text:
Coca Carola

Coca Carola is a Swedish punk band formed in Åkersberga in 1986. Signed to the Swedish record label Beat Butchers, they were active and recognized in the Swedish punk scene during the years 1986–2004. In 2023, after a long hiatus, the band has begun to tour again.

History
At the beginning, the band consisted of musicians who had been fired from their previous bands: Curt "Curre" Sandgren (former member of the band Rolands Gosskör), Mårten Tolander, "Ostron" Åkerholm, and Björn Gullberg. in 1987, Coca Carola released a self-titled single, and after this Gullberg left the band. For a short time, Putte Strand joined the band as a guitarist. He was replaced by Conny Melkersson, who had played with Sandgren in the band Rolands Gosskör.

In 1988, Coca Carola, together with the band 23 Till, recorded their first LP titled Anabola Melodier. Coca Carola was responsible for the first half of the album.

In the summer of 1991, Coca Carola released the single C/C 7 Inch; the three songs on this single used English-language lyrics. Busy with his other band NEIN, Melkersson left the band. Coca Carola found a new guitarist, Jonas Winberg. In the fall of 1992, the band's debut album titled Tigger & Ber was released, and the band toured Germany. They played at the Hultsfred Festival and toured with De lyckliga kompisarna.

In January 1994, Coca Carola released their second album, Läckert. Later that year, bassist Åkerholm left the band because it was taking too much of his time. Since the band had planned to release a single in the fall of 1994, Åke Noring (also from Rolands Gosskör) temporarily played bass in Coca Carola. Jonas Mellberg, who was a member of the band Håkan sover naken joined Coca Carola, and remained their bass player until 2004.

In September 1994, Coca Carola released the EP Klubben för inbördes beundran, whose first track, "Inbördes beundran", reached 37th place on the Swedish singles charts.

In early 1996, Coca Carola released their third album, Dagar kommer. For the band's 10th anniversary later that year, the collection ...Fem år till moped was released. This album contained several new songs, as well as older rare tracks. In 1997, the band took a hiatus, but returned in 1998 with a new album Så Fel Som Bara Vi Kan Ha.

In December 1998, Coca Carola received a request to write songs for a compilation album for the soccer team Djurgården's IF. The band accepted, and wrote nine new songs, including Min tröja.

In 2000, their fifth studio album was recorded and released, Feber. In 2004, Coca Carola released their most recent album Kryp din Jävel and played their final gig at Fregatten in Åkersberga. Soon after that, Curre Sandgren and Jonas Mellberg started a new band, called Den sista leken, which released an album the same year.

In 2012, Coca Carola reunited for a gig at the festival Peace & Love in Borlänge and for an unpublicized concert at Kafé 44 in Stockholm. The following year, they played with Charta 77 and Johan Johansson, among others, at an event in Stockholm; this performance was dedicated to the Peace & Love festival, which had gone bankrupt.

In 2023, Coca Carola started to play again regularly. Their reunion concert took place at The Public in their hometown of Åkersberga in January 2023. In February 2023, the band participated in "Ögir will arise" at Frimis in Örebro, a benefit to support the rebuilding of Per Granberg's music bar Ögir, which burned down in 2022. Coca Carola's next gig in August 2023 was even larger: at Gröna Lund. This performance included Jonas Mellberg was a guest artist. In 2024, Coca Carola have committed to a benefit concert for guitarrist Micke Blomquist, organized by Asta Kask.

Band members
Current line-up:
 * Curt Erik Torvald "Curre" Sandgren – guitar & vocals
 * Jonas Winberg – guitar & backing vocals
 * Åke Norling – bass & backing vocals
 * Mårten Tolander – drums

Previous/temporary members:
 * Jonas Mellberg – bass & backing vocals
 * Ostron Åkerholm – bass
 * Björn Gullberg – guitar, vocals
 * Conny Melkersson – guitar
 * Putte Strand – guitar

Studio albums

 * 1) 1988: Anabola Melodier (Studio Otukt)
 * 2) 1992: Tigger & Ber (Beat Butchers)
 * 3) 1994: Läckert (Beat Butchers)
 * 4) 1996: Dagar kommer (Beat Butchers)
 * 5) 1998: Så fel som bara vi kan ha (Beat Butchers)
 * 6) 2000: Feber (Beat Butchers)
 * 7) 2004: Kryp Din Jävel (Beat Butchers)

Live albums

 * 2000: Live i Kalmar, Järnkaminerna
 * 2001: Kom och slå mig igen
 * 2002: Live i Sundsvall
 * 2004: En sista gång på 44:an

Compilations

 * 1996: ...Fem år till moped
 * 2012: Samlade

EPs

 * 1994: Dimmornas land
 * 1994: Klubben för inbördes beundran

Singles

 * 1987: Kom till kriget (self-titled 7inch)
 * 1991: C/C 7"
 * 2015: Tack För Festen

Samplers (selected)

 * 1995: Äggröran 1 (Ägg Tapes & Records)
 * 1998: Definitivt 50 Spänn 7
 * 1998: Definitivt The Greatest Hits
 * 2001: Definitivt 50 Spänn 10
 * 2007: Svenska Punkklassiker Vol. 2

title: Coca Carola


text:
List of television stations in Baja California Sur

The following is a list of all IFT-licensed over-the-air television stations broadcasting in the Mexican state of Baja California Sur. There are 22 television stations in Baja California Sur.

List of television stations

Defunct stations

 * XHK-TV 10, La Paz (1968–2015)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User:LtPylon

Hi... I'm Pylon... or you can call me Mike. I don't know why you're here, but let me say that you would be much better off spending your time reading something else. In any case, I was an E2 convert to Wikipedia back in 2000. I have yet to do anything of significance here and may do so when I'm not so engaged with work, volunteer pursuits, friends, family, and travel. /Sandbox

"A man may die, nations may fall, but an idea lives on..." -J.F.K.

Sic Transit Gloria Mundi.

title: User:LtPylon


text:
User:EvanProdromou/Sandbox

 * User:EvanProdromou/WikiProject Wine interview

I'm dicking around with the new table formatting instructions.

 * User:EvanProdromou/DivIndent

wikitravel:en:Main Page wikitravel:Main Page wikitravel:nl:hoofdpagina wikitravel:de:Hauptseite

Kopf

title: User:EvanProdromou/Sandbox


text:
Bengali poetry

Bengali poetry is a rich tradition of poetry in the Bengali language and has many different forms. Originating in Bengal, the history of Bengali poetry underwent three successive stages of development: poetry of the early age (like Charyapad), the Medieval period and the age of modern poetry. All ages have seen different forms of poetry and poetical tradition. It reached the pinnacle during the Bengali Renaissance period although it has a rich tradition and has grown independent of the movement. Major Bengali Poets throughout the ages are Chandidas, Alaol, Ramprasad Sen, Michael Madhusudan Dutt, Nabinchandra Sen, Rabindranath Tagore, Dwijendralal Ray, Satyendranath Dutta, Kazi Nazrul Islam, Jibanananda Das, Jasimuddin, Sukanta Battacharya, Al Mahmud.

Introduction
Poetry in the colloquial dialect of Bengal first originated from Prakrit, and based upon local socio-cultural traditions. It was antagonistic towards Vedic rituals and laws as opposed to the sahajaya traditions of the poets themselves - who were mainly Buddhist sages.

The medieval period marked the introduction of, which played an important role in Muslim life and brought much Persian and Arabic influence to the poetic lexicon. Shah Gharibullah was said to have initiated this trend with his epic "Amir Hamza". Many, based on battles, were written during this time. Jongonamas were generally elegiac in tone. Works relating to Karbala were called (meaning 'grief' in Arabic) literature. Both janganama and marsiya literature first developed in Arabia and later Persia. Muslim Sufis and soldiers introduced this form of poetry in the Bengali language to the masses in Bengal and Arakan. Well-known poems of include Zainab's Chautisha by Sheikh Faizullah, Maqtul Husayn by Mohammad Khan and Qasim-er Lodai O Fatima-r Surotnama by Sherbaz. The works mixed Bengali folk poetry with Perso-Arabian stories and themes, and are considered an important part of the Muslim culture of Bengal.

Ancient and medieval eras
Charyapad is the oldest poetry and literary specimen of Bengali language. It is also the oldest work in the neo-Indian Aryan language. The composers of these hymns, composed between the tenth and twelfth centuries AD, were easily Buddhist Siddhacharyas. The 24 major Charyapadas were by Lui Pa, Kukkuripa, Biruapa, Gundaripa, Chatilpa, Bhusuk Pa, Kahnpa, Kambalambarpa, Dombipa, Shantipa, Mahittapa, Veenapa, Sarhapa, Shabar Pa, Azdebpa, Dhenpana, Dankapa, Darikpa Dhampa, Tantripa, Laridombipa. 1200-1350 AD of Bengali literature has been marked as the "dark age of Bengali Literature." During this time Srikrishnakirtan was composed by Baduchandidas.The other remarkable poets of Vaishnava literature of this era were Vidyapati, Jnandas, Govindadas, Yashoraj Khan, Chandkaji, Ramchandra Basu, Balram Das, Narhari Das, Vrindavan Das, Basudavas, Bansibad. Syed Sultan, Harhari Sarkar, Fateh Paramananda, Ghanshyam Das, Gayas Khan, Alaol, Deen Chandidas, Chandrashekhar, Haridas, Shivram, Karam Ali, Pir Muhammad, Hiramani, Bhavananda. Famous poets of Mangal Kavya are Kanahari Dutt, Narayan Dev, Vijaygupta, Bipradas Piplai, Madhav Acharya, Mukundaram Chakraborty, Ghanram Chakraborty, Srishyam Pandit, Bharat Chandra Roy Gunakar, Khemananda, Ketka Das Khemananda, Dwij Madhav, Adi Rupram, Manik Ram, Mayur Bhatt, Khelaram, Rupram, Sitaram Das, Shyamj.

The 18th-century two brilliant Shakta bhakti court poets were Bharatchandra Ray and Ramprasad Sen.

Modern poets (1800–present)
The poets who built the bridge between the Middle and Modern Ages is the poet of the Ages:Ishwar Chandra Gupta (1812–1859), he is also regarded as the Father of the Modern Bengali Language. Michael Madhusudan Dutt (1824–1873) broke the medieval paradigm and Bengali poetry entered free verse - he is especially popular for his Bengali sonnets. Biharilal Chakraborty (1835–1894) who was a romantic and lyric poet of European style also contributed to the shift.

Kabiguru Rabindranath Tagore (1861–1941) was arguably the most revolutionary poets of Bengali Literature. He contributed to every branch of Bengali Literature including poetry. He broke down the old custom of writing in "Sadhu" language in poetry and introduced an aspect of poetry which had more freedom to the poet. He composed over two-thousand songs - which are known as "Rabindra Sangeet" and they are popular in West Bengal and Bangladesh even today. His songs "Jana Gana Mana" and "Amar Sonar Bangla" are the National Anthems of India and Bangladesh respectively. His impact on Bengali Literature is so immense that all Bengali Literature during his lifetime is known as the "Rabindric Era", after his name. The contemporary poets of the Rabindric era were Satyendranath Dutta, Kamini Ray, Updendrakishore Raychowdhury, Sukumar Ray, Jatindra Mohan Bagchi among others. The death of Rabindranath Tagore is often considered to be the end of Classical Poetry and the start of Modern Poetry.

At the beginning of the twentieth century, there was a new change in Bengali style of poetry, with poets adopting an even liberal style of poetry, bringing an end to Bengali Romanticism. Poets such as Mohitlal Majumder, Kazi Nazrul Islam and Jatindranath Sengupta beginning contributed to this period. "Bidrohi Kavi" Kazi Nazrul Islam is considered to be the bridge between the Anti-Romantic Period and the Modern Period of Poetry. He had composed a vast oeuvre of poetry and songs – the latter of which are known as "Nazrul Geeti" which are sung even today. Nazrul is also the National Poet of Bangladesh. His poem "Bidrohi" had a big impact on the Indian Freedom Movement.

The death of Tagore led to the beginning of the Golden Age of Modern Poetry, which lasted from the 1940s to the 60s. Numerous poets left their profound impact on poetry in this era, such as:Amiya Chakraborty (1901–1966), Jibanananda Das (1899–1954), Buddhadeb Basu (1906–1964), Bishnu Dey (1909–1962), Sudhindranath Dutta (1901–1960), Sablu Shahabuddin (1957–1999).They were followed by the generation of Shakti Chattopadhyay (1933–1995), Sankha Ghosh (1932–2021),Tarapada Ray (1936–2007) Sunil Gangopadhyay (1935–2012) – known as Krittibas group of poets.

Jasimuddin's work of Pastoral mood led him to be known as Palli Kabi. He was one of the major poets.

"Hungryalist movement" also emerged in Bengal that was consisted with new bunch of poets in Bengal, inducing Malay Roy Choudhury and Purusottam Kumar Debnath

Poets of West Bengal and Bangladesh
Following the Partition of India in 1947, Bengali poets were divided along nationalistic lines of India and Pakistan. West Bengal still had a great literary culture around it. Many Poets like Shakti Chattopadhyay, Binoy Majumdar, Samir Roychoudhury, Malay Roy Choudhury, Subimal Basak, Pranabkumar Chattopadhyay,  Subodh Sarkar emerged.

The poets of East Bengal took different yet similar path, many taking inspiration from Kazi Nazrul Islam and other major poets. Some of the popular Bengali Muslim poets of this period included Farrukh Ahmad, Talim Hossain, Golam Mostofa and Raushan Yazdani. The Bangladesh Liberation War and struggle for freedom became a popular theme in the years that followed. Al Mahmud is considered one of the greatest Bengali poets to have emerged in the 20th century.

title: Bengali poetry


text:
Talk:Marbled Godwit

Untitled
Does anyone know how to stop the image at the bottom overlapping the taxotable (probably only a problem on a 800x600 screen, but still... seglea 02:54, 29 Nov 2003 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Grey Plover which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 09:45, 4 May 2014 (UTC)

Move discussion in progress
There is a move discussion in progress on Talk:Grey plover which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 10:00, 5 May 2014 (UTC)

title: Talk:Marbled Godwit


text:
Fourier inversion theorem

In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function $$f:\R \to \Complex$$ satisfying certain conditions, and we use the convention for the Fourier transform that

 * $$(\mathcal{F}f)(\xi):=\int_{\mathbb{R}} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,$$

then

 * $$f(x)=\int_{\mathbb{R}} e^{2\pi ix\cdot\xi} \, (\mathcal{F}f)(\xi)\,d\xi.$$

In other words, the theorem says that

 * $$f(x)=\iint_{\mathbb{R}^2} e^{2\pi i(x-y)\cdot\xi} \, f(y)\,dy\,d\xi.$$

This last equation is called the Fourier integral theorem.

Another way to state the theorem is that if $$R$$ is the flip operator i.e. $$(Rf)(x) := f(-x)$$, then

 * $$\mathcal{F}^{-1}=\mathcal{F}R=R\mathcal{F}.$$

The theorem holds if both $$f$$ and its Fourier transform are absolutely integrable (in the Lebesgue sense) and $$f$$ is continuous at the point $$x$$. However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.

Statement
In this section we assume that $$f$$ is an integrable continuous function. Use the convention for the Fourier transform that

 * $$(\mathcal{F}f)(\xi):=\int_{\mathbb{R}^n} e^{-2\pi iy\cdot\xi} \, f(y)\,dy.$$

Furthermore, we assume that the Fourier transform is also integrable.

Inverse Fourier transform as an integral
The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function $$g$$ and all $$x \in \mathbb R^n$$ set

 * $$\mathcal{F}^{-1}g(x):=\int_{\mathbb{R}^n} e^{2\pi ix\cdot\xi} \, g(\xi)\,d\xi.$$

Then for all $$x \in \mathbb R^n$$ we have

 * $$\mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x).$$

Fourier integral theorem
The theorem can be restated as

 * $$f(x)=\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} e^{2\pi i(x-y)\cdot\xi} \, f(y)\,dy\,d\xi.$$

By taking the real part of each side of the above we obtain

 * $$f(x)=\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \cos (2\pi (x-y)\cdot\xi) \, f(y)\,dy\,d\xi.$$

Inverse transform in terms of flip operator
For any function $$g$$ define the flip operator $$R$$ by

 * $$Rg(x):=g(-x).$$

Then we may instead define

 * $$\mathcal{F}^{-1}f := R\mathcal{F}f = \mathcal{F}Rf.$$

It is immediate from the definition of the Fourier transform and the flip operator that both $$R\mathcal{F}f$$ and $$\mathcal{F}Rf$$ match the integral definition of $$\mathcal{F}^{-1}f$$, and in particular are equal to each other and satisfy $$\mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x)$$.

Since $$Rf=R\mathcal{F}^{-1}\mathcal{F}f =RR \mathcal{FF}f$$ we have $$R=\mathcal{F}^2$$ and

 * $$\mathcal{F}^{-1}=\mathcal{F}^3.$$

Two-sided inverse
The form of the Fourier inversion theorem stated above, as is common, is that

 * $$\mathcal{F}^{-1}(\mathcal{F}f)(x) = f(x).$$

In other words, $$\mathcal{F}^{-1}$$ is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.

 * $$\mathcal{F}(\mathcal{F}^{-1}f)(\xi) = f(\xi).$$

Since $$\mathcal{F}^{-1}$$ is so similar to $$\mathcal{F}$$, this follows very easily from the Fourier inversion theorem (changing variables $$\zeta := -\zeta$$):

 * $$\begin{align}

f & =\mathcal{F}^{-1}(\mathcal{F}f)(x)\\[6pt] & =\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{2\pi ix\cdot\xi}\,e^{-2\pi iy\cdot\xi}\, f(y)\, dy\, d\xi\\[6pt] & =\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{-2\pi ix\cdot\zeta}\,e^{2\pi iy\cdot\zeta}\, f(y)\, dy\, d\zeta\\[6pt] & =\mathcal{F}(\mathcal{F}^{-1}f)(x). \end{align}$$

Alternatively, this can be seen from the relation between $$\mathcal{F}^{-1}f$$ and the flip operator and the associativity of function composition, since

 * $$f = \mathcal{F}^{-1}(\mathcal{F}f) = \mathcal{F}R\mathcal{F}f = \mathcal{F} (\mathcal{F}^{-1}f).$$

Conditions on the function
When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything "behaves nicely". In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed. However, there is no "best" class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.

Schwartz functions
The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it is an elementary direct statement about the function (as opposed to imposing a condition on its Fourier transform), and the integral that defines the Fourier transform and its inverse are absolutely integrable. This version of the theorem is used in the proof of the Fourier inversion theorem for tempered distributions (see below).

Integrable functions with integrable Fourier transform
The Fourier inversion theorem holds for all continuous functions that are absolutely integrable (i.e. $$L^1(\mathbb R^n)$$) with absolutely integrable Fourier transform. This includes all Schwartz functions, so is a strictly stronger form of the theorem than the previous one mentioned. This condition is the one used above in the statement section.

A slight variant is to drop the condition that the function $$f $$ be continuous but still require that it and its Fourier transform be absolutely integrable. Then $$f = g$$ almost everywhere where $f$ is a continuous function, and $$\mathcal{F}^{-1}(\mathcal{F}f)(x)=g(x)$$ for every $$x \in \mathbb R^n$$.

Integrable functions in one dimension
If the function is absolutely integrable in one dimension (i.e. $$ f \in L^1(\mathbb R)$$) and is piecewise smooth then a version of the Fourier inversion theorem holds. In this case we define
 * Piecewise smooth; one dimension

 * $$\mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{-R}^R e^{2\pi ix\xi}\,g(\xi)\,d\xi.$$

Then for all $$ x \in \mathbb R$$

 * $$\mathcal{F}^{-1}(\mathcal{F}f)(x) = \frac{1}{2}(f(x_-) + f(x_+)),$$

i.e. $$\mathcal{F}^{-1}(\mathcal{F}f)(x)$$ equals the average of the left and right limits of $$ f$$ at $$ x$$. At points where $$ f$$ is continuous this simply equals $$ f(x)$$.

A higher-dimensional analogue of this form of the theorem also holds, but according to Folland (1992) is "rather delicate and not terribly useful".

 * Piecewise continuous; one dimension

If the function is absolutely integrable in one dimension (i.e. $$ f \in L^1(\mathbb R)$$) but merely piecewise continuous then a version of the Fourier inversion theorem still holds. In this case the integral in the inverse Fourier transform is defined with the aid of a smooth rather than a sharp cut off function; specifically we define

 * $$\mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{\mathbb{R}} \varphi(\xi/R)\,e^{2\pi ix\xi}\,g(\xi)\,d\xi,\qquad\varphi(\xi):=e^{-\xi^2}.$$

The conclusion of the theorem is then the same as for the piecewise smooth case discussed above.

 * Continuous; any number of dimensions

If $$ f$$ is continuous and absolutely integrable on $$\mathbb R^n$$ then the Fourier inversion theorem still holds so long as we again define the inverse transform with a smooth cut off function i.e.

 * $$\mathcal{F}^{-1}g(x):=\lim_{R\to\infty}\int_{\mathbb{R}^n} \varphi(\xi/R)\,e^{2\pi ix\cdot\xi}\,g(\xi)\,d\xi,\qquad\varphi(\xi):=e^{-\vert\xi\vert^2}.$$

The conclusion is now simply that for all $$x \in \mathbb R^n$$

 * $$\mathcal{F}^{-1}(\mathcal{F}f)(x)=f(x).$$

 * No regularity condition; any number of dimensions

If we drop all assumptions about the (piecewise) continuity of $$f$$ and assume merely that it is absolutely integrable, then a version of the theorem still holds. The inverse transform is again defined with the smooth cut off, but with the conclusion that

 * $$\mathcal{F}^{-1}(\mathcal{F}f)(x) = f(x)$$

for almost every $$x \in \mathbb R^n.$$

Square integrable functions
In this case the Fourier transform cannot be defined directly as an integral since it may not be absolutely convergent, so it is instead defined by a density argument (see the Fourier transform article). For example, putting
 * $$g_k(\xi):=\int_{\{y\in\mathbb{R}^n:\left\vert y\right\vert\leq k\}} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,\qquad k\in\mathbb{N},$$

we can set $$\textstyle\mathcal{F}f := \lim_{k\to\infty}g_k$$ where the limit is taken in the $$L^2$$-norm. The inverse transform may be defined by density in the same way or by defining it in terms of the Fourier transform and the flip operator. We then have

 * $$f(x)=\mathcal{F}(\mathcal{F}^{-1}f)(x)=\mathcal{F}^{-1}(\mathcal{F}f)(x)$$

in the mean squared norm. In one dimension (and one dimension only), it can also be shown that it converges for almost every $f$- this is Carleson's theorem, but is much harder to prove than convergence in the mean squared norm.

Tempered distributions
The Fourier transform may be defined on the space of tempered distributions $$\mathcal{S}'(\mathbb{R}^n)$$ by duality of the Fourier transform on the space of Schwartz functions. Specifically for $$f\in\mathcal{S}'(\mathbb{R}^n)$$ and for all test functions $$\varphi\in\mathcal S(\mathbb{R}^n)$$ we set
 * $$\langle \mathcal{F}f,\varphi\rangle := \langle f,\mathcal{F}\varphi\rangle,$$

where $$\mathcal{F}\varphi$$ is defined using the integral formula. If $$f \in L^1(\mathbb R^n) \cap L^2(\mathbb R^n)$$ then this agrees with the usual definition. We may define the inverse transform $$\mathcal{F}^{-1}\colon\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$$, either by duality from the inverse transform on Schwartz functions in the same way, or by defining it in terms of the flip operator (where the flip operator is defined by duality). We then have

 * $$\mathcal{F}\mathcal{F}^{-1} = \mathcal{F}^{-1}\mathcal{F} = \operatorname{Id}_{\mathcal{S}'(\mathbb{R}^n)}.$$

Relation to Fourier series
The Fourier inversion theorem is analogous to the convergence of Fourier series. In the Fourier transform case we have
 * $$f\colon\mathbb{R}^n\to\mathbb{C},\quad\hat f\colon\mathbb{R}^n\to\mathbb{C},$$
 * $$\hat f(\xi):=\int_{\mathbb{R}^n} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,$$
 * $$f(x)=\int_{\mathbb{R}^n} e^{2\pi ix\cdot\xi} \, \hat f(\xi)\,d\xi.$$

In the Fourier series case we instead have
 * $$f\colon[0,1]^n\to\mathbb{C},\quad\hat f\colon\mathbb{Z}^n\to\mathbb{C},$$
 * $$\hat f(k):=\int_{[0,1]^n} e^{-2\pi iy\cdot k} \, f(y)\,dy,$$
 * $$f(x)=\sum_{k\in\mathbb{Z}^n} e^{2\pi ix\cdot k} \, \hat f(k).$$

In particular, in one dimension $$k \in \mathbb Z$$ and the sum runs from $$- \infty$$ to $$\infty$$.

Applications

In applications of the Fourier transform the Fourier inversion theorem often plays a critical role. In many situations the basic strategy is to apply the Fourier transform, perform some operation or simplification, and then apply the inverse Fourier transform.

More abstractly, the Fourier inversion theorem is a statement about the Fourier transform as an operator (see Fourier transform on function spaces). For example, the Fourier inversion theorem on $$f \in L^2(\mathbb R^n)$$ shows that the Fourier transform is a unitary operator on $$L^2(\mathbb R^n)$$.

Properties of inverse transform
The inverse Fourier transform is extremely similar to the original Fourier transform: as discussed above, it differs only in the application of a flip operator. For this reason the properties of the Fourier transform hold for the inverse Fourier transform, such as the Convolution theorem and the Riemann–Lebesgue lemma.

Tables of Fourier transforms may easily be used for the inverse Fourier transform by composing the looked-up function with the flip operator. For example, looking up the Fourier transform of the rect function we see that $$f(x) = \operatorname{rect}(a x) \quad \Rightarrow \quad (\mathcal{F}f)(\xi)=\frac{1}{|a|} \operatorname{sinc}\left(\frac{\xi}{a}\right),$$ so the corresponding fact for the inverse transform is $$g(\xi)=\operatorname{rect}(a \xi) \quad \Rightarrow \quad (\mathcal{F}^{-1}g)(x)=\frac{1}{|a|} \operatorname{sinc}\left(-\frac{x}{a}\right) .$$

Proof
The proof uses a few facts, given $$f(y)$$ and $$\mathcal{F}f (\xi) = \int_{\mathbb{R}^n} e^{-2\pi i y\cdot\xi} f(y)\,dy$$.

Since, by assumption, $$\mathcal{F}f\in L^1(\mathbb{R}^n)$$, then it follows by the dominated convergence theorem that
 * 1) If $$x \in \mathbb R^n$$ and $$g(\xi) = e^{2 \pi \mathrm{i}x \cdot \xi} \psi(\xi)$$, then $$(\mathcal{F}g)(y) = (\mathcal{F}\psi)(y - x)$$.
 * 2) If $$\varepsilon \in \mathbb R$$ and $$\psi(\xi) = \varphi(\varepsilon\xi)$$, then $$(\mathcal{F}\psi)(y) = (\mathcal{F}\varphi)(y/\varepsilon)/|\varepsilon|^n$$.
 * 3) For $$f, g \in L^1(\mathbb R^n)$$, Fubini's theorem implies that $$\textstyle\int g(\xi) \cdot (\mathcal{F}f)(\xi)\,d\xi = \int(\mathcal{F}g)(y) \cdot f(y)\,dy$$.
 * 4) Define $$\varphi(\xi) = e^{-\pi \vert \xi \vert^2}$$; then $$(\mathcal{F}\varphi)(y) = \varphi(y)$$.
 * 5) Define $$\varphi_\varepsilon(y) = \varphi(y/\varepsilon)/\varepsilon^n$$. Then with $$\ast$$ denoting convolution, $$\varphi_\varepsilon$$ is an approximation to the identity: for any continuous $$f \in L^1(\mathbb R^n)$$ and point $$x \in \mathbb R^n$$, $$\lim_{\varepsilon \to 0} (\varphi_\varepsilon \ast f)(x) = f(x)$$ (where the convergence is pointwise).

 * $$\int_{\mathbb{R}^n} e^{2\pi i x\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi = \lim_{\varepsilon \to 0}\int_{\mathbb{R}^n} e^{-\pi\varepsilon^2|\xi|^2 + 2\pi i x\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi.$$

Define $$g_x(\xi) = e^{-\pi\varepsilon^2\vert \xi \vert^2 + 2 \pi \mathrm{i} x \cdot \xi}$$. Applying facts 1, 2 and 4, repeatedly for multiple integrals if necessary, we obtain
 * $$(\mathcal{F}g_x)(y) = \frac{1}{\varepsilon^n}e^{-\frac{\pi}{\varepsilon^2}|x - y|^2}=\varphi_\varepsilon(x-y).$$

Using fact 3 on $$f$$ and $$g_x$$, for each $$x\in\mathbb R^n$$, we have

 * $$\int_{\mathbb{R}^n} e^{-\pi\varepsilon^2|\xi|^2 + 2\pi i x\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi = \int_{\mathbb{R}^n} \frac{1}{\varepsilon^n}e^{-\frac{\pi}{\varepsilon^2}|x - y|^2} f(y)\,dy = (\varphi_\varepsilon * f)(x),$$

the convolution of $$f$$ with an approximate identity. But since $$f \in L^1(\mathbb R^n)$$, fact 5 says that

 * $$\lim_{\varepsilon\to 0}(\varphi_{\varepsilon} * f) (x) = f(x).$$

Putting together the above we have shown that

 * $$\int_{\mathbb{R}^n} e^{2\pi i x\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi = f(x). \qquad\square$$

title: Fourier inversion theorem


text:
List of television stations in Durango

The following is a list of all IFT-licensed over-the-air television stations broadcasting in the Mexican state of Durango. There are 21 television stations in Durango.

List of television stations

title: List of television stations in Durango


text:
Trivial topology

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.

Details
The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space X with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space.

Other properties of an indiscrete space X—many of which are quite unusual—include:

 * The only closed sets are the empty set and X.
 * The only possible basis of X is {X}.
 * If X has more than one point, then since it is not T0, it does not satisfy any of the higher T axioms either. In particular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable.
 * X is, however, regular, completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.
 * X is compact and therefore paracompact, Lindelöf, and locally compact.
 * Every function whose domain is a topological space and codomain X is continuous.
 * X is path-connected and so connected.
 * X is second-countable, and therefore is first-countable, separable and Lindelöf.
 * All subspaces of X have the trivial topology.
 * All quotient spaces of X have the trivial topology
 * Arbitrary products of trivial topological spaces, with either the product topology or box topology, have the trivial topology.
 * All sequences in X converge to every point of X. In particular, every sequence has a convergent subsequence (the whole sequence or any other subsequence), thus X is sequentially compact.
 * The interior of every set except X is empty.
 * The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, a property that characterizes trivial topological spaces.
 * As a result of this, the closure of every open subset U of X is either ∅ (if U = ∅) or X (otherwise). In particular, the closure of every open subset of X is again an open set, and therefore X is extremally disconnected.
 * If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.
 * X is a Baire space.
 * Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.

In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.

The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.

Let Top be the category of topological spaces with continuous maps and Set be the category of sets with functions. If G : Top → Set is the functor that assigns to each topological space its underlying set (the so-called forgetful functor), and H : Set → Top is the functor that puts the trivial topology on a given set, then H (the so-called cofree functor) is right adjoint to G. (The so-called free functor F : Set → Top that puts the discrete topology on a given set is left adjoint to G.)

title: Trivial topology


text:
Engativá

Engativá is the 10th locality of Bogotá. It is located in the west of the city. This district is mostly inhabited by lower middle and working class residents.

Etymology
Engativá is either derived from Ingativá; cacique Inga; "Land of the Sun", or from the Chibcha words Engue-tivá; engue is "delicious" and tivá is "captain"; "captain of the delicious [people]".

Geography
Engativá is limited to the north by the Salitre River with Suba, to the east by Avenida Carrera 68 and Bosa, to the south by Avenida El Dorado and Fontibón, and to the west by the Bogotá River.

History
Engativá was a village in the confederation of the Muisca. Modern Engativá was founded in 1537. It has become a rural territory, people used to work as farmers of Bogotá in 1571. The church of the town was built in honor of the pope Clemente XII in 1638 and from 1737 it was named the Nuestra Señora de los Dolores's Sanctuary. The church was destroyed by earthquakes, but was rebuilt in 1960.

title: Engativá


text:
Primrose Hill

Primrose Hill is a Grade II listed public park located north of Regent's Park in London, England, first opened to the public in 1842. It was named after the 64 m natural hill in the centre of the park, the second highest natural point in the London Borough of Camden. The hill summit has a clear view of central London, as well as Hampstead and Belsize Park to the north and is adorned by an engraved quotation from William Blake. Based on the popularity of the park, the surrounding district and electoral ward were named Primrose Hill. The Primrose Hill district is in the London Borough of Camden, England.

Amenities of the park include an outdoor gym known as the Hill Trim Trail, a children's playground, and toilets, all located on the south side near Primrose Hill bridge which connects to London Zoo and Regent's Park.

History
Like the Regent's Park, Primrose Hill was once part of a great chase appropriated by Henry VIII. Later, in 1841, it became Crown property and in 1842 an Act of Parliament secured the land as public open space. The name "Primrose Hill" has been in use since the 15th century, giving the lie to later claims that it was named after Archibald Primrose, 5th Earl of Rosebery, whose premiership witnessed the rapid expansion of London underground rail network London.

In October 1678, Primrose Hill was the scene of the mysterious murder of Sir Edmund Berry Godfrey. In 1679 three Catholic labourers, Robert Green, Henry Berry and Lawrence Hill were found guilty of the murder (though subsequently posthumously exonerated) and hanged at the top of the hill. For a few years after the hanging, Primrose Hill was known as Greenberry Hill.

In 1792, Primrose Hill was the site of the first meeting of the "Gorsedd of the Bards of the Isles of Britain", organised by the Welsh radical poet and antiquarian Iolo Morganwg. The Gorsedd (a community of bards) took place at a ceremony on 21 June (the summer solstice) and was designed to replicate ancient Druidic rites. A plaque commemorating this event was unveiled in 2009 by Huw Edwards on behalf of the London Welsh Association.

The canal through the area was completed in 1816, and the railway, running under the hill, was completed in 1838; this was the first rail tunnel in London. By that time, the area was considered to be a "prime development opportunity" according to one source. In 1840, Charles FitzRoy, 3rd Baron Southampton, sold the land that he owned and new villas were built over the subsequent years. Other land was still owned by Eton College but transferred to the government in 1841. The Crown drained and leveled the land after 1851 and began adding park features, to turn it into "park for the people". The park was historically split between the ancient parishes (which later became metropolitan boroughs) of Marylebone (now part of the City of Westminster), St Pancras and Hampstead (in the modern London Borough of Camden), with the hill itself a part of Hampstead.

The area east of the park was developed and became known as Primrose Hill, after the park. The Primrose Hill district is surrounded by St John's Wood to the west, Swiss Cottage to the northwest, Belsize Park to the north, Chalk Farm to the northeast, Camden Town to the east and Regent's Park itself lies adjacent to the south of the hill itself. The nearest stations to Primrose Hill are Chalk Farm tube station to the northeast and Swiss Cottage tube station to the northwest. The defunct Primrose Hill railway station, now housing a business, sits on the railway lines that separate the Primrose Hill area from Camden Town. Primrose Hill Tunnel, the first railway tunnel in London, has had its eastern portals Grade II*- and its western portals Grade II* listed since 1974. Beginning in the late 1960s several of the roads were closed to motor traffic in response to an unacceptable level of collisions and consequent loss of life. The changes were carefully designed to render the area largely free of through motor traffic.

Crowds flocked to Primrose Hill following the loosening of lockdown restrictions due to the COVID-19 pandemic in London. Large amounts of litter were left around the park and social distancing was not observed. This resulted in the closure of Primrose Hill temporarily over a weekend.

, the Primrose Hill Open Space is managed by The Royal Parks.

Notable buildings and residents
The Primrose Hill district is an archetypal example of a successful London urban village, due to the location and the quality of its socio-historical development, and is home to many prominent residents. There are seven English Heritage blue plaques in Primrose Hill commemorating the historic personalities that have lived there. The plaques mark the residences of poet Sir Hugh Clough, historian and broadcaster A. J. P. Taylor and painter William Roberts at 11, 13, and 14 St Mark's Crescent respectively; revolutionary socialist and philosopher Friedrich Engels at 122 (and later 41) Regent's Park Road; photographer Roger Fenton at 2 Albert Terrace; poet and novelist Sylvia Plath at 3 Chalcot Square; and poet William Butler Yeats at 23 Fitzroy Road.

Stanley Johnson and Lukas Heller each lived at different times at the 'Rocking Horse House' on Regent's Park Road. Among those living in the area are broadcasters Joan Bakewell and Nicholas Crane, as well as actors Daisy Ridley, Derek Jacobi, and Brian Cox.

Elliott Square is a grouping of modernist 1960s houses by Douglas Streeter, built as part of the Chalcot Estate on land owned by Eton College.

In popular culture
Primrose Hill is commonly referred to as an 'iconic', and 'famous' location, and features significantly in popular British culture.

 * The band Madness have a song titled "Primrose Hill" on their album The Rise & Fall.
 * The songs "The Promise" and "Hoxton Heroes" by Girls Aloud mention Primrose Hill in the lyrics: "Here I am, walking Primrose, wondering if I'm gonna see you again" and "You've got the Primrose Set on your cell phone".
 * The song "For Tomorrow" by Blur references Primrose Hill, and the extended version's title includes the line "(Visit to Primrose Hill Extended)".
 * The song "Upfield" by Billy Bragg on his album William Bloke mentions "seeing angels up on Primrose Hill".
 * In the song "Girl VII" by Saint Etienne, the lyrics include a list of locations, with Primrose Hill included.
 * The film Bridget Jones: The Edge of Reason was partially filmed on Primrose Hill.
 * Primrose Hill is a key location in the live action/animated comedy films Paddington and Paddington 2.
 * In the film One Hundred and One Dalmatians, Pongo and Perdita meet a neighbouring Great Dane at Primrose Hill, learning from him that their puppies are located in Suffolk, before setting out to find them.
 * In Jackass: The Movie, Primrose Hill is used for the "Rocket Skates" stunt, wherein cast members strap bottle rockets to rollerblades and attempt to skate through the park.
 * James McCartney and Sean Ono Lennon, sons of Paul McCartney and John Lennon, respectively, released their first joint collaboration, a song titled "Primrose Hill", in April 2024.

During the 1990s Primrose Hill was a popular place to live with some who worked in the film, television, music and fashion industries and who were referred to as the Primrose Hill set in the media.

title: Primrose Hill