The fourier-analysis tag has no wiki summary.

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458 views

### In what ways is the standard Fourier basis optimal?

Is there any convincing sense in which the standard trigonometric basis for
the space $V$ of square-integrable real-valued functions on $[-\pi,\pi]$ is
optimal among all the orthonormal bases?
(If ...

**2**

votes

**0**answers

58 views

### Is there a Fourier transform for principal r-discrete groupoids?

I have been looking for an analog of the Fourier transform for groupoids coming from tilings (which are generally principal and r-discrete), however all the generalizations I have found assume that ...

**0**

votes

**1**answer

102 views

### local moments of measures whose Fourier transform vanish in an interval

Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form
\begin{equation*}
\int_{-\delta}^{+\delta}|h|(dt)\le ...

**3**

votes

**2**answers

199 views

### On lower bounds of exponential frames in l1 norm

Let $\{t_k\}_{k=-\infty}^\infty$ be a sequence of real numbers.
I'm interested in finding the largest number A such that
\begin{equation*}
\int_{-\Omega}^\Omega|\sum_{k=-\infty}^{+\infty}c_ke^{2\pi i ...

**1**

vote

**0**answers

64 views

### Symmetry of Sundry Planar Convex Sets of Constant Width & Minimal Area

In a much broader paper in “Optimization Methods & Software 27,6 (2012) pp1073-1099” Bayen & Henrion consider planar compact, convex sets with support functions which are finite Fourier ...

**11**

votes

**1**answer

383 views

### Is there something wrong with Hörmander's theorem on stationary phase method

It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - ...

**2**

votes

**1**answer

157 views

### Interpolating delta like functions by trigonometric polynomials of bounded modulus and fast decay

Consider a grid of points $T=\{t_0,t_1,\ldots,t_m\}$ with $0\le t_i\le 1$. I would like to find a function $f(t):[0,1]\rightarrow \mathbb{C}$ of the form
\begin{equation*}
f(t)=\sum_{k=-n}^n c_k ...

**2**

votes

**1**answer

243 views

### bounding the absolute value of a trigonometric polynomial

Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$
\begin{equation*}
f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})}
...

**2**

votes

**0**answers

205 views

### computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...

**14**

votes

**2**answers

737 views

### A question in Fourier analysis

I recently came across this problem:
Let $T=\mathbb R /2 \pi \mathbb Z$ the circle, with its proabability Haar measure $\mu$. Any integrable function $f : T \rightarrow \mathbb C$
has Fourier ...

**5**

votes

**2**answers

481 views

### Fourier series representing a continuous function?

This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either.
Fourier series define an isometry $L^2(\mathbb{Z}) ...

**10**

votes

**2**answers

387 views

### Uncertainty principle on finite groups

For a finite group $G$ with normal subgroup $H$, the induced representation $\text{Ind}_H^G(1)$ decomposes as a sum of irreducibles with the multiplicities equal to the dimensions, because it is is ...

**2**

votes

**0**answers

166 views

### Fourier : from analyticity to exponential decay ; what prevents optimal decay ?

Hello everyone !
In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) :
if ...

**45**

votes

**5**answers

4k views

### Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line:
$$ \zeta(\frac{1}{2} + i k_j) = 0 $$
Let $f$ be the Fourier transform of the sum ...

**0**

votes

**1**answer

135 views

### upper bound on product of distances from points on a circle

Let C be a circle of radius 1 in the complex plane with n points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other n points. The goal ...

**2**

votes

**0**answers

205 views

### An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like,
\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...

**0**

votes

**1**answer

259 views

### Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1.
It is claimed that Lemma 2 is ...

**1**

vote

**0**answers

277 views

### A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...

**2**

votes

**2**answers

303 views

### A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$.
Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ?
I agree this question is too vague, ...

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**0**answers

164 views

### On a differential inequality

The question has probabilistic origins, but it would take too long to elaborate. $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\eS}{\mathscr{S}}$
Fix a nonnegative ...

**1**

vote

**1**answer

311 views

### Pointwise convergence of double Fourier series

I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function.
Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of ...

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votes

**2**answers

522 views

### Schönhage–Strassen algorithm

After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...

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**2**answers

726 views

### Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...

**1**

vote

**1**answer

252 views

### Is this set a Riesz Basis of $L^2(0,\pi)$

Let $A=\{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\} _{n=1}^\infty$, where $a$ is a positive real number. Is $A$ a Riesz Basis of $L^2(0,1)$?

**3**

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**1**answer

151 views

### Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...

**1**

vote

**1**answer

112 views

### Almost Everywhere Convergence of Walsh Series of $L^2$ functions

I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is any notes which presents his argument more ...

**13**

votes

**1**answer

746 views

### On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO.
Let $S$ be a finite set of integers. For $P$ a subset of ...

**3**

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**1**answer

246 views

### About the boundedness of a multiplication operator.

Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator:
\begin{equation}
Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}.
...

**6**

votes

**1**answer

284 views

### For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...

**3**

votes

**1**answer

1k views

### The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on ...

**0**

votes

**1**answer

406 views

### nonnegative Fourier Transform

Let $\widehat{f}(\xi)$ be Fourier transform of $f$ given by
\begin{align}
\widehat{f}(\xi)=\int_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)dx.
\end{align}
Suppose that $\widehat{f}(\xi)$ is nonnegative and ...

**4**

votes

**1**answer

390 views

### Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain.
Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...

**5**

votes

**2**answers

346 views

### Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...

**8**

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**0**answers

283 views

### Uncertainty principle in Entropy terms

Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
...

**6**

votes

**4**answers

891 views

### fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...

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**2**answers

340 views

### Sufficient condition for $L^p$ multiplier on the torus

Hello,
I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from ...

**6**

votes

**3**answers

1k views

### Fourier Coefficients and Hölder Continuity

Suppose we are given the Fourier coefficients of an $L^2$ function on the circle. Are there necessary and sufficient conditions on the coefficients that allow us to determine that $f$ is Hölder ...

**4**

votes

**1**answer

259 views

### Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$

Assume that P is a real valued strong elliptic polynomial, then what do we know about the following
$$
K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0
$$
The ...

**0**

votes

**2**answers

873 views

### Derivative indicator function

I am wondering what is the derivative of the following function with respect to $x(t)$ in sense of distributions.
$$
I\left(\int_0^t x(\tau)d\tau \leq c\right)
$$
where $I$ is the indicator function ...

**6**

votes

**1**answer

314 views

### faces in the discrete cube

This arose from a question Gil Kalai asked about a problem I posed involving the Fourier transform on the discrete cube. Maybe it is more tractable. I'm afraid I'm not sure how to do this kind of ...

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**0**answers

1k views

### Fourier transform on the discrete cube

Notation: identify an element of $\{-1,1\}^n$ with the set $S \subseteq \{1, \ldots, n\}$ on which it takes the value $-1$.
The following is an asymptotic question. "Close to one" means "more than ...

**1**

vote

**1**answer

99 views

### Asymptotics of a one-parameter family of Schwartz functions

For $\tau > 0$ define $\theta_{\tau}(x) = e^{\tau(x-x^{2})}$. I am curious about the asymptotics of $\widehat{\theta}_{\tau}(\tau)$, that is
$\int_{\mathbb{R}} e^{\tau(x - x^{2})}e^{-2\pi i ...

**0**

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**0**answers

298 views

### Help with an irregular integral

I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...

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**2**answers

161 views

### Taylor expansion convergence relation to power-spectrum

Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty ...

**3**

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**1**answer

170 views

### Approximation power of wavelets

The Wikipedia article on Wavelet Transform states that:
Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, ...

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vote

**3**answers

592 views

### Fourier Transform, for entire function

On THIS site, Alexandre used Fourier transform to solve the problem.
If we use Fourier transform, how to define it to ensure any entire function has a FT?
Classical FT is defined by
$$ ...

**3**

votes

**3**answers

1k views

### Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of ...

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**1**answer

611 views

### Questions about the Fourier expansion of $e^{iz\cot(x)}$

Referring to a question I posted on MS, I post it here, as I didn't get an answer:
By analogy with the Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form :
...

**2**

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**1**answer

232 views

### Fourier transform of a particular function

In order to estimate the fundamental solution of some particular types of differential operators,I need estimates on some kind of oscillatory integrals.For simplicity, consider the Fourier transform ...

**0**

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**2**answers

168 views

### Convolutional Equation

Hi,
Given C>0. Let $f,g,h$ be $L^2$ functions such that $f,g,h$ have a compact and finite-measure support, and $f*f(x)=g*g(x)=h*h(x)=f*g(x)=g*h(x)=f*h(x)=0$ (where $*$ is the convolution) for all ...