Questions tagged [fourier-analysis]

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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Multidimensional improper Riemann integrals with oscillatory kernels: Existence

I have asked this question three weeks ago here https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930 but received no relevant answers. Let $n\geq 2$ ...
user130023's user avatar
11 votes
0 answers
570 views

Fourier transforms and nontrivial vector bundles

We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
user131711's user avatar
3 votes
1 answer
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What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
Merry's user avatar
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7 votes
1 answer
326 views

Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?

Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$? It is known that for $n = 2$, the function $\displaystyle ...
sciona's user avatar
  • 211
6 votes
1 answer
387 views

Positive-definiteness of radial sinc function in three dimensions

In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
Capublanca's user avatar
11 votes
1 answer
656 views

Poisson summation formula for number fields

Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...
Automorphic's user avatar
14 votes
2 answers
3k views

Differentiability of Fourier series

Consider the function defined by the Fourier series $$ f(x;\alpha) = \sum_{n=1}^\infty \frac{1}{n^\alpha} \exp(i n^2 x ) , $$ where $\alpha >1 $. For what values of $\alpha $ is $f$ ...
pie's user avatar
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8 votes
0 answers
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A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters

Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$? (...
Lviv Scottish Book's user avatar
9 votes
1 answer
388 views

The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that $\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that $(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$. It is known ([1,2,3]...
user130023's user avatar
1 vote
1 answer
279 views

Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation

This is a restated version of my original very broad question. Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
David Hongxiang QIU's user avatar
3 votes
1 answer
200 views

Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?

I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection. As we ...
ABB's user avatar
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2 votes
1 answer
166 views

$L^2(X) \cong L^2(X',\xi)$

Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example: Let X=A\G, X'=N\G, where G=PGL(2), A={ $\left[\begin{array}{...
Cooler Panda's user avatar
1 vote
1 answer
130 views

Maximum Magnitude Deviation between DFT and DTFT

This is a cross-post from signal processing forum as it was not conclusive. Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then $$ X(\omega) = \sum_{n = 0}^...
Jiro's user avatar
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16 votes
3 answers
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Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle?

Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by $$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$ Clearly, $(...
Benoît Kloeckner's user avatar
0 votes
1 answer
224 views

A question about the convolution theorem

I have the following "argument" about Fourier series, which I know is wrong because it yields a ridiculous conclusion. However, I don't know where the mistake is, and need to know which step is the ...
Gabe K's user avatar
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3 votes
1 answer
211 views

Does zero Fourier dimension imply there is no Rajchman measure?

Let $K\subset R^d$ be a compact set. It is well known that its Fourier dimension is defined by $$\dim_F K=\sup\{s\ge 0: \exists \mu \in M_1(K) s.t. \hat{\mu}(x)=O(|x|^{-s/2})\}(|x|\to\infty),$$ where $...
ljjpfx's user avatar
  • 175
12 votes
1 answer
549 views

Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
H A Helfgott's user avatar
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12 votes
3 answers
954 views

Connections between martingales and Fourier analysis

I have had this strange feeling recently that somehow, the theory of martingales we study in probability, and the theory of Fourier analysis are very alike. But I am not able to formalize my thoughts. ...
Landon Carter's user avatar
4 votes
0 answers
564 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
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3 votes
1 answer
566 views

Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples? This seems to be a well-known result, but I can ...
Tomas's user avatar
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Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?

Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$. Is it true ...
GabrieleBenedetti's user avatar
-2 votes
1 answer
144 views

Relationship between "Radial" Fourier transform and Fourier transform, especially at infinity

Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support. What is the relationship between $$ \widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
MichaelGaudreau's user avatar
6 votes
1 answer
121 views

Multi-parameter stationary phase asymptotic expansion

I am looking for an asymptotic expansion of the oscillatory integral of the form $$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$ as $\lambda_i\to \infty$ ...
Subhajit Jana's user avatar
6 votes
0 answers
214 views

Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler

Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
Vesselin Dimitrov's user avatar
3 votes
1 answer
297 views

Detecting if a series represents a rational function

A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to ...
Mehdi Yazdi's user avatar
1 vote
1 answer
137 views

Uniqueness of solution to system of integral equations

Given the following system of integral equations for an integrable function $f(x)$: For all integers $k \ge 1$ holds $\int_{0}^{2\pi} [f(x)]^k e^{(ikx)} dx = 0$. If $f(x)$ is real-valued and non-...
Schmelli's user avatar
1 vote
3 answers
305 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
asv's user avatar
  • 21.1k
7 votes
0 answers
604 views

Convolution theorem on a non-abelian Lie group

Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
onamoonlessnight's user avatar
10 votes
1 answer
559 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
user avatar
1 vote
1 answer
263 views

Calculus of variation with discontinuous solutions?

I'm thinking of the following question: Consider a function $f: U\rightarrow\mathbb{R}$ where $U=[0,L_1)\cup(L_1,L]$, and an energy functional $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
hsnms's user avatar
  • 9
7 votes
2 answers
209 views

Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions

I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
Saal Hardali's user avatar
  • 7,549
0 votes
1 answer
265 views

The minimum of the maximum of a sequence of sinc functions

I apologise if this is trivial or well known to be impossible: Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$ such that for the function defined as $$ f_{a_1,\...
kodlu's user avatar
  • 10.1k
2 votes
0 answers
97 views

Star-convex curve and Fourier series

Let x(t) be a periodic function on [0, 2$\pi$]. I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
DrT's user avatar
  • 21
1 vote
0 answers
301 views

Conditions for Poisson summation (for discontinuous functions)

Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
Tian An's user avatar
  • 3,689
13 votes
1 answer
618 views

A question on the sine function

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ Here I ask the ...
Zhi-Wei Sun's user avatar
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4 votes
1 answer
278 views

$L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
Mostafa's user avatar
  • 4,454
10 votes
0 answers
172 views

A combinatorial proof of the Harrow--Kolla--Schulman theorem

Let $Q^n := \{0,1\}^n$ be the Hamming cube with the Hamming metric. (Recall that the Hamming is defined by the distance $d(x,y) := \# \{ i : x_i \neq y_i \}$. For integers $0 \leq k \leq n$, define a ...
K Hughes's user avatar
  • 579
12 votes
1 answer
435 views

Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?

One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...
K Hughes's user avatar
  • 579
2 votes
0 answers
144 views

Theory of distributions on various domains

The prototypical example of a distribution is the Dirac delta function, defined as a linear functional taking a well behaved test function $\phi:\mathbb{R} \to \mathbb{R}$ and returning its value at ...
Fizikus's user avatar
  • 29
8 votes
1 answer
2k views

graph signal processing

I have read this article https://arxiv.org/abs/1307.5708 about vertix-frequency analysis on graph. David IShuman in this article claims that,"we generalize one of the most important signal ...
niloofar jamshidi's user avatar
2 votes
1 answer
116 views

Are the Prolate Spheroidal Wave Functions absolutely integrable?

I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability. ...
Iconoclast's user avatar
2 votes
0 answers
284 views

Average of irrational flow on the torus

Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$ defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that $$ \lim_{T\...
Sean's user avatar
  • 311
4 votes
0 answers
276 views

A uniform Riemann sum approximation of the integral of the Fejer kernels

Let $F_N(t)$ denote the Fejer kernel $$F_N(t):={1\over N+1}{\sin^2\big({(N+1)}{t\over2}\big)\over \sin^2\big( {t\over2}\big)}\ .$$ Consider Riemann sums approximation for $\int_{-\pi}^\pi F_N(t) dt$ ...
M.Mancino's user avatar
  • 136
1 vote
0 answers
107 views

Discrete Wavelet Transform and Gaussian decay

I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
S. Montaner's user avatar
10 votes
2 answers
7k views

About the Fourier transform of the logarithm function

I want to calculate / simplify: $$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$ where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
Bertrand's user avatar
  • 1,121
2 votes
0 answers
98 views

A specific Schwartz function $f$ on $\mathbb C^2$

Choose a Schwartz function on $\mathbb C$ of the form $f(z)=f(r e^{i\theta})= f_0(r) e^{in\theta}$. Then $$(*) \quad f(e^{i\alpha} z)= e^{in\alpha} f(z), \quad \forall z\in \mathbb C.$$ Now, let $f$ ...
Z. Alfata's user avatar
  • 640
5 votes
1 answer
167 views

$L^p$ estimates and functions with positive Fourier transform

For $f\in\mathcal{S}$ a Schwartz function on $\mathbb{R}^n$ and $m$ a bounded function, define $Tf$ by $\widehat{T f}=m\cdot \widehat{f}$. Fix $1<p<\infty$, $p\not=2$. Suppose we have proved ...
IdiotMathematician's user avatar
1 vote
1 answer
468 views

Which Fourier transform is the correct one?

Given $H(x)$ is the Heaviside Theta, the tables give the following Fourier transforms for it: $$ H(x+a)\to -PV\frac{i e^{i a w} }{w}+\pi \delta (w)$$ while from Sokhotski–Plemelj theorem it follows ...
Anixx's user avatar
  • 9,306
1 vote
0 answers
236 views

On the bound of the Stein-Wainger oscillatory integral

Let $\lambda\in \mathbb{R}$, $\phi\in C^\infty(\mathbb{R})$. We define the Stein-Wainger oscillatory integral by $$I=p.v.\int_\mathbb{R} e^{i\lambda\phi(t)}\frac{dt}{t}.$$ Stein-Wainger [1] showed ...
orange's user avatar
  • 11
5 votes
0 answers
260 views

Is there a Lie group Fourier transform for D-modules?

Given a D-module $\mathcal{M}$ on $\mathbb{C}^n$, its Fourier transform $\widehat{\mathcal{M}}$ is equal to $\mathcal{M}$ as a set, but its module structure over $\mathbb{C}[x_1,...,x_n,\partial_1,...,...
Pulcinella's user avatar
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