The fourier-analysis tag has no wiki summary.

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### 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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### 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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### 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 ...

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250 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 $L^2(0,1)$?

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### 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 ...

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### 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 ...

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707 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 ...

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239 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}.
...

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### 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 ...

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### 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 ...

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### 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 ...

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376 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. ...

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### 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 ...

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### 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) =
...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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816 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 ...

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### 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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### 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 ...

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### 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 ...

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293 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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### 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 ...

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### 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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### 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
$$ ...

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### 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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### 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 :
...

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### 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 ...

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### 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 ...

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### Weakest assumption for pointwise convergence of Fourier series

This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS!
Suppose ...

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### Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...

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### Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that
(a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$,
(b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...

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### How well can you approximate a function by a band-restricted function?

Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$.
Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) ...

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### Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?

Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...

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### Berry Esseen inequality for multidimensional distributions

The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary
$$
\sup_{t \in ...

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### Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view.
For ...

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288 views

### Estimates for Fourier transform

Let $f(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Let $\mu$ be a compactly supported Borel measure (not necessarily positive) on $\mathbb{R}^n$. Define
$$
\tilde{\mu}(\xi) = \int ...

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### Minimizing |FT(X)|_{\infty} by permutation of X_i - question on Fourier transform related to engineering problem (peak factor of OFDM system)

Consider vector X =( X_1 ... X_N), consider the discrete Fourier transform $Y=F(X)$.
I am interested to minimize $|Y|_{\infty}$, by permutation of numbers X_i, how to do it ?
Here $|Y|_{\infty}$ is ...

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### Norm concentration of trigonometric polynomials - Uncertainty principle

Hi all,
I am interested in the following question (which is quite similar to one I posed a long while ago): Let $P_{N}(t)=\underset{k=-N}{\overset{N}{\sum}}c_{k}e^{ikt}$ be a unit norm trigonometric ...

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### What properties should a transform have to deserve the descriptor Fourier?

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions:
1) What properties do you feel are ...

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### Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency?
Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...

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### On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is :
...

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### A fourier series related to spin Chern numbers almost commuting matrices

Let
$$
f(x)=\sin(x)\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}.
$$
In my study of almost commuting unitary matrices, $U$ and $V$, I
have need for a bound like
$$
\left\Vert ...

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### Bound on trigonometric sum

I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have
$\sum_{k=1}^n -1+\cos(k\theta) \leq C(n)$.
If we rewrite this ...

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### How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...

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### Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$.
It ...

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566 views

### Estimating parameters of a mixture of normal distributions.

I want to estimate the parameters $\mu_i$ and $\sigma^2_i $ of a countable mixture of Gaussians with assumed equal weights, variance and identically spaced means. I intially thought that the Fourier ...

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### Kahn-Kalai-Linial for intersecting upsets

Is there any known improvement on the Kahn-Kalai-Linial inequality (on the influences of boolean functions) in the special case in which $f$ is the indicator function of an intersecting monotonic set ...

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### History of the Fourier transform

Does anyone know a good book or article on the History of the Fourier transform? It's first appearance (of the transform) and use in particular? Or at least some source with some historical ...