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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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39 views

Short-time Fourier transform of $f\ast g (y)- f\ast g(x)$?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$ It is also known that the ...
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votes
0answers
48 views

Fourier series of a heat equation [closed]

I am solving this PDE $$2u_{xx}=u_t$$ $$u(x,0)=x$$ $$u(0,t)=u_x(4,t)=0$$. I found this solution $$u(x,t)=\sum_0^{\infty}\left(D_n\sin\left(\frac{n\pi+\frac{\pi}{2}}{4}x\right)\exp\left(\left(\frac{4}{...
7
votes
0answers
113 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 ...
10
votes
2answers
814 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 $f$ ...
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0answers
64 views

Speed of divergence series

Given a positive sequence $\{a_k\}$ and an $s>0$ such that $\sum_{k=1}^\infty k^sa_k = \infty$, but $\sum_{k=1}^\infty k^s a_k/(\log k)^2 = 1$. For any $t\in [0,b]$, where $b$ is a fixed number, ...
8
votes
0answers
115 views

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 $\mathbb T^\omega$? (...
9
votes
1answer
189 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]...
1
vote
1answer
79 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 $\...
2
votes
1answer
129 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 ...
2
votes
1answer
101 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}{...
1
vote
1answer
66 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}^...
12
votes
3answers
500 views

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, $(...
0
votes
1answer
89 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 ...
3
votes
1answer
117 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 $...
9
votes
1answer
306 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 ...
0
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0answers
19 views

Bound for nonhomogeneous low-frequency cut-off operator in littlewood-paley decomposition

In Remark 2.11 of Fourier analysis and nonlinear PDEs written by Bahouri and chemin, it suggests nonhomogeneous low-frequency cut-off operator $S_j$ is bound operator from $L_p$ space to $L_p$ space, ...
6
votes
3answers
324 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. ...
4
votes
0answers
70 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 ...
2
votes
1answer
360 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 ...
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0answers
68 views

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 ...
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1answer
118 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 \...
3
votes
0answers
32 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 $\lambda_i\to \infty$ ...
6
votes
0answers
134 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 ...
3
votes
1answer
256 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 ...
1
vote
1answer
105 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-...
0
votes
0answers
53 views

Initial value theorem for Fourier transform

Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
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2answers
170 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 ...
5
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0answers
96 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 ...
11
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1answer
383 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 ...
1
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0answers
111 views

Calculus of variation with discontinuous solutions?

I'm thinking of the following question: Consider a function $f: [0,L]\rightarrow\mathbb{R}$ and an energy functional $$F=\int_{0}^{L}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\mathrm{d}x.$$ The ...
7
votes
2answers
168 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 ...
0
votes
1answer
68 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,\...
2
votes
0answers
50 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(...
1
vote
0answers
61 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 $...
13
votes
1answer
420 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 ...
4
votes
1answer
163 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 ...
10
votes
0answers
105 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 ...
11
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1answer
262 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 ...
2
votes
0answers
118 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 ...
7
votes
1answer
565 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 ...
2
votes
1answer
73 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. ...
2
votes
0answers
117 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\...
3
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0answers
146 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 $\int_{-\pi}^\pi F_N(t) dt$ ...
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0answers
40 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 ...
0
votes
0answers
39 views

Composing functions whose derivatives have $L^1$ Fourier transforms

Let $f, g \in C^n(\mathbb{R})$. Given bounds $\|f^{(i)}\|_\infty \le a_i, \|g^{(i)}\|_\infty \le b_i \forall i \in \{1, ..., n\}$, we can derive bounds for $\|(g \circ f)^{(i)}\|_\infty$ using the ...
2
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1answer
188 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\...
2
votes
0answers
81 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 $f$ ...
6
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1answer
110 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 ...
1
vote
1answer
315 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 ...
0
votes
0answers
42 views

Positivity of an integral with product of functions and their Fourier transforms

What condition two functions $f$ and $g$ defined on $\mathbb{R^+}$ must fulfilled to have: $$\int\limits_{0}^\infty \ln(x) (fg+ \hat{f} \hat{g} ) dx >0$$ Where we note $\hat{f}(x)= 2\int\...