious
	Preoccupation	Control	27.70 (6.70)	26.57 (6.84)	25.29 (6.62)	Inter-subject	0.352	0.01
						Time x group	0.183	0.01
	EORTC	Group	1.63 (0.53)	-	1.56 (0.46)	Intra-subject	0.027	0.03
	Physical
	Functioning	Control	1.57 (0.54)	-	1.51 (0.48)	Inter-subject	0.43	0.00
						Time x group	0.74	0.00
	EORTC	Group	1.84 (0.83)	-	1.55 (0.66)	Intra-subject	0.141	0.01
	Role
	Functioning a	Control	1.63 (0.74)	-	1.59 (0.70)	Inter-subject	0.372	0.01
						Time x group	0.017	0.03

a Inter-subject effect: group effect. b Intra-subject effect: time effect. Time x group: time/group interaction effect. Test controlled with 'age' as covariates;

(b)

Test controlled with 'presence of chemotherapy' as covariates. The EORTC QLQ-C30 and BR23 were administered at E1 and E3 only. EORTC QLQ-BR23 hair loss and sexual satisfaction items were not included in the analyses because of missing responses (in both cases) or irrelevance (no more alopecia). Non-standardized rough scores are used to maintain a uniform significance of the mean values.

Acknowledgements

This research received a grant from a French PHRC (Programme Hospitalier de Recherche Clinique (Hospital Programme of Clinical Research).


Professor Ahmed Abouelaz
email: ah.abouelaz@gmail.com

This book contains two parts: Part 1: Radon transforms on some Riemannian spaces. Part 2: Diophantine integral geometry in the lattice Z n . Our work was focused on study of the Radon transform on several spaces: Euclidean space R n , sphere S d , the n-dimensional torus T n , lattice Z n and Damek-Ricci space (N A -spaces). The Paley-Wiener theorems, Paley-Wiener-Schwartz theorems, inversion formulas for the Radon transforms, support theorems for the Radon transforms, Plancherel formulas for the Radon transform, are discussed in details in these spaces. The complex analog of the Radon transform is known as the Penrose transform. The Penrose transform be considered in the forthcoming book. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross -selectional scans of an object. It is known that the Radon transform has many applications precisely in geology, astronomy, medecine and physics. We study also the discrete Radon transform and the d-plane Radon transform on the lattice Z n . Precisely We characterize the image of exponential type functions under the discrete Radon transform R on the lattice Z n of the Euclidean space R n , (n ≥ 2). We also establish the generalization of Volberg' s uncertainly principle on Z n , which is proved by means of this characterization. It is noteworthy that this work is the fruit of several papers of the author and also by his collaborators (see the references of this book).

I am grateful to Professor François Rouvière for its advices and suggestions.

Notations and conventions

-N = {0, 1, 2, 3, . . .} -(-N) = {. . . , -3, -2, -1, 0} -Z = N ∪ (-N) -R is the real line -R + = {x ∈ R | x ≥ 0} . -C is the complex line, that is: C = a + ib | (a, b) ∈ R 2 .
-S (Z n ) is the space consisting of all complex-valued rapidly decreasing f defined on Z n and called the Schwartz space. -C (Z n ) is the subspace of S (Z n ) consisting of all complex-valued functions f defined on Z n with finite support. -l p (Z n ), 1 ≤ p < +∞, be the space of all complex-valued functions f on Z n such that m∈Z n |f (m)| p < +∞. -l ∞ (Z n ) be the space of all complex-valued functions f on Z n such that sup m∈Z n |f (m)| < +∞. -W is a Lie group and w is its Lie algebra. -δ j,k is the Kronecker delta: δ j,k = 0 for j = k and δ j,j = 1.

-δ x the delta-measure at the point x : δ x (φ) = φ (x) .

-If G is a group, we denote by G the dual of G, i.e the set of all continuous unitary representations of G. -GL (n, C) the set of all complex matrices of order n with nonzero determinant. -SL (n, C) the set of all complex matrices of order n with determinant equal to 1. -SL * (n, Z) the special linear group of degree n over a ring Z, is the set of all n × n matrices with determinant equal ±1. -d (a) the greatest common divisor of the the integers a 1 , . . . , a n .

-D (W ) = C ∞ c (
-P the set of all element a = (a 1 , . . . , a n ) ∈ Z n {0} such that d (a) = 1.

-Let G be a group and K is the subgroup of G, K acts on G. If this action is transitive, then G K is homogenuous space: example the Steifel variety. -R s the spherical Radon transform on the sphere S d .

-The Spherical Fourier transform associated to R s is given by

f (n) = π 0 cos [(n + λ) t] R s f (t) dt, for all n ∈ N.
-B being the Euler Beta function.

-C λ n (cos t) is the Gegenbauer polynomial. -C ∞ (G K) the space of all complex-valued functions defined on G = SO (d + 1) which are K = SO (d) bi-invariants and C ∞ on G. -The Spherical Fourier transform on S d can be written as

f (n) = n! (2λ)! B (λ + 1 2, 1 2) π 0 f (a t ) C λ n (cos t) (sin t) 2λ dt,
for all f ∈ C ∞ (G K), -R α,β is the generalized the spherical Radon transform on the sphere S d .

-The generalized spherical Fourier transform, associated to R α,β , on compact symmetric space is defined by (cos t) the Jacobi polynomial with α > β > - 1 2 . -We define the classical Radon transform as follows:

f (n) = 1 Γ (α + 1) π 0 f (t) γ (α,
R c f (H (ω, t)) = H(ω,t) f (x) dµ(x), where H (ω, t) = {x ∈ R n | ωx = t} is the hyperplane of R n . ω = (ω 1 , . . . , ω n ) ∈ S n-1 and ωx = x 1 ω 1 + • • • + x n ω n .
-Let G a Lie group and g its Lie algebra, we denote U g the universal enveloping algebra. -D g (G) be the algebra of the invariants differential operators by the left translation.

-Let N be the real vector space of the finite dimensional, W a Lie group and G = W N the semidirect product, we define the representation Π λ of G as follows

Π λ (g) ϕ (w) = λ wnw -1 ϕ (wv) for all (v ,w) ∈ W 2 , g = nv, w ∈ W
-N * is the complexified of the dual N (the set of the characters of N ); then

Π λ = Ind G N λ. -E (G)
the space of all distributions with compact support on G = W N.

-For λ ∈ N * and S ∈ E (G) , we define the Fourier-Laplace transform FS (λ) as operator with dense domain on H by

FS (λ) ϕ (w) =< S(g), Π λ (g) ϕ (w) >, with ϕ ∈ H ∞ ⊂ H and g = nw ∈ G = W N.
-H ∞ is the Garding espace included in the Hilbert space H.

-In sections 2.7 and 3.2, we define the classical Fourier transform as follows

ψ (λ) = N ψ (n) λ (-n) dn for all ψ ∈ C ∞ c (N ) ,
where C ∞ c (N ) is the space of all functions infinitely differential functions on N with compact support. -F I is the one-dimensional Fourier transform given by

F 1 f (t) = R f (x) exp (-2iπxt) dx, for all t ∈ R.
-The analogue of the Helgason'operator is defined by

F 1 (-y ) k ϕ (ω, s) = |s| 2k F 1 ϕ (ω, s) exp (sωy) , for all y ∈ R n
and (ω, s) ∈ S n-1 × R, ϕ belongs to S S n-1 × R . -We denote F c the classical Fourier transform defined by

F c f (λ) = R n f (x) exp (-2iπλx) dx for all (λ, f ) ∈ R n × C ∞ c (R n ) .
-We define the Cauchy-Lipshitz's class Lip (α, Z n , 2) by

Lip (α, Z n , 2) = f ∈ l 1 (Z n ) | Ff (x + h) -Ff (x) L 2 (T n ) = O (|h| α ) , h -→ 0
-For the n-dimensional torus, we define the Cauchy-Lipshitz's class as follows

Lip (α, T n , 2) = f ∈ L 2 (T n ) | f (x + h) -f (x) L 2 (T n ) = O (|h| α ) , h -→ 0 .
-In section 3.5, we denote f (π) the Fourier operator of the goup G, where π ∈ G ( G being the unitary dual of G, i.e the set of all continuous unitary irreductible representations of the group G.

-L is the Laplace-Beltrami operator in S d . -For d ∈ N {0} such that 1 ≤ d ≤ n, the set consisting of all integers d × n matrices is designated by M d,n (Z) . -P d,n is the subset of all elements A of M d,n (Z) such that A = QD 0 V ,
where Q ∈ SL * (d, Z) and V ∈ SL * (n, Z) and D 0 is the matrix given by

D 0 =      1 0 • • • 0 0 • • • 0 0 1 • • • 0 0 • • • 0 . . . . . . . . . . . . . . . • • • . . . 0 0 • • • 1 0 • • • 0     
(see section 4.19) -The d-plane Radon transform on the torus T n is defined by

Rf (D (x, A)) = 1 0 • • • 1 0 f (x + pr ((t 1 , . . . , t d ) A) dt 1 • • • dt d .
-For x ∈ T n and A ∈ P d,n , we denote by D (x, A) the d-plane in the torus T n given by

D (x, A) = x + pr ((t 1 , . . . , t d ) A) | (t 1 , . . . , t d ) ∈ R d .
-P k is the set defined by

P k = {A ∈ P d,n | Ak = 0} , where k = (k 1 , • • • , k n ) ∈ Z n . -We denote δ (α, β) the function defined by δ (α, β) = 1 if α = β 0 otherwise , where (α, β) ∈ Z 2 . -For (k, A) ∈ Z n × P d,n , we denote Rf (k, A) by Rf (k, A) = f (k) if Ak = 0 0 if Ak = 0. ,
-The Damek-Ricci space is a semi-direct product of R + and a two step nilpotent group, i.e η, η, η = 0 and η, η = 0, where η is the Lie algebra of the nilpotent group N. The Lie algebra can be decompose in the orthogonal sum of two spaces p and z, η = p⊕ z, where z is the center of η.

-We denote k the dimension of z and m = dim p = 2m . We designate A the multiplicative group isomorph to R {0} . Then the Damek-Ricci space is denoted by S = N A.

-Denote Q = 1 2 m + k = .
The multiplicative of S is defined by na.n a = n an a -1 aa , where (n, a) ∈ N × A and (n , a ) ∈ N × A.

-c (λ) is the generalized Harish-Chandra function defined by

c (λ) = Γ (2iλ) Γ ((m + k + 1) 2) Γ (Q 2 + iλ) Γ ((m + 2) 4 + iλ)
.

-For n 1 fixed in N, we define the Poisson kenel on N A by

P (., n 1 ) : N A -→ R na -→ P a n -1 1 n ,
where, for a > 0, P a (n) is the function on N defined by

P a (n) = P a (X, Z) = a Q a + |X| 4 2 + |Z| 2 -Q
.

-The kernel P λ (λ ∈ C) on N A × N is defined by

P λ : N A × N -→ R (na, n) -→ P λ (na, n) = P a n -1 n 1 2-(iλ) Q -For f ∈ D (N A)
, the Fourier-Helgason transform is defined by

f (λ, n) = N A f (x) P λ (x, n) dx, for all (λ, n) ∈ C × N.
-The horocyclic Radon transform is

Rf (λ, n) = F 1 f (λ, n) = R n f (λ) = e λQ 2 N f (nσ (n 1 exp (λH))) dn 1 ,
where σ is the geodesic inversion defined by

σ (V, Z, a) = 1 a + |V | 4 2 2 + |Z| 2 -a + |V | 4 2 + J Z V, -Z,
f (m) , for all m ∈ l 1 (Z n ) .
-For α > 0, we define the Fourier-Hermite transform as follows

H α f (λ) = R n f (x)exp -α||x -λ|| 2 -2iπλx dx
We denote the Hermite-Radon transform, by

R α,t f (ω, p) = xω=p f (x) e α,ωt (x) dµ (x)
where e α,ωt (x) = exp -α ||x -ωt|| 2 and (ω, t, p) ∈ S n-1 × R × R.

Contents

Introduction

The main purpose of this book is to study several integral transforms, precisely: Radon transforms, discrete Radon transforms on the lattice Z n and the discrete Fourier-Bessel transform on Z n . Our study was focused on the study of the Radon transform on several spaces such that: Euclidean space R n , compact symmetric spaces S d = (SO (d + 1) /SO (d)), the n-dimensional torus T n = (R n /Z n ), the lattice Z n , the Damek-Ricci space (N A -space) and the finite Radon transform. Note that these transformations have many applications in physic, also in chemistry . . . etc. The Radon transform and discrete Radon transform are quite studied in this book; since these transformations have the great importance in the field of medicine also in geology. This work is divided in two parts: Part I: Radon transform on certain Riemannian spaces. PartII: Study of certain transformations on Z n . The first part is constituted by five chapters: The Chp. 1 contains basic concepts of the theory of groups and homogeneous spaces. In Chap. 2, we study in a detailed manner the harmonic analysis of the semi direct product G = N W ; where N is a vector space a finite dimension and W is the connected Lie group. The main results of this chapter are theorems of Paley-Wiener on the group G = N W and also the Paley-Wiener-Schwartz on G. Precisely, in this work we characterize the Fourier image of several distribution spaces on some Lie groups and we give applications. The theorems of Paley-Wiener described with respect to a fundamental family of compact and multiplication operators. These objects are constructed via a sub-multiplicative function own continuing. The Paley-Wiener space distributions of compact support becomes the operators of the space on a Sobolev type space, checking some properties. The theorems of Paley-Wiener on the space of indefinitely differentiable functions with compact support (resp the algebra of square integrable functions of compact support) are studied. For particular cases of Lie groups, simplifications intervenes and the operators studied are then the Hilbert Schmidt operators . Chap. 3 contains important and new results for exemples: Inversion theorem for the classical Radon transform R c (see Theorem 3.3) , Gutzmer formula for the classical Radon transform ( see Theorem 3.7); Hardy's inequality for the operator R c (see Theorem 3.8) and finally, we give a characterization of space R c E (R n ) , where R c is the classical Radon transform(see Theorem 3.11) Note that the proof of Theorem 3.3 is based on the development of plane waves of Dirac measure; precisely, we use the Galderon identity. The idea is at the origin of Theorem 3.5 (see [START_REF] Faraut | Formule de Gutzmer pour la complexification d 'un espace symetrique[END_REF]). The Theorem 3.7 generalizes the Helgason Theorem (see [70, page 116]). Theorem 3.11 gives another statment of Helgason Theorem [71, Theorem 2.6] . In section 3.5, we give the Titchmarsh's Theorem for the classical Radon transform (see Theorem 3.17). We give also certains discrete Titchmarsh' Theorems for the lattice Z n and its dual torus T n . Let be N the vector space N R n and W is a connected Lie group which acts by a right action on W. We denote by G = N W the semidirect product relatively to this action ( see chapter 2). In the subsection 3.2.1, we establish a local inversion formula for the Radon transform on G = N W. In addition, we define and prove an inversion formula for the local Radon transform on the group S 3 R 4 (S 3 R 4 is the semidirect product of the unit sphere and R 4 ). We complete this chapter by study the Fourier -Hermite transform and revised Hardy's uncertainty principle; precisely, we study the eigenfunctions of Fourier transform in the goal to establish the revised Hardy uncertainty principle (see Theorem 3.22). We give the inversion theorem and Plancherel formula for this transform;( see Theorems 3.28 and 3.25). It is clear that the Fourier-Hermite transform generalizes the classical Fourier transform. The Radon-Hermite transform (associated to Fourier-Hermite transform) generalizes the classical Radon transform. In Chap. 4, we studied the integral geometry of the sphere S d . This chapter contain important results such as the geometry approach to the Radon transform of S d and its dual Radon transform. In addition we establish some inversion formulas for this transform also its dual in S d . The Plancherel formula and support theorem are also studied on S d . We finish this chapter by defining and studying the Radon transform on the n-dimensional torus T n . We give the interesting results for example (range characterization of the image of Radon transform on T n , inversion formula and support theorem for the Radon transform in T n ). We generalize this results to the d-plane Radon transform on T n . We study, in Chap. 5, the harmonic analysis and integral geometry on Damek-Ricci space N A. We define and study the Radon transform in N Aspace. The important results are: the inversion formulas and Plancherel formula for the Radon transform and its dual on N A-space. In addition, we study the spectral theory for the Strichartz 's operator. Precisely, we gener-alize Strichartz 's theorem (see Chap. 5) in the case of Damek-Ricci space N A. These results generalize the case of a Riemannian symmetric space of the non compact type of rank one. In the part II, we begin by studying the integral geometry in the finite set X. The analogue definition of finite Radon transform consists the make the average of function f over the subset of a finite set X (see [START_REF] Bolker | The Finite Radon Transform[END_REF],[118], [108]). In the sequel, we adopt the definition of [108]) for two reasons: The Strichartz 's definition [108]) is more natural and similar to the classical case. Indeed, in the study (see [108]) the author has used the finite plane geometry to define the finite Radon transform. This definition can be extended (see [108]) to the finite k-plane transform R k with k ∈ {2, 3, . . . , card(X)} with card(X) is the cardinal of the set X. More precisely, Let X be a finite set of points and N be the cardinal of X. Let Y be the set of lines of X, each line y ∈ Y being a subset of X subject to single axiom "two points determine a unique line", which is equivalent to: (A) "For any two points x 1 , x 2 ∈ X, there exists a unique y ∈ Y such that x 1 ∈ y and x 2 ∈ y ". We say that Y is simple if for all lines y ∈ Y , card(y) = 2. Y is not simple if there exists y 0 ∈ Y such that card(y 0 ) > 2. Let l 2 (X) (resp. l 2 (Y )) be the space of all complex-valued functions on X (resp. on Y ). The finite Radon transform is defined (see [108]) as the operator R on l 2 (X) to l 2 (Y ) given by the formula

Rf (y) = x∈y f (x) , y ∈ Y ;
and the dual Radon transform of R is the operator R * of l 2 (Y ) into l 2 (X) given by R * F (x) = y x F (y) for all F ∈ l 2 (Y ) and x ∈ X.

In the following, we seek a kernel G (y, x) which is solution of these equations

RG (y 0 , ) = β y0 (y 0 ∈ Y ) and R * G (., x 0 ) = χ x0 (x 0 ∈ X) ,
where χ x0 is the characteristic function of the set {x 0 }, also β y0 is the function defined on Y such that β y0 (y) = 0 if y = y 0 and β y0 (y 0 ) = 1. Consequently, we calculate explicitly the solution G (y, x) of the functional equations above, for which we establish the inversion formulas for R and its dual R * (see expression of this kernel in this chapter). We establish then the inversion formulas for R and R * .

In Chap. 7, we define and study the Radon transform R on discrete hyperplane in the lattice Z n (n ≥ 2) defined by linear diophantine equations. More precisely, we study carefully various natural questions in this context: specific properties of the discrete Radon transform R and R * , inversion formula for R and also an appropriate support theorem in the discrete case. The purpose of Chap. 8 is to extend carefully the discrete Radon transform, studied in the above chapter, to Radon transform R on the discrete Grassmannian G (d, n) (n ≥ 3 and 1 ≤ d < n-1) consisting of all discrete d-planes in the lattice Z n defined by systems of linear diophantine equations. By analogy with the integral geometry on Grassmann manifolds and projective spaces, which was developed by many authors, (see [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF], [START_REF] Helgason | Geometry Analysis on Symmetric Spaces[END_REF], [START_REF] Helgason | Integral Geometry and Radon Transforms[END_REF], [START_REF] Helgason | The Radon Transform[END_REF], [START_REF] Helgason | A duality for symmetric spaces with applications to group representations[END_REF] ). Our study in the present work with various natural questions in this context: specific properties of the discrete Radon d-plane transform R and its dual R * , inversion formula for R and also an appropriate support theorem for this Radon transform. In Chap. 9, We give a discrete Paley-Wiener theorem relatively at the discrete Radon transform, which characterizes the image of functions on Z n with finite support. Let K = {x 1 , . . . , x n } be a finite set in Z n . We denote by C

(Z n ) K the subspace of S (Z n ) consisting of all complex-valued functions on Z n such that suppf ⊂ K. Let G K = {H ∈ G |H ∩ K = ∅}
(where G the discrete Grassmannian on Z n ). We denote by D K (G) the subspace of S (G) consisting of all complex-valued functions F on G such that suppF ⊂ G K . We define D * ,K (G) as the subspace of D K (G) constituted by the elements F verifying the moment condition and for each m ∈ Z n , there exists j m ∈ N for which F (H (a j , a j m)) = F (H (a jm , a jm m)) for all j ≥ j m , where a j = 1, j, j 2 , . . . , j n-1 . we have then

R is a bijection of C (Z n ) K onto D * ,K (G) and R C (Z n ) K = D * ,K (G).
Let S * (G) denote a subspace of S (G) consisting of all F ∈ S (G) such that there exists a G ∈ C ∞ (T n ) for which F 1 F (H (a, .)) (θ) = G (θa). We obtain the following result: R is a bijection of S (Z n ) onto S * (G) and R (S (Z n )) = S * (G). We give also a characterization of the image of discrete Hardy space under the discrete Radon transform R (see Theorem 9.23). We end this chapter by establishing other inversion formulas for the discrete Radon transform. In Chap.10, we characterize the image of exponential type functions under the discrete Radon transform R on the lattice Z n of the Euclidean space R n (n ≥ 2). we also establish the generalization of Volberg 's uncertainty principle on Z n , which is proved by means of this characterization. We give also an analogue of Nazarov 's uncertainty inequality for n-dimensional Fourier series from the one for n-dimensional Fourier transform. Some inequalities are new and better than ones deduced from a classical local uncertainty inequality. We introduce and study the Sobolov 's spaces l 2 (Z n , τ ) (τ > 0), which allows us to deduce the Gutzmer formula ( see Theorem10.29).

We end this introduction by studying the discrete convolution with Bessel functions, precisely, we define and study of Fourier-Bessel transform. We establish an inversion formula and Plancherel theorem for this transform (see Theorems 10.16, 10.17).

Part I RADON TRANSFORMS ON CERTAIN RIEMANNIAN SPACES

Chapter 1 FUNDAMENTAL CONCEPTS OF GROUP THEORY.

Definition of a group.

A Nonempty set G is called a group if a product x 1 x 2 is defined for every two elements x 1 , x 2 ∈ G, for which the following conditions holds:

a) x 1 x 2 ∈ G, for all x 1 , x 2 ∈ G; b) (x 1 x 2 ) x 3 = x 1 (x 2 x 3 ), for all x 1 , x 2 , x 3 ∈ G; c) there exists a unique element e in G such that ex = xe = x for all x ∈ G;
e is called the identity element of the group G; d) for every element x ∈ G there exists a unique element, designated x -1 for which xx -1 = x -1 x = e; the element x -1 is called the inverse of x. It is clear that x is the inverse of x -1 , so that x -1 -1 = x.

A group G is called commutative ( or abelian) if x 1 x 2 = x 2 x 1 for all every x 1 , x 2 ∈ G and non commutative in the opposite case.

Examples

1) The set of all complex matrices of order n with nonzero determinant is a group if multiplication is defined as multiplication of matrices; this group is usually denoted as GL (n, C). Its identity element is the identity matrix and the inverse element of the matrix A is the matrix A -1 . The group GL (n, R) of all real matrices of order n with nonzero determinant is defined similarly. If n ≥ 2 These groups are non commutative. 2) Let SL (n, C) be the set of all complex matrices of order n with determinant equal to 1. We define a product in SL (n, C) as the product of matrices. Then SL (n, C) is a group, since in multiplication of matrices determinants also multiply. The group SL (n, R) of all real matrices of order n with determinant equal to 1 is defined similarly.

Chapter 2

PALEY WIENER THEOREMS FOR G = W N .

Introduction and notations.

We characterize the Fourier's image of many spaces of distributions on certain Lie groups, we give also some applications. The groups considered are the semidirect product of a real vector space by any connexe Lie group.

In the following we fix a vectoe space N and a connexe Lie group W , this group acts on N by a left action, and G = N W their semi-direct product relatively to this action. We introduce a Riemannian distance, and with this distance we prove the existence of a fundamental sub-multiplicative function Λ which is continuous and proper function on W . The topology defined by this function on W (that is by the balls:

Q r = {w ∈ W | Λ (w) ≤ r}, r > 0) is equivalent to the initial topology of W .
We characterize the Fourier transform of the distributions with compact support in G. They are the families of the operators on a space of the Sobolev type satisfying certain properties. The support of a distribution can be read on these properties. We characterize also the Fourier transform for the square integrable functions with compact support. When G = N × N , the simplifications intervene and therefore we find the classical Paley Wiener theorem.

Chapter 3

GUTZMER FORMULA AND RADON TRANSFORM ON R n .

3.1 Introduction.

In 1917 Johann Radon [START_REF] Radon | Uber die Bestmmung von Funcktionen durch ihre Integralwerte langs gewisser Mannigflatigkeiten[END_REF] solved the following problem: find a function f on the Euclidean plane R 2 knowing its integrals

R c f (ξ) = ξ f,
along all lines ξ in the plane. The operator R c is now called the Radon transform. This problem was generalized and studied seriously in R n (and in several varieties) by [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF], [START_REF] Helgason | Geometry Analysis on Symmetric Spaces[END_REF], [START_REF] Gelfand | Selected Topics in integral Geometr[END_REF]. The study of the operator R c is an important part of the integral geometry. The essential pillars of the integral geometry are 1) Inversion formula: knowing R c f (and its properties) determine f (and its properties) ?

2) Support theorem of R c : knowing the properties the support of f, can we find the properties the support for R c and vice versa ?. Let P n be the differentiable manifold constituted by all hyperplanes of Euclidean space R n . It is well know that the Euclidean Radon transform associates to a function f defined and integrable on R n , a function R c f on P n given by the formula

R c f (H (ω, t)) = H(ω,t) f (x) d µ (x) ,
where d µ (x) is the Lebesgue measure on the hyperplane

H (ω, t) = {x ∈ R n | xω = t} ;
where ωx denotes the usual inner product of x and ω regarded as two vectors of the Euclidean space R n . ω being the element of the unit sphere S n-1 .

Chapter 5

INTEGRAL GEOMETRY AND SPECTRAL THEORY ON NA-SPACE.

Introduction.

We say that M is a harmonic variety if, for every point m ∈ M (m origin), the Laplacian £ admits a elementary solution around m, that is, the solution is function of the geodesic distance d (m, x) alone.

The question that was put to the mathematicians is, the harmonic varieties are they symmetrical?. The response to this question is given by Ewa Damek and Fulvio Ricci. They gave a great class of harmonic varieties which does not are symmetrical. Precisely, in order to generalize Heisenberg group, Kaplan (see [START_REF] Kaplan | Fondamental solution for a class of hypoelliptic PDE generated by composition of quadratic forms[END_REF]introduced Lie groups of H-type. Damek and Ricci ( see [START_REF] Damek | Harmonic Analysis on solvable extension of H-type groups[END_REF], [START_REF] Damek | A class of nonsymmetric Harmonic Riemannian spaces[END_REF]) defined and studied a new class of groups called " extension of Lie group of H-type". Damek-Ricci space is a semi-direct product of R + and a two step nilpotent group. More precisely, let η be a two-step real nilpotent Lie algebra (i.e. η, η, η = 0 and η, η = 0) endowed with an inner product <, > such that η decompose in the orthogonal sum of two spaces p and z η = p ⊕ z, where z is the center of η . We denote by k the dimension of z.

Let N be the connected and simply connected group of Lie algebra η. Since η is nilpotent, the exponential map is surjective, we may therefore parametrize N by p ⊕ z and write (V, Z) for exp (V + Z) where V ∈ p and Z ∈ z. By the Baker-Campbell-Hausdorff formula, the product law in N is given by the formula

(V, Z) . V , Z = V + V , Z + Z + 1 2 [V, V ] ,
for all V , V ∈ p and for all Z, Z ∈ z. Let dV and dZ be the Lebesgue measures on p and z respectively, the measure dV dz is the Haar measure on N which will be denoted later by dn. Let <, > η be an inner product on η. The space η is called an H-type algebra if for every Z ∈ z we have J 2 Z = -|Z| 2 I p , where I p is identity on p and J Z is a map of p into p defined by equality < J Z X, Y > η =< [X, Y ] , Z > η , for all X, Y ∈ p see [START_REF] Abouelaz | Horocyclic Radon Transform on Damek-Ricci space[END_REF]. Note that for each unitary Z ∈ p, the map J Z define a complex structure on p, then

Part II STUDY OF CERTAIN TRANSFORMATIONS ON Z n . Chapter 6 THE FINITE RADON TRANSFORM

Introduction

As we said before, the Radon transform is defined in R 2 by John Radon [START_REF] Radon | Uber die Bestmmung von Funcktionen durch ihre Integralwerte langs gewisser Mannigflatigkeiten[END_REF] and generalized in R n by several authors particularly [START_REF] Gelfand | Géomètrie intégrale et théorie des représentations[END_REF], [START_REF] Helgason | The Radon Transform[END_REF]. The Radon transform in Euclidean space R n associates to a function f on R n a function Rf on P n (P n denotes the space of all hyperplanes H (ω

; t) in R n ) by the formula Rf (ω, t) = H(ω,t) f (x) dµ (x) ,
where dµ (x) is the Euclidean measure on the hyperplane

H (ω, t) = {x ∈ R n | xω = t} ,
ωx denotes the usual inner product of ω and x regarded as two vectors of the Euclidean space R n . The vector ω being an element of the unit sphere S n-1 .

In the case of the finite set, the analogue of this definition consists the make the average of function f over the subset of a finite set X (see [START_REF] Bolker | The Finite Radon Transform[END_REF], [118]).

In the sequel, we adopt the definition of [108] for two reasons: The Strichartz's definition [108] is more natural and similar to the classical case. Indeed, in the study (see [108]) the author has used the finite plane geometry to define the finite Radon transform. This definition can be extended (see [108]) to the finite k-plane transform R k with k ∈ {2, 3, . . . , card (X)} (card (X) is the cardinal of the setX). More precisely, Let X be a finite set of points and N be the cardinal of X.

Let Y be the set of lines of X, each line y ∈ Y being a subset of X subject to single axiom " two points determine a unique line", which is equivalent to: (A) "For any two points x 1 , x 2 ∈ X, there exists a unique y ∈ Y such that x 1 ∈ y and x 2 ∈ y " . We say that Y is simple if for all lines y ∈ Y , card (y) = 2. Y is not simple if there exists y 0 ∈ Y such that card (y 0 ) > 2.

Let l 2 (X) resp.l 2 (Y ) be the space of all complex-valued functions on X Chapter 7

DIOPHANTINE INTEGRAL

GEOMETRY IN Z n .

Introduction

Let f : R n → C be a function integrable on each hyperplane in R n . Let P n denote the differentiable manifold

R n × S n-1 /± of all hyperplane H (ω, t) in R n (H (ω, t) = {x ∈ R n | xω = t}), with (ω, t) ∈ S n-1 × R n , S n-1
being the unit sphere. The Radon transform of f is defined as the function R c f :

P n → C given by R c f = H(ω,t) f (x) dµ (x) , for all (ω, t) ∈ S n-1 × R n , (7.1)
where dµ is the Euclidean measure on the hyperplane H (ω, t). In the case of a finite set E, the analogue of this definition consists in making the average of a function f : E → C over the non-empty subsets of E, see [4], [118].

In the case of the lattice Z n , we give in this section an analogue of the definition (7.1), which consists in making the average of a suitable complex-valued function f on Z n over discrete hyperplane H (a, k) = {x ∈ Z n | ax = k} defined by linear diophantine equations, with (a, k) ∈ P × Z, where P designates the set of all elements a = (a 1 , . . . , a n ) ∈ Z n \ {0} such that d (a) = 1, d (a) being the greatest common divisor of the integers a 1 , . . . , a n , and ax denotes the usual inner product of a and x regarded as two vectors of the Euclidean space R n . Now, we note that one of the difficulties of diophantine integral geometry, relatively to its Euclidean homologue, is the problem of the existence of solutions in Z n for linear diophantine equation ax = c, with a = (a 1 , . . . a n ) ∈ Z n \ {0} and c ∈ Z, therefore the arithmetic of the integers a 1 , . . . a n and c is essential for our study.

We will deepen the study of the discrete Radon transform on Z n , and we generalize it to the discrete grassmannians associated with Z n .

Chapter 8

DIOPHANTINE INTEGRAL

GEOMETRY ON G (d, n).

Introductions and preliminaries

In

R c f (ξ) = ξ f (x) dm (x) , for all (f, ξ) ∈ D (R n ) × G (d, n) , (8.1)
where dm is the Euclidean measure on the d-plane ξ and D (R n ) denotes the space of all complex-valued C ∞ functions on R n with compact support. On the other hand, the dual Radon d-plane transform R * c is given by

R * c ϕ (x) = ξ x ϕ (ξ) dµ (ξ) , for all (ϕ, x) ∈ E (G (d, n)) × R n , (8.2)
where E (G (d, n)) denotes the space of all complex-valued C ∞ functions on G (d, n) and dµ is the measure on the set of all d-planes through x which is invariant under the group of rotations around x and such that µ {ξ ∈ G (d, n) | x ∈ ξ} = 1. Now, we note that one of the difficulties to extend the discrete Radon transform, to the Radon transform on the discrete Grassmannian G (d, n) consist-Exercices 1. Let Z n be the lattice of the Euclidean space R n . For n ≥ 2, define the following lattice

Z n 2 n = m 2 n | m ∈ Z n . It is clear that Z n ⊂ Z n 2 ⊂ Z n 2 2 ⊂ • • • ⊂ Z n 2 n ⊂ • • • Study the diophantine

The d-plane Radon transform on Lie groups of exponential type

In the following, we will define the d-plane Radon transform on solvable Lie groups of exponential type . We establish an inversion theorem and Plancherel formulas of this transform.

Introduction

Among the many works that study of integral geometry on symmetric spaces (see Helgason's works [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]); the case of the integral geometry on solvable Lie groups of exponential typ is not however studied. We give some properties concerning the exponential function. Let G be the connected Lie group and simply connected and g its Lie algegra. The Lie group GL (n, C) is exponential i.e exp(M n×n (C)) = GL (n; C). The nilpotent groups are exponential. Let S n be the space of the symmetric matrices and S n,+ the space of symmetric matrices defined positive. the expponential is surjective of S n onto S n,+ . The Lie goup SL (n, R) is not exponential, more generally, the linear group GL (n, R) is not exponential, that is exp g = G. In the following: for any arbitrary Lie group G and x ∈ G, we define

Ind G (x) = min k ∈ N | x k ∈ E (G) If
f • exp (x 1 X 1 + x 2 X 2 ) dµ(x 1 , x 2 ),
where We complete this chapter by study the Fourier -Hermite transform and revised Hardy's uncertainty principle; precisely, we study the eigenfunctions of Fourier transform and Radon transform, we establish the inversion theorem and Plancherel formula for this transform; ( see Theorems 3.28 and 3.25). Note that the Fourier-Hermite transform and the revised Hardy uncertainty principle are defined and studied by Abouelaz and Attioui [START_REF] Abouelaz | Uncertainty Principle and Radon Transform[END_REF].

A (ω) = ω 1 ω 2 0 0 But, A(ω)X = ω 1 ω 2 0 0 x 1 x 2 = x 1 ω 1 + x 2 ω 2 0 1 = x 1 ω 1 + x 2 ω 2 , ω = (ω 1 , ω 2 ) ∈ S 1 . Then, H (t, A(ω)) = x = (x 1 , x 2 ) ∈ R 2 | x 1 ω 1 + x 2 ω 2 =
It is clear that the Fourier-Hermite transform generalizes the classical Fourier transform. The Radon-Hermite transform (associated to Fourier-Hermite transform) generalizes also the classical Radon transform. We conclude this Chapter 3 by defining and studying the Radon transform on the solvable Lie groups( in particular the nilpotent Lie groups). More precisely: we establish the inversion theorems and Plancherel formulas of this transform for the nilpotent Lie groups.

  W ) and D (R) = C ∞ c (R) are spaces of smooth compactly supported functions. -C c (W ) is the space of continuous functions on W with compact support. -C 0 (W ) is the space of continuous functions on W vanishing at infinity. -< ., . > denotes the distributional duality. -M (w) = sup ||w|| , w -1 , where ||w|| = sup {|wn| | n ∈ N } , wn is the action of the group W on N . Only in the Chapter 2, N is a real vector space of finite dimensional and n is an element of N. -S d is the unit sphere of dimensional d and SO (d + 1) be the orthogonal special group. -V n,k = SO (n) SO (n -k) the Steifel variety.

2

  PALEY WIENER THEOREMS FOR G = W N . . . . . . . . 2.1 Introduction and notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fundamental function on the semi-direct product G = W N 2.3 Examples of groups G whose the function M is not proper. . . 2.4 Charaterization of Fourier-Laplace of the space D(G). . . . . . . . 2.5 Invariant differential operators on G. . . . . . . . . . . . . . . . . . . . . . . 2.6 Fourier-Laplace transform on E (G). . . . . . . . . . . . . . . . . . . . . . . 2.7 Paley-Wiener-Schwartz theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Plan of this theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.8 Statement topological of Paley-Wiener-Schwartz's theorem. . . 51 2.9 Consequence of the main theorem. . . . . . . . . . . . . . . . . . . . . . . . . 54 2.10 Applications of the Paley-Wiener-Schwartz theorem. . . . . . . . . 56 2.11 Characterization of the Fourier image of L 2 c (G). . . . . . . . . . . . . 58 2.11.1 Topolgical statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.12 Characterization of Fourier image of the space C ∞ c (G). . . . . . 63 GUTZMER FORMULA AND RADON TRANSFORM ON R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Inversion theorem and Gutzmer formula for R c . . . . . . . . . . . . . 68 3.2.1 Applications: Local Radon Transform on G = N W . 74 3.3 The Hardy-Littlewood inequality of the Radon transform . . . . 80 3.4 Radon transform on compactly supported distributions . . . . . . 82 3.5 The Titchmarsh Theorem for the Radon transform. . . . . . . . . . 84 3.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.7 Hermite functions and Hardy uncertainty principle . . . . . . . . . 89 3.8 Hardy 's uncertainty principle for the Radon transform . . . . . . 94 3.9 Fourier-Hermite-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.10 The Hermite-Radon transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.11 Inversion formula for the Hermite Radon transform . . . . . . . . . 97 3.12 Proof of Titchmarsh 's Theorems on the lattice Z n and its dual T n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.13 The d-plane Radon transform on Lie groups of exponential type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.13.2 Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 104 3.13.3 The exponential functions and their properties . . . . . . . 105 3.13.4 New parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.13.5 The inversion formulas for the d-plane

  Since W acts on N , for all element w ∈ W we define an endomorphism of N, denote ||w|| = sup {|w.n| : |n| ≤ 1, n ∈ N } where |.| is the Euclidean norm on N, we also denote |.| the dual norm on N * complexified of the dual N (the set of the characters of N ) and we design by M (w) the real number defined by M (w) = sup ||w|| , w -1 , for all w ∈ W .

  the following section, we extend carefully the discrete Radon transform to the Radon transform R on the discrete Grassmannian G(d, n), n ≥ 3 and 1 ≤ d < n -1 consisting of all discrete d-planes in the lattice Z n defined by systems of linear diophantine equations. By analogy with the integral geometry on Grassmann manifolds and projective spaces, which was developed by many authors, particularly C. A. Berenstein and E. Casadio Tarabulsi[START_REF] Berenstein | Inversion formulas for thek-dimensional Radon transform in real hyperbolic spaces[END_REF], F. Gonzalez[START_REF] Gonzalez | On image of Radond-plane transform and its dual[END_REF],[START_REF] Gonzalez | Radon transforms on Grassmann manifolds[END_REF], S. Helgason[START_REF] Helgason | Integral Geometry and Radon Transforms[END_REF] and B. Rubin [101]. We briefly recall the definition of the classical Radon d-plane transform R c on the Euclidean space R n as well as its dual R * c , d being an integer such that 0 < d < n, with n ≥ 2. We denote by G (d, n) the Grassmann manifold consisting of all affine d-dimensional planes in R n . The Radon d-plane transform R c is defined by

= a b 0 1 ∈

 1 Lie group is exponential if exp(g) = G = E (G). The examples of this Lie groups are: the Lie groups such that: the ax + b line group i.e the group G GL (2, R) | a > 0, b ∈ R ; this group is the semi-direct product whose the law is given by (b, a) (b , a ) = (ab + b, aa ) .

  the minimum exists ∞ otherwiseInd G (G) = {Ind G (x) | x ∈ G} Now, we have the important theorem R * c R c f (x) = Ω d Ω n R n f (x + y) |y| d-n dyΩ n is the volume of the unit sphere S n-1 .3.13.5The inversion formulas for the d-plane Radon transform on nilpotent groups and certain solvable Lie groups.Troughout G would be the solvable connected and simply connected Lie group and g be the Lie algebra.of G.We will start with the next groupLet G = a b 0 1 | a > 0, b ∈ R ,the ax+b group G of affine transformations of real line R e f (H (t, A (ω))) = A(ω)X=t

  t is the hyperplane, it follows thatR e f (H (t, A(ω)) = x1ω1+x2ω2=t f • exp (x 1 X 1 + x 2 X 2 ) d µ (x 1 , x 2 ). = g f • exp (x 1 X 1 + x 2 X 2 ) δ (t -(x 1 ω 1 + x 2 ω 2 )) dx 1 dx 2 f ∈ D (G) and δ Dirak measure. Then f = (4π) c R c (f • exp) • Log,since here exp g = G. More generally, we consider the goup , A] | det (A) > 0 , b being a vector that has the same numbers of rows as the matrix A. The law of G is [b, A] . [b , A ] = [Ab + b, AA ]. The Lie algebra of G is given by the Radon transform on the group S 3 R 4 (S 3 R 4 is the semidirect product of the unit sphere and R 4 ).

  m i with a = (a 1 , . . . , a n ) ∈ P and m = (m 1 , . . . , m n ) ∈ Z n .-We define and denote the discrete Radon transform on Z n by

	H (a, k) = {m ∈ Z n | am = k} ,
	where am = i=1 a Rf (H (a, k)) = n
	m∈H(a,k)
	a ,
	see section 5.4.
	-In the Chapters 6, 7, 8, 9,10. the Radon transform is denoted by R.
	-For (a, k) ∈ P × Z, we denote H (a, k) the discrete hyperplane of Z n de-
	fined by

i

  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of a group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Subgroup, cosets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Part I RADON TRANSFORMS ON CERTAIN RIEMANNIAN
	SPACES
	1 FUNDAMENTAL CONCEPTS OF GROUP THEORY. . .
	1.1 1.3 Normal divisor, Factor group. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.4 The center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.5 Study of Orbits and Stabilizers. . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.5.1 Fundamentals Examples. . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.5.2 Examples (Homogeneous Spaces) . . . . . . . . . . . . . . . . . . .
	1.6 Semi direct Products of Groups . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	1.6.

2 Others examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Spherical Fourier transform and Radon transform on the sphere S d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.4 A geometric approach on the Radon transform on S d . . . . . . . . 123 4.5 A geometric approach to the dual Radon transform on S d . . . . 127 4.6 Transmutation formula of operators. . . . . . . . . . . . . . . . . . . . . . . 128

Radon transform on nilpotent groups and certain solvable Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.13.6 Plancherel formulas for the nilpotent groups and affine group of the line for the Radon transform . . . . . . 114 INTEGRAL GEOMETRY ON THE SPHERE AND THE TORUS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.3

  integral geometry of the lattice Z n 2 n ? 2. Can be inverted the transform Rf (H(a, k)) = am=k,m∈Z n f (m), for n > 2, where (a, k) ∈ P × Z? 3. Let Z n + iZ n be the Gauss 's lattice. Study the diophantine integral geometry associated to the Gauss 's lattice? Remark 3.32 If α = 0, we find the Helgason Theorem (see [72, Theorem 2.1 page 10]).

STUDY OF CERTAIN DISCRETE TRANSFORMS.

The goal of this section is to reconstruct a function of C (Z n ) from its d-planes Radon transform. where C (Z n ) is the space of all functions f defined on Z n with finite support. We establish also a Plancherel formula relatively at discrete Radon transform for the functions f of C (Z n ) . Afterwards, we characterize some discrete functional spaces by the Radon transform (Paley-Wiener theorem and Paley-Wiener-Schwartz theorem relatively to the discrete Radon transform). Let f ∈ l 1 (Z n ), we define the Strichartz's operator as follows (see [108])

f (kj) for all k ∈ N \ {0} .

(9.1)

The operator R can be generalized by

for all m = (m 1 , . . . , m n ) ∈ Z n \ {0},with f is a function of S (Z n ) such that f (0) = 0. We denote by µ the Mobius function defined on N by µ

n and u j = u 1 u 2 . . . u j-1 u j+1 . . . u n for all j = 1, 2, . . . , n. The function T (u1,...,un) will be defined as follows:

We denote by S 0 (Z n ) the subspace of the Schwartz space S (Z n ) consisting of all complex-valued functions f defined on Z n such that f (0) = 0. We design by S 0,+ (Z n )the subspace of S 0 (Z n ) such that the function t → f (0, . . . , 0, t)

n and u j = u 1 u 2 . . . u j-1 u j+1 . . . u n , we design by u the vector of Z n defined by u = ( u 1 , . . . , u n ) ∈ Z n . Recall that P s,n is the Chapter 10 STUDY OF CERTAIN DISCRETE TRANSFORMS.

Introduction and preliminaries.

We recall briefly that the uncertainty principle states, roughly speaking, that a non zero function and its Fourier transform cannot both be sharply localized, which can be interpreted topologically by the fact that they cannot have simultaneously their supports in a same too small compact, see the Heisenberg uncertainty principle in [START_REF] Havin | The Uncertainty Principle in Harmonic Analysis[END_REF]. Several versions of the uncertainty principle on have been established by many authors in the last few decades. Among the contributions dealing with this important topic, let us quote principally [START_REF] Astengo | An uncertainty principle on homogeneous trees[END_REF], [START_REF] Price | Local uncertainty inequalities for Fourier series[END_REF]. On the the other hand, we note that the uncertainty principle is on of the major themes of the classical Fourier analysis as well as its neighbouring parts of the mathematical analysis. In this section, we characterize the image of exponential type functions under the discrete Radon transform R on the lattice Z n of the Euclidean space R n (n ≥ 2). We also establish the generalization of Volberg' s uncertainty principle on Z n , which is proved by means of this characterization. Also we shall obtain an analogue of Nazarov's uncertainty inequality for n-dimensional Fourier series from the one for n-dimensional Fourier transform. Some inequalities are new and better than ones deduced from a classical local uncertainty inequality (see [START_REF] Abouelaz | On the Range of the Radon Transform onZ n and the Related Volberg's Uncertainty Principle[END_REF], for more details). However, as for the Nazarov uncertainty principle, it is known only for Fourier transform. In this subsection we shall obtain an analogous uncertainty inequality for n-dimensional Fourier series. Nazarov' s uncertainty inequality is originally appeared in [START_REF] Nazarov | Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type[END_REF] and [START_REF] Jaming | Nazarov's uncertainty principle in higher dimension[END_REF], Jaming extends it to a higher dimensional Fourier transform. The techniques of which we made use essentially the diophantine integral geometry as well as the Fourier analysis. We consider here the lattice where

In particular, we are studying Cartan displacement group. Let G be the Cartan's group defined by:

, the Radon transform of the Cartan displacement group, can be inverted by (-∆)

where (b, a) = a b 0 1 , and

Plancherel formula for the Radon transform in the nilpotent groups Theorem 3.45 Let G be the nilpotent Lie group connected and simply connected and f ∈ D (G). Then, we have the following formula

From section 3.2.1 (see formulas (B) , (C)), we have

Let G be a connected and simply connected real solvable Lie group and g its Lie algebra. Let Λ 2 (G) be the set defined by

the point . is the Law of the group G. The exponential is it surjective of g onto Λ 2 (G)?. Note that the proof of Theorem3.3 is based on the development of plane waves of Dirac measure, precisely, we use the Galderon identity. The idea is at the origin of Theorem 3.5 (see [START_REF] Faraut | Formule de Gutzmer pour la complexification d 'un espace symetrique[END_REF]). The Theorem 3.7 generalizes the Helgason Theorem (see [71, page 116]. Theorem 3.11 gives another statement of the Helgason Theorem [72, Theorem 2.6]. In section 3.5, we give the Titchmarsh's Theorem for the classical Radon transform (see Theorem 3.17). We give also certains discrete Titchmarsh' Theorems for the lattice Z n and its dual torus T n . Let be N the vector space N R n and W is a connected Lie group which acts by a right action on W. We denote by G = N W the semidirect product relatively to this action ( see chapter 2). In the subsection 3.2.1, we establish a local inversion formula for the Radon transform on G = N W. In addition, we give a local Radon transform for Chapter 4

BIBLIOGRAPHICAL NOTES

INTEGRAL GEOMETRY ON THE SPHERE AND THE TORUS.

In the following, we will deeply study the integral geometry in the unit sphere and we give certain applications.

Introduction.

Among the many works that study of integral geometry on a symmetric space (See [START_REF] Gelfand | Géomètrie intégrale et théorie des représentations[END_REF], [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF], [START_REF] Helgason | The Radon Transform[END_REF], [START_REF] Helgason | A duality for symmetric spaces with applications to group representations[END_REF],[109]), the case of a symmetric space of compact type is often considered that the non-compact type. The compact case has however been raised on several occasions especially S. Helgason (See [START_REF] Helgason | A duality for symmetric spaces with applications to group representations[END_REF], [START_REF] Helgason | Groups and Geometric Analysis. Integral Geometry, Invariant Differential Operators, and Spherical Functions[END_REF]), on the other hand, T. Sherman in ( See [102]) has developed a Fourier analysis on the sphere analogous to the abelian Lie groups. Note that Strichartz in [110] has established a local Fourier analysis in S d . In the following, we define and study a Radon transform on the sphere S d = SO (d + 1) /SO(d) of dimension d. This transform R s is obtained by integration a given function over the sphere of dimension (d -1) passing through a fixed point a π = (0, . . . , 0, -1) of S d called south pole. Applying the functions on S d that are invariant under rotation around this point, an interesting analogy is obtained with the Abel 's transform of the hyperbolic space SO 0 (d, 1) /SO (d)(of dimension d) which consists of integrate over the horocycles(of dimension d -1). Our transform R s on the sphere is thus-like the one of the hyperbolic spacerelated simply to spherical Fourier transform. This transform can be reversed by means of a differential operator(if d is odd) or Integro-differential(if d is even). We adopt the geometric approach, based upon the Radon transform R s on the sphere S d , also the dual Radon transform (R s )

* on S d . We generalize [17, Theorem 2.2], by replacing in this theorem the assumption f vanishes in a neighbourhood of the south pole with f vanishes to second order at the south pole. We give a theorem of transmutation formula of operators. The Paley Wiener theorems on the sphere S d are also studied.