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25 votes
3 answers
13k views

Fourier transform of the unit sphere

The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
Francois Ziegler's user avatar
23 votes
6 answers
4k views

Anti-delta function?

Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
Anixx's user avatar
  • 10.1k
11 votes
1 answer
691 views

Reference request: Fourier transform on the multiplicative group of real numbers

Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
Jochen Glueck's user avatar
8 votes
1 answer
1k views

Who introduced the discrete Fourier transform?

I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...
ACR's user avatar
  • 879
8 votes
1 answer
2k views

General Fourier inversion formula (Gil-Pelaez)

Gil-Pelaez (1951) proves the Fourier inversion formula \begin{align*} F(x) &= \frac{1}{2} + \frac{1}{2\pi} \int_0^\infty \frac{e^{itx}\phi(-t)-e^{-itx}\phi(t)}{it}dt \\ &= \frac{1}{2} - \frac{...
Alex's user avatar
  • 255
4 votes
0 answers
595 views

On smoothness of a function and decay of its Fourier transform

I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here. I am interested in a quantitative version of the principle that smoothness of ...
Durac's user avatar
  • 41
4 votes
0 answers
226 views

Any references on infinite-dimensional Fourier-Plancherel theory?

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
Alexander Shamov's user avatar
3 votes
2 answers
487 views

Where to find a table of fair Fourier transforms? [closed]

I am greatly dissatisfied with those tables of Fourier transforms, available online. I simply have no guess what method they use to derive their tables, but it seems completely off to me. For ...
Anixx's user avatar
  • 10.1k
3 votes
1 answer
305 views

What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?

I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks. ...
Taras Banakh's user avatar
  • 41.9k
3 votes
1 answer
157 views

How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?

I have a question about the completeness of complex exponentials in function spaces. For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
Zhang Yuhan's user avatar
2 votes
0 answers
80 views

Prove uniqueness of Radon transform without using Fourier transform

The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity): If a continuous function with compact support has zero ...
Zhang Yuhan's user avatar
2 votes
0 answers
56 views

Inequality for a weighted bilinear form in Fourier variables

Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$. Consider the ...
Guy Fsone's user avatar
  • 1,101
2 votes
0 answers
120 views

request for any expository works in pointwise convergence of double Fourier series and especially a paper by Hardy

Quart. J. Math. Volume 37, Issue 1, Pages 53-79 On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Hardy, G.H. I am not ...
Rajesh D's user avatar
  • 698
1 vote
1 answer
117 views

Positivity of exponentially bounded characteristic functions

I've noticed that for the classical examples of exponentially bounded, symmetrical distributions (Gaussian, Laplace, Double Exponential, Uniform), their characteristic functions are positive for all ...
Rono's user avatar
  • 73
1 vote
1 answer
1k views

Fourier approximation error in L^2 for piecewise continuous functions

Let $u:[0,2\pi)\to \mathbb{R}$ be the step function $$u(x) = \begin{cases} 1 & \text{if } x \in [0,\pi), \\ 0 & \text{if } x \in [\pi,2\pi) \end{cases}$$ By a direct computation, one ...
Paglia's user avatar
  • 837
1 vote
1 answer
229 views

Result of Beurling concerning absolute convergence of Fourier series of |f|

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put, $$A(\mathbb T):= \{f\in L^{1}(\...
Inquisitive's user avatar
  • 1,051
1 vote
0 answers
86 views

Fourier transform relation for spherical convolution

Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$. The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \...
Goulifet's user avatar
  • 2,306
1 vote
0 answers
146 views

Proof for $\Phi(t)$ is strictly decreasing for $t>0$ in Riemann's zeta function

I am looking for reference for proof that $\Phi(t)$ is strictly decreasing for $t>0$ and the first derivative of $\Phi(t)$ is negative for $t>0$ (see Page 5 in Conrey's article below) Conrey ...
John's user avatar
  • 11
1 vote
0 answers
74 views

Calculation of a multi-dimensional Fourier transform

I am interested in the following multi-dimensional Fourier transform: $$ \int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
Y.Okuyama's user avatar
  • 373
1 vote
0 answers
151 views

Fourier transforms exhibiting symmetries about their critical points

Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
John Clever's user avatar
1 vote
0 answers
50 views

Comparison of (square) of a function and its Fourier transform in an integral

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
Bertrand's user avatar
  • 1,199
1 vote
0 answers
157 views

Technical question about a Fourier transform

I would like to know if there is an explicit expression for the Fourier transform of the following function: $$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$ or to know where I can find some techniques to ...
Felice Iandoli's user avatar
1 vote
0 answers
204 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
user48365's user avatar
  • 113
0 votes
2 answers
180 views

Inversion formula for discrete sine and cosine transforms

$\newcommand{\wh}[1]{{\widehat{#1}}} \newcommand{\R}{{\mathbb{R}}} $I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
Bettina's user avatar
  • 113
0 votes
1 answer
71 views

Asymptotic expansion inverse discrete Fourier transform

Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series ...
Scottish Questions's user avatar
0 votes
1 answer
273 views

Fourier transform of measures on $\mathbb{T}$

I'm currently working with Fourier transforms of measures on the $\mathbb{T}^n$ (more specifically in dimension two), i.e. $$ \hat{\mu}(k) = \int_{\mathbb{T}^n} e^{i k \cdot x} d\mu(x) $$ or something ...
spaceman's user avatar
  • 595