Nicolas Cheifetz
email: nicolas.cheifetz@veolia.com

Zineb Noumir

Allou Samé

Anne-Claire Sandraz

Cédric Féliers

Véronique Heim

Modeling and Clustering Water Demand Patterns from Real-World Smart Meter Data

Nowadays, drinking water utilities need an acute comprehension of the water demand on their distribution network, in order to efficiently operate the optimization of resources, the management of billing and to propose new customer services. With the emergence of smart grids, based on Automated Meter Reading (AMR), a better understanding of the consumption modes is now accessible for smart cities with more granularities. In this context, this paper evaluates a novel methodology for identifying relevant usage profiles from the water consumption data produced by smart meters. The methodology is fully data-driven using the consumption time series which are seen as functions or curves observed with an hourly time step. First, a Fourierbased additive time series decomposition model is introduced to extract seasonal patterns from time series. These patterns are intended to represent the customer habits in terms of water consumption. Two functional clustering approaches are then used to classify the extracted seasonal patterns: the functional version of K-means, and the Fourier REgression Mixture (FReMix) model. The K-means approach produces a hard segmentation and K representative prototypes. On the other hand, the FReMix is a generative model and produces also K profiles as well as a soft segmentation based on the posterior probabilities. The proposed approach is applied to a smart grid deployed on the largest Water Distribution Network (WDN) in France. The two clustering strategies are evaluated and compared. Finally, a realistic interpretation of the consumption habits is given for each cluster. The extensive experiments and the qualitative interpretation of the resulting clusters allow to highlight the effectiveness of the proposed methodology.

Introduction

All modern cities need to deal with increasing populations and climate change, while maintaining adequate water services for consumers. Here, the climate change is only mentioned to emphasize the systemic changes inherent in any smart city.

Until now, the water or energy consumption reading were traditionally collected once or twice a year on large territories (for example, regions or nations). With the arrival of smart grid meters, this situation has changed and indexes can now be collected automatically with more granularities. The management of smart cities [START_REF] Giffinger | Smart Cities -Ranking of European medium-sized cities[END_REF][START_REF] Nam | Conceptualizing smart city with dimensions of technology, people, and institutions[END_REF]) is based on automated electronic meters that are deployed on the distribution network and are used to handle billing and customer 1 services. The first researches performed in the area of demand patterns classification belong to the electricity network fields [START_REF] Irwin | Statistical electricity demand modelling from consumer billing data[END_REF][START_REF] Hernández | Classification and clustering of electricity demand patterns in industrial parks[END_REF]. Most of the research in water field is focused on demand forecasting [START_REF] Donkor | Urban water demand forecasting: review of methods and models[END_REF]. Several approaches have been proposed for this purpose, including statistical prevision models [START_REF] Adamowski | Peak daily water demand forecast modeling using artificial neural networks[END_REF][START_REF] Blokker | Simulating residential water demand with a stochastic end-use model[END_REF]. The emergence of smart meters shifts this research to classification of water demand [START_REF] Aksela | Demand estimation with automated meter reading in a distribution network[END_REF]. [START_REF] Mckenna | Water demand pattern classification from smart meter data[END_REF] proposed a procedure for classification of water demands recorded from smart meters using a Gaussian mixture model for feature selection then the classical K-means algorithm for clustering [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF].

In various applications, the data to be analyzed are not multivariate observations, but these can be seen as functions or curves either continuous or discrete, namely functional data. Such studies usually refer to Functional Data Analysis (FDA) when data are varying in a continuum and potentially infinite dimensional [START_REF] Ramsay | Functional data analysis[END_REF][START_REF] Wang | Review of functional data analysis[END_REF]. Example of functional data encompass longitudinal data, responses in medical treatments and objects in video sequences. In the current case of smart meters, each signal is seen as a temporal function and collected intermittently at discrete time points. Analyzing smart meter consumption is useful for water utilities in order to develop innovative capabilities in terms of grid management, planning and customer services. Functional clustering aggregates data mining techniques, which aims to identify homogeneous groups among functional data without using prior knowledge about their group labels (unknown cluster membership). Aiming to analyze household consumption, Cardell-Oliver (2013) introduces a methodology to cluster daily water use signature patterns based on expert rules and a classical K-means. Many functional clustering methods have been developed over the last decade.

These methods can usually be separated into two categories: nonparametric methods using specific distances or dissimilarities between curves [START_REF] Dabo-Niang | On the using of modal curves for radar waveforms classification[END_REF], and mixture-model-based methods [START_REF] Samé | Model-based clustering and segmentation of time series with changes in regime[END_REF][START_REF] Jacques | Model-based clustering for multivariate functional data[END_REF].

The collected curves can be multivariate leading to a large representation space like in [START_REF] Cheifetz | A Sequential Testing Procedure for Multiple Change-Point Detection in a Stream of Pneumatic Door Signatures[END_REF] for change-point detection based on a specific curve modeling. The approach of the regression mixture model proposed by [START_REF] Gaffney | Curve Clustering with Random Effects Regression Mixtures[END_REF] motivated the focus of this article. This paper is organized as follows: the overall methodology is described in section 2. This methodology is decomposed in two consecutive steps that is to say the extraction of seasonal patterns from time series in section 3, and the identification of clusters with their profiles in section 4 based on two clustering strategies: a functional version of K-means and a dedicated Expectation Maximization (EM) algorithm. The section 5 introduces the experimental data set, and an analysis of the clustering results is given. Finally, the article ends with a conclusion and some perspectives.

General Methodology

The aim of this paper is to identify automatically the major water usage patterns in a set of time series recorded by smart water meters. A multi-step methodology is formulated to address this problematic, as illustrated in Figure 1. The first step consists in extracting the seasonal part of each time series, which represents the habits of water consumption for each meter, using a Fourier-based time series decomposition. Then, these seasonal components are normalized and used as input data by clustering algorithms. Two algorithms are used to classify the functional data into various water usage clusters. The first one consists in using the K-means jointly with the Functional Principal Component Analysis (FPCA) method and the second one is based on a Fourier regression mixture model recently introduced by [START_REF] Samé | Décomposition et classification de données fonctionnelles pour l'analyse de la consommation d'eau potable[END_REF]. Both the seasonal extraction and clustering approaches are described in the next sections. 3 Extracting seasonal patterns from time series Let (y 1 , . . . , y n ) denote n time series, where each one of them y i = (y i1 , . . . , y iT ) corresponds to hourly consumptions recorded by a single water meter, that is to say y i is a univariate time series and y it is a real valued scalar. It is implicitly assumed that all the series are recorded over the same time grid indexed by the ordered times {1, . . . , T } for all n curves.

Fourier-based time series decomposition

The methodology developed in this paper is based on the following classical additive decomposition:

y it = f it + x it + d it + ε it , (1)
where f it is the global trend of the time series which is modeled in a non parametric way using moving averages [START_REF] Gourieroux | Time series and dynamic models[END_REF].

x it is the seasonal component. As the studied water consumption time series are subject to daily and weekly seasonality, a Fourier basis decomposition [START_REF] De Livera | Forecasting time series with complex seasonal patterns using exponential smoothing[END_REF] is formulated:

x it = q1 j=1 α (1) j cos 2πjt 24 + α (2) j sin 2πjt 24 + q2 j=1 α (3) j cos 2πjt 168 + α (4) j sin 2πjt 168 , (2)
where q 1 and q 2 are the respective numbers of trigonometric terms used to handle the daily and weekly seasonality, and the α

(1)

j , α (2) j , α (3) j , α (4)
j are the coefficients to be estimated. This trigonometric modeling has the advantage of requiring considerably less parameters compared to an approach based on dummy variables [START_REF] De Livera | Forecasting time series with complex seasonal patterns using exponential smoothing[END_REF].

d it is a component devoted to capture the effect of exceptional public non-working days in France (eg. January 1 st , May 1 st , Christmas Day . . . ). The following decomposition is used: d it = 24 j=1 γ j δ tj , where δ tj = 1 if t corresponds to the hour j of a non-working day and δ tj = 0 otherwise.

3

ε it is a centered Gaussian noise.

For compliance with the additivity and gaussianity assumptions of this decomposition model, each time series (y it ) t=1,...,T was replaced by (log(y it + λ)) t=1,...,T , where λ is a small positive number preventing degeneracy caused by null consumptions.

Note that this transformation is used in the same way as the well known Box-Cox transformation [START_REF] Box | An analysis of transformations[END_REF].

Parameters estimation and practical use of the model

Given a time series y i recorded by a smart meter, the trend f i is estimated using a simple moving average [START_REF] Gourieroux | Time series and dynamic models[END_REF][START_REF] Shumway | Time series analysis and its applications: with R examples[END_REF]. As the daily and weekly periodicities (24 and 168) should be removed from the univariate time series, a centered moving average of order 168 is performed.

After estimating the trend and given a couple (q 1 , q 2 ), the coefficients α 1j , α 2j , α 3j , α 4j and γ j are simultaneously identified by performing a multiple linear regression of (y it -f it ) over the variables cos 2πjt 24 , sin 2πjt 24 , cos 2πjt 168 , sin 2πjt 168 and δ tj .

Selecting the couple (q 1 , q 2 ) remains a sensible point which can ideally be addressed by choosing the couple which optimizes a model selection criterion such as the Akaike information criterion (AIC) introduced by [START_REF] Akaike | A new look at the statistical model identification[END_REF] or the Bayesian information criterion (BIC) introduced by [START_REF] Schwarz | Estimating the dimension of a model[END_REF]. In this paper, several combinations of (q 1 , q 2 ) were tested and the couple (4, 24) has been selected leading to a good compromise between visual representation of seasonal patterns and modeling accuracy. An example of decomposition of a time series is shown in Figure 2. The trend is displayed together with the complete time series while the seasonal component is displayed with the weekly sub-series. From each time series y i , the model parameters defined by Equation ( 1) are thus identified, and the periodic seasonal pattern defined by x i = (x i1 , . . . , x im ), with m = 168, is extracted. Due to the periodicity of the series (x it , . . . , x iT ) defined by Equation ( 2), it should be noted that the first terms m = 168 are sufficient to characterize the time series. Then, the set of seasonal patterns (x 1 , . . . , x n ) are standardized as suggested by [START_REF] Gaffney | Probabilistic curve-aligned clustering and prediction with regression mixture models[END_REF]

x it ← x it -(1/m) m j=1 x ij σ(x i )
, ∀i, t, where σ(x i ) is the standard deviation of x i . The set of normalized seasonal patterns is used as input data for the clustering step which will be described in the following section. It is worth noting that the proposed decomposition can also be used to fill missing values that may occur along the time series. The reconstruction formula is y it = f it + x it + d it , where f it , x it , d it are the estimated components.

Clustering seasonal profiles

In order to extract relevant usage profiles from water consumption time series, two functional data clustering approaches are considered in this paper: the first one is the functional version of the K-means algorithm and the second one is based on a specific Fourier regression mixture model.

Functional clustering based on FPCA

In this subsection, the clustering method [START_REF] Peng | Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions[END_REF][START_REF] Sood | Functional regression: A new model for predicting market penetration of new products[END_REF] is inspired by functional data analysis [START_REF] Ramsay | Functional Data Analysis[END_REF][START_REF] Wang | Review of functional data analysis[END_REF] which assumes that data are functions or curves. This clustering approach is mainly based on Functional Principal Component Analysis (FPCA) and can be summarized into the following two consecutive steps:

1. Smoothing and dimension reduction: this step consists in converting the n time series (x 1 , . . . , x n ) into functional objects (x 1 (t), . . . , x n (t)), and then applying the classical PCA on the multivariate data obtained by discretizing the functions

x i (t) over the temporal grid {1, . . . , m}. In this paper, the PCA is directly performed on the seasonal patterns (x 1 , . . . , x n ) that are based on trigonometric (smooth) functions, and the principal components are selected such as 95% of the data variance is explained.

2. Clustering: in this step consists, a classical clustering method is performed on the principal component scores estimated previously. The well-known K-means algorithm [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] is applied using several random initializations and the partition with the lowest intra-cluster inertia is selected.

The resulting functional clustering strategy is called FPCA-KM. The number of cluster K has been selected by minimizing the BIC-like penalized criterion BIC(K) = C + ν K log(n), where C is the intra-cluster inertia optimized by the K-means algorithm and ν K = Kq is the number of parameters to be estimated with q the number of selected principal components.

The general idea of PCA is to create a small number of uncorrelated variables with maximal variance. The extension of this technique for functional data is proposed in the work of [START_REF] Ramsay | Functional Data Analysis[END_REF]; [START_REF] Ferraty | Nonparametric functional data analysis: theory and practice[END_REF]. The FPCA is an efficient tool providing common functional components explaining the structures of individual trajectories.

Fourier regression mixture model

Inspired by the polynomial regression mixture model formulated by [START_REF] Gaffney | Trajectory clustering with mixtures of regression models[END_REF], this subsection introduces a Fourier regression mixture model, called the FReMix model. The Fourier regression mixture was preferred to polynomial and spline regression mixtures, for its compliance with the modeling adopted in the first step (seasonal pattern extraction).

Moreover, Fourier polynomial is a universal approximator of functions and remains a good candidate in modeling clusters whose prototypes are non linear and potentially periodic functions.

Model definition

Unlike standard vector-based mixture models, the density of each component of the FReMix model is represented by a trigonometric prototype function that is parameterized by regression coefficients and a noise variance. The prototype functions represent the class conditional expectations of x i . The Fourier regression mixture model therefore assumes that each time series x i is distributed according to the following density

f (x i ; θ) = K k=1 π k N x i ; Uβ k , σ 2 k I , (3)
where θ = (π 1 , . . . , π K , β 1 , . . . , β K , σ 2 1 , . . . , σ 2 K ) is the complete parameter vector. The probabilities π k are the proportions of the mixture satisfying q1+q2) is the coefficient vector of the k-th regression model and σ 2 k > 0 is the associated noise variance. The matrix U = [u 1 , . . . , u m ] is a regression matrix of size m × 2(q 1 + q 2 ),

K k=1 π k = 1, β k = (β k,1 , . . . , β k,2(q1+q2) ) ∈ R 2(
where the vector u t ∈ R 2(q1+q2) is defined by (∀t = 1, . . . , m)

u t =

EM algorithm and practical issues

Assuming that the n seasonal time series (x 1 , . . . , x n ) are independent, the parameter vector θ is estimated in the same way as for the classical Gaussian mixture model [START_REF] Mclachlan | The EM algorithm and extensions[END_REF] and the polynomial regression mixture model [START_REF] Gaffney | Trajectory clustering with mixtures of regression models[END_REF], by maximizing the log-likelihood

L(θ) = n i=1 log K k=1 π k N x i ; Uβ k , σ 2 k I
, via the Expectation-Maximization (EM) procedure [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF][START_REF] Gaffney | Trajectory clustering with mixtures of regression models[END_REF][START_REF] Mclachlan | The EM algorithm and extensions[END_REF].

The pseudo-code can be found in the paper proposed by [START_REF] Samé | Décomposition et classification de données fonctionnelles pour l'analyse de la consommation d'eau potable[END_REF]. As a reminder, the couple (q 1 , q 2 ) = (4, 24) is selected in the seasonal pattern extraction step (cf. subsection 3.2). The algorithm is initialized as follows: the initial regression coefficients and variances are obtained by performing a Fourier regression separately on K seasonal series randomly drawn into the data set (x 1 , . . . , x n ) and the initial proportions of the latent classes are set to π k = 1 K . This process is repeated 20 times and the parameters with the highest log-likelihood are selected. The number of clusters is selected through the BIC criterion [START_REF] Schwarz | Estimating the dimension of a model[END_REF] defined by BIC(K) = -2L( θ) + ν K log(n), where θ is the parameter vector estimated by the EM algorithm, and ν K is the number of free parameters of the model: ν K = 2K(q 1 + q 2 + 1) -1.

After estimated the parameter vector θ, a time series partition is obtained by assigning each series x i to the cluster having the highest posterior probability These series are used as input data for the clustering algorithms.

τ ik = π k N (xi;Uβ k ,σ 2 k I) K =1 π N (xi;Uβ ,σ 2 I) • 5

Selecting the number of clusters

The number of clusters for the two methods was selected by running the algorithms with several values of K and then choosing the value which minimizes the BIC criterion. Figure 3 shows the evolution of this criteria for the two clustering algorithms in relation to the number of clusters. For both methods, the BIC criterion exhibits a decrease continuously while the K value increases. Nevertheless, it can be seen that the variation of BIC is not significant when the number of clusters is above 8.

Therefore, the number of clusters is selected such as K = 8.

Results interpretation and discussion

The seasonal time series are classified into K = 8 clusters, using functional K-means (FPCA-KM strategy) and Fourier regression mixture (FReMix model) as illustrated respectively by Figures 4a and4b. For each cluster, the weekly prototype is displayed in orange (sub-figures on the left). Moreover, the right plots of Figures 4a and 4b display the cluster profiles using a daily representation, the colors (from blue to red) indicating the day of the week (from Monday to Sunday). The percentage of input time series belonging to each cluster is also provided. It can be observed that the consumption profiles are quite similar for the two methods, despite the diffences of the cluster percentages. As no socio-demographic data about customers was available at this stage of the study, a qualitative evaluation of the results is performed and the pattern repartition shown in Figure 4 can be explained by the following realistic categories:

-Office or industrial use: cluster 1. One can observe an active water consumption from Monday to Friday (workdays) -Residential use: clusters 3, 4, 5. The temporal dynamic of these clusters corresponds to customers who wake up between 6 AM and 8 AM, take a shower and then go to work. This habit is characterized by a consumption peak around 10 AM in the morning. The other peak, observed in the evening at around 20 PM, corresponds to the return at home. The minimum consumption level between these two peaks can be attributed to persons in households who stay at home during working hours.

-Commercial use: clusters 6, 7, 8. This category corresponds to a set of customers whose consumption habits are the same during working days and weekends. It may correspond, for example, to small businesses or medical centers that stay open every day and have the same daily consumption profile. It should be noticed that clusters 6, 7 differ from the other clusters by their smaller evening peak.

-Noise cluster: cluster 2. This cluster, which has the largest variance, groups a set of atypical patterns which does not match with the other clusters. It can be considered as a noise cluster.

Note that a functional clustering scheme is adopted because this is suitable to deal with the analysis of our consumption curves. Indeed, these real-valued data can be seen as the realizations of a one-dimensional stochastic process, recorded on the same time grid (hourly spaced) of ordered times. In practice, data frames are frequently sent by modules which are physically connected to the meters ; each consumption time series can be re-constructed based on a sequence of the data frames. Exogenous variables (e.g. weather inputs or meter localization) are not considered in this work due to a non-significant improvement in the results but might be used in a future work. In other words, the time series x i have a length of 168 due to the trigonometric modeling of the chosen Fourier basis decomposition. The Fourier coefficients are identified performing a multiple linear regression on the global time series y i detrended which limits the effect of a long seasonality. Finally, our sample has a duration of 15 months (with 5 seasons at most) and we are more interested about identifying the major mode of consumption than estimating water demand profiles with local changes and a fine granularity.

However, we wanted to integrate some prior knowledge about day/week seasonality and exceptional public non-working days. A Fourier-based decomposition has the capacity to easily take into account this prior knowledge and this decomposition is consistent with our probabilistic FReMix model definition. A wavelet-based analysis could also be used for decomposition (keeping some local properties along temporal patterns) but integrating such prior knowledge might not be straightforward and the number of parameters in this case should be not be reduced significantly. In this article, we evaluate a probabilistic method and a geometrical approach. This second method is based on a Kmeans and minimizes the intra-cluster inertia which can be seen as an aggregated distance over the water time series. As our knowledge, the complexity of time series distances (e.g. dynamic time warping) can be prohibitive with a large time series dataset. This paper deals with an unsupervised classification problem based on water consumption time series. Water demand forecasting is not the issue in this paper ; nevertheless, the resulting segmentation of water consumption time series can be used for several scientific problems including sequential detection, predictive classification or demand forecasting. We assume no supervision in our setting due to a partial and uncertain knowledge of the usage labels ; users do not inform systematically their water utilities when businesses change or people come in / leave a home. This is why there is no quantitative accuracy about clustering. Each log-consumption time series is standardized before clustering which leads to a discrimination in term of seasonal patterns and not based on water volume. This explains why we entitled the cluster 1 as "'office and industrial usage". Of course, industrial usage might produce erratic water patterns which would be classified in cluster 2. Partitioning the 8 clusters in 4 categories would suggest non negligible variations in residential use as well as commercial use, and extra investigations about the users which should not be underestimated in term of time and cost. The EM algorithm used to fit the Fourier REgression Mixture (FReMix) model is flexible and can be reformulated in a future work with a semi-supervision (by fixing a set of posterior probabilities) or a partial supervision (e.g. using belief functions).

Identifying the major usage profiles from water consumption is an interesting topic to water utilities. Indeed, the resulting segmentation helps the water companies to gain a better knowledge about users consuming the distributed water. The user is having a better experience with the tools developed by their water utility. For instance, users at Veolia Eau d'Ile de France (Paris area in France) can already monitor their water index / consumption on a dedicated website for free. Using our clustering results, people could compare with similar patterns and adapt their consumptions according to their needs. In addition, the resulting clusters are used by an early warning system which alerts the user when a leakage occurs into the private network. An erratic water pattern (like in cluster 2) can be the sign of a leakage and might initiate a corrective action. Concerning the grid management, each prototype can be used to represent the water behavior of users belonging to the same cluster. Sampling a large amount of water meters is useful for several topics (e.g. tracking the meter metrology, estimating the global consumption modes based on a limited number of meters) ; such sampling analysis is straightforward using our meter segmentation.

Conclusions and perspectives

A general methodology is introduced in this paper for automatically discriminating several water usages and extracting relevant water consumption profiles from time series recorded by smart meters. Considering that the consumption habits are of interest and not the consumption levels, the first step of the method consists in extracting the seasonal part of time series using an additive classical decomposition model. This modeling of the seasonal component is based on a specific Fourier expansion which takes into account daily and weekly periodicities. As this study aims to identify relevant water usage profiles, two functional clustering techniques are used to classify the seasonal patterns extracted from the water consumption time series: a functional variant of the K-means algorithm and a specific EM algorithm based on a Fourier regression mixture model (FReMix). The FReMix model is richer than the other clustering approach in that the Fourier basis decomposition is fully integrated in the modeling and each cluster is described by its first two moments while the K-means only extracts the mean curves. Furthermore, the K-means produces a hard segmentation while the FReMix creates a soft partition where each cluster membership is weighted by a posterior probability. Eight clusters are then identified for the two clustering methods. The resulting prototypes are quite similar for the two approaches and a realistic category is given each cluster. More investigations are in progress with the water utility Veolia Eau d'Ile de France in order to refine the clustering results and the proposed methodology is also being applied to a new large scale database.

Figure 1 .

 1 Figure 1. Block diagram describing the global methodology.

Figure 2 .

 2 Figure 2. Extraction of periodic seasonal patterns using Fourier-based time series decomposition. The trend is displayed with the complete time series (a) and the seasonal component is displayed with the weekly time series (b).

  Experimental study using real data5.1 Description of the data setThe experimental data set represents the water consumption recorded by a few smart meters deployed on the network of Syndicat des Eaux d'Ile-de-France (SEDIF). The SEDIF is a large association including 150 municipalities which provides drinking water for more than 4 million inhabitants of suburban Paris. This is the largest drinking WDN in France with about 8,000 km of pipes and more than 750,000 m3 of water produced each day. The consumption is measured hourly (in liter) by 10, 233 meters during 15 months (from Nov-2013 toMar-2015). The resulting data set is then made of univariate time series (y 1 , . . . , y n ), where n = 10, 233 and the length of each time series y i is T = 11, 016. After the extraction of periodic seasonal patterns (cf. section 3), a new set of time series (x 1 , . . . , x n ) is built where the length of each seasonal patterns x i is m = 168.

Figure 3 .

 3 Figure 3. Evolution of the BIC criteria according to the number of clusters.

Figure 4 .

 4 Figure 4. Clustering results obtained with the FPCA-KM (4a) and the FReMix (4b). For each side, the subfigures on the left represent a weekly view of the clusters with their prototypes displayed in orange. The subfigures on the right are daily prototypes resulting from the segmentation of the weekly orange curves and colors (from blue to yellow to red) indicate the day of the week (from Monday to Sunday).

during the business hours, and a very low consumption during the weekend.

Acknowledgements. The study presented in the paper was part of a previous work between IFSTTAR and Veolia Eau d'Ile de France. This work is now part of the French-German collaborative research project ResiWater (http://www.resiwater.eu/) that is funded by the French National Research Agency (ANR; project: ANR-14-PICS-0003) and the German Federal Ministry of Education and Research (BMBF; project: BMBF-13N13690).


Lazhar Dhaouadi
email: lazhardhaouadi@yahoo.fr

Functions of q-positive type

Functions of q-positive type

Introduction and Preliminaries

Given a positive finite Borel measure µ on the real line R, the Fourier transform Q of µ is the continuous function

Q(x) = R e -itx dµ(t).
The function Q is a positive definite function, i.e for any finite list of complex numbers z 1 , . . . , z n and real numbers x 1 , . . . , x n n r=1 n l=1 z r z l Q(x r -x l ) ≥ 0.

Bochner's theorem says the converse is true, i.e. every positive definite function Q is the Fourier transform of a positive finite Borel measure. In q-Fourier analysis, semelar phenomenon will appear. It is the subject of our article.

In the following we consider 0 < q < 1 and we adopt the standard conventional notations of [START_REF] Gasper | Basic hypergeometric series, Encycopedia of mathematics and its applications 35[END_REF]. We put

R + q = {q n , n ∈ Z},
the set of q-real numbers and for complex a

(a; q) 0 = 1, (a; q) n = n-1 i=0 (1 -aq i ), n = 1...∞.
Jackson's q-integral (see [START_REF] Jackson | On a q-Definite Integrals[END_REF]) in the interval [0, ∞[ is defined by

∞ 0 f (x)d q x = (1 -q) ∞ n=-∞ q n f (q n ).
Let C q,0 and C q,b denote the spaces of functions defined on R + q continued at 0, which are respectively vanishing at infinity and bounded. These spaces are equipped with the topology of uniform convergence, and by L q,p,v the space of functions f defined on R + q such that

f q,p,v = ∞ 0 |f (x)| p x 2v+1 d q x 1/p < ∞.
The q-exponential function is defined by

e(z, q) = ∞ n=0 z n (q, q) n = 1 (z; q) ∞ , |z| < 1.
The normalized Hahn-Exton q-Bessel function of order v > -1 (see [START_REF] Swarttouw | The Hahn-Exton q-Bessel functions[END_REF]) is defined by

j v (z, q) = ∞ n=0 (-1) n q n(n-1) 2
(q, q) n (q v+1 , q) n z n .

The q-Bessel Fourier transform F q,v introduced in [START_REF] Dhaouadi | Inequalities in q-Fourier Analysis[END_REF][START_REF] Koornwinder | On q-Analogues of the Hankel and Fourier Transform[END_REF] as follow

F q,v f (x) = c q,v ∞ 0 f (t)j v (xt, q 2 )t 2v+1 d q t,
where

c q,v = 1 1 -q (q 2v+2 , q 2 ) ∞ (q 2 , q 2 ) ∞ .
Define the q-Bessel translation operator as follows:

T v q,x f (y) = c q,v ∞ 0 F q,v (f )(t)j v (xt, q 2 )j v (yt, q 2 )t 2v+1 d q t, ∀x, y ∈ R + q , ∀f ∈ L q,v,1 .
Recall that T v q,x is said positive if T v q,x f ≥ 0 for f ≥ 0. In the following we tack q ∈ Q v where

Q v = {q ∈]0, 1[, T v q,x is positive for all x ∈ R + q }.
The q-convolution product of both functions f, g ∈ L q,1,v is defined by

f * q g(x) = c q,v ∞ 0 T v q,x f (y)g(y)y 2v+1 d q y.
In the end we denote by A q,v the q-Wiener algebra

A q,v = {f ∈ L q,1,v , F q,v (f ) ∈ L q,1,v } .
The followings results in this sections was proved in [START_REF] Dhaouadi | Inequalities in q-Fourier Analysis[END_REF].

Proposition 1 Let n, m ∈ Z and n = m, then we have

c 2 q,v ∞ 0 j v (q n x, q 2 )j v (q m x, q 2 )x 2v+1 d q x = q -2n(v+1) 1 -q δ nm . Proposition 2 |j v (q n , q 2 )| ≤ (-q 2 ; q 2 ) ∞ (-q 2v+2 ; q 2 ) ∞ (q 2v+2 ; q 2 ) ∞ 1 if n ≥ 0 q n 2 +(2v+1)n if n < 0 .
Proposition 3 The q-Bessel Fourier transform

F q,v : L q,1,v → C q,0 , satisfying F q,v (f ) C q,0 ≤ B q,v f q,1,v ,
where

B q,v = 1 1 -q (-q 2 ; q 2 ) ∞ (-q 2v+2 ; q 2 ) ∞ (q 2 ; q 2 ) ∞ .
Theorem 1 Given f ∈ L q,1,v then we have

F 2 q,v (f )(x) = f (x), ∀x ∈ R + q . If f ∈ L q,1,v and F q,v (f ) ∈ L q,1,v then F q,v (f ) q,2,v = f q,2,v . Proposition 4 Let f ∈ L q,1,v then T v q,x f (y) = ∞ 0 f (z)D v (x, y, z)z 2v+1 d q z,
where

D v (x, y, z) = c 2 q,v ∞ 0 j v (xt, q 2 )j v (yt, q 2 )j v (zt, q 2 )t 2v+1 d q t.
Proposition 5 Given two functions f, g ∈ L q,v,1 then

f * q g ∈ L q,v,1 , and
F q,v (f * q g) = F q,v (f ) × F q,v (g).
Proposition 6 The q-Gauss kernel

G v (x, t, q 2 ) = (-q 2v+2 t, -q -2v /t; q 2 ) ∞ (-t, -q 2 /t; q 2 ) ∞ e(- q -2v t x 2 , q 2 ), satisfying F q,v e(-ty 2 , q 2 ) (x) = G v (x, t, q 2 ),
and for all function

f ∈ C q,b lim a→0 c q,v ∞ 0 f (x)G v (x, a 2 , q 2 )x 2v+1 d q x = f (0).
Theorem 2 Given 1 < p, p ′ , r ≤ 2 and

1 p + 1 p ′ -1 = 1 r .
If f ∈ L q,p,v and g ∈ L q,p ′ ,v then f * q g ∈ L q,r,v .

2 Functions of q-positive type Definition 1 A function φ is of q-positive type if

φ ∈ C q,b ∩ L q,1,v
and for any finite list of complex numbers z 1 , . . . , z n and q-real numbers

x 1 , . . . , x n n r=1 n l=1 z r z l T v q,xr φ(x l ) ≥ 0. ( 1
)
Proposition 7 Let φ ∈ A q,v of q-positive type then F q,v φ is of q-positive type.

Proof. From Proposition 3 and the definition of the q-Wiener algebra

F q,v (φ) ∈ C q,b ∩ L q,1,v .
On the other hand, with the inversion formula in Theorem 1 we get

T v q,x F q,v (ξ)(y) = ∞ 0 j v (tx, q 2 )j v (ty, q 2 )t 2v+1 φ(t)d q t, then n r=1 n l=1 z r z l T v q,xr F q,v ξ(x l ) = c q,v ∞ 0 n r=1 n l=1 z r z l j v (x r t, q 2 )j v (x l t, q 2 ) t 2v+1 φ(t)d q t = c q,v ∞ 0 n r=1 z r j v (x r t, q 2 ) n l=1 z l j v (x l t, q 2 ) t 2v+1 φ(t)d q t = c q,v ∞ 0 n r=1 z r j v (x r t, q 2 ) 2 t 2v+1 φ(t)d q t ≥ 0.
This finish the proof.

Proposition 8 If φ is of q-positive type and f ∈ L q,2,v then φ * q f ∈ L q,2,v , and φ * q f, f ≥ 0.

Proof. From Theorem 2 we see that φ * q f ∈ L q,2,v . On the other hand

φ * q f, f = c 2 q,v ∞ 0 ∞ 0 T v q,x φ(y)f (y)y 2v+1 d q y f (x)x 2v+1 d q x = (1 -q) 2 c 2 q,v ∞ r=1 ∞ l=1
q (2v+2)r f (q r )q (2v+2)l f (q l )T v q,q r φ(q l ) ≥ 0.

This finish the proof.

Corollary 1 If φ is of q-positive type then

F q,v φ(x) ≥ 0, ∀x ∈ R + q .
Proof. Given x ∈ R + q and let

f x : t → c q,v j v (xt, q 2 ),
then with Proposition 2 we see that f x ∈ L q,2,v and by Proposition 1

F q,v f x (y) = δ q,v (x, y),
which implies (see [START_REF] Dhaouadi | Inequalities in q-Fourier Analysis[END_REF])

F q,v φ × F q,v f x , F q,v f x = F q,v φ(x)δ q,v (x, x) = 1 (1 -q)x 2v+2 F q,v φ(x).
From Proposition 8 F q,v φ × F q,v f, F q,v f = φ * q f, f ≥ 0, this leads to the result.

Proposition 9 If φ is of q-positive type then F q,v φ ∈ L q,1,v .

Proof. From Proposition 6 lim a→0 ∞ 0 e(-a 2 x, q 2 )F q,v φ(x)x 2v+1 d q x = lim a→0 c q,v ∞ 0 G v (x, a 2 , q 2 )φ(x)x 2v+1 d q x = φ(0).
By the monotone convergence theorem and the preview corollary we see that

∞ 0 |F q,v φ(x)|x 2v+1 d q x = ∞ 0 F q,v φ(x)x 2v+1 d q x = φ(0).
This finish the proof.

Theorem 3 Let (ξ n ) n≥0 be a sequences of probability measures of M + q such that lim

n→∞ F q,v (ξ n )(x) = ψ(x),
then there exists ξ ∈ M + q such that the sequence ξ n converge strongly toward ξ and F q,v (ξ) = ψ.

In the following we consider the function ψ defined by

ψ(x) = 1 -x if x < 1 0 otherwise .
Now we are in a position to state and prove the q-analogue of the Bochner's theorem Theorem 4 Let φ be a function defined on R + q continued at 0. Assume that the following function

φ n : x → φ(x)ψ(q n x), satisfy (1)
for all n ∈ N then there exist ξ ∈ M + q such that F q,v (ξ) = φ.

Proof. The function φ n is of q-positive type. From Corollary 2 there exist ̺ n a positive function of A q,v such that

F q,v (̺ n ) = φ n .
The measure ξ n defined by d q ξ n (x) = ̺ n (x)d q x, belong to M + q and ∞ 0

x 2v+1 d q ξ n (x) = F q,v (̺ n )(0) = φ n (0) = φ(0).

Assume that φ(0) = 1. On the other hand

lim n→∞ F q,v (ξ n )(x) = lim n→∞ φ n (x) = φ(x).
From Theorem 3 there exists ξ ∈ M + q such that the sequence ξ n converge strongly toward ξ, and F q,v (ξ) = φ, which leads to the result.

Corollary 2 If φ is of q-positive type then there exist a positive function ξ ∈ A q,v such that φ(x) = F q,v ξ(x), ∀x ∈ R + q .

Proof. From the inversion formula in theorem 1

Define the function ξ as follows

By the use of Corollary 1 and Proposition 9 we see that ξ is a positive function of A q,v .

Proposition 10 Suppose φ is of q-positive type. If f ∈ L q,1,v is positive function then the product φF q,v f is of q-positive type.

Proof. Proposition 5 and Proposition 9 give

From the definition of the q-convolution product we write

Proposition 4 give

This implies with Corollary 1

which leads to the result.

Corollary 3 Given two functions φ 1 , φ 2 which are of q-positive type then the product φ 1 × φ 2 is also of q-positive type.

Proof. Let ξ = F q,v φ 2 then with the inversion formula in theorem 1 we see that F q,v ξ = φ 2 . Proposition 9 give

and by Proposition 10 we achieved the proof.

3 q-Bochner's Theorem

We consider the set M + q of positives and bonded measures on R + q . The q-Bessel Fourier transform of ξ ∈ M + q is defined by

The q-convolution product of two measures ξ, ρ ∈ M + q is given by

T v q,x f (t)t 2v+1 d q ξ(x)d q ρ(t), and we have F q,v (ξ * q ρ) = F q,v (ξ)F q,v (ρ).

The following Theorem (see [START_REF] Dhaouadi | Inequalities in q-Fourier Analysis[END_REF]) is crucial for the proof of our main result.