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5
votes
0answers
115 views

Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces? Examples for what I have in mind: Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and ...
5
votes
0answers
268 views

converse of Weyl criterion

Let $f∈L^1([0,1))$,suppose for all equi-distributed sequence $\{a_n\}^∞_{n=1}$ in $[0,1)$,we have $$\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{k=1}^Nf(a_k)=\int_0^1f.$$ Do we have that $f$ is Riemann ...
2
votes
0answers
189 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

Short version of question: I'm trying to understand the extent to which a function is prevented from being "flat" as a result of being the Fourier transform of a positive function. That is, the extent ...
2
votes
0answers
188 views

Is $f$ an absolutely continuous function? [closed]

Let $$ f(x)=\sum_{n=1}^{\infty}\frac{\sin(2^{n}\pi x)}{n\cdot2^{n}}, \,\,\,\,\,\, x\in [-1, 1]. $$ Is $f$ an absolutely continuous function? If yes how can I show it? If not how about on total ...
-2
votes
1answer
287 views

Inverse Laplace transform in a signal that is a sum of exponentials - MATLAB [closed]

I want to analyze a discrete signal in time, that is sum of exponentials distributed around two distinct values (T1, T2). My goal is to calculate these two different distributions, T1, T2, weights and ...
1
vote
0answers
91 views

Differentiability of $f*g$ on the circle, for integrable f, bounded g, and some decay of the Fourier coefficients of f

If $f\in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$ where $\mathbb{T}$ is the circle, such that $\hat{f}\in L^{p}(\mathbb{Z})$ for some $1\leq p<\infty$, do we have that $f*g$ is ...
5
votes
2answers
559 views

What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$

It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
4
votes
1answer
181 views

If $f$ is non-prime, can we say $|f|$ is also a non-prime; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < ...
1
vote
2answers
196 views

Behavior of the Fourier transform (FT) of a function and FT of its absolute function

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}.$ We define the Fourier transform of $f$ as follows: ...
1
vote
0answers
145 views

Constructing an $\epsilon$-net for a Lipschitz subspace of $L^2$

Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function. Also, the member of $X$ are uniformly bounded $$ |x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}. $$ Let $F: X ...
3
votes
2answers
611 views

Extension of Poisson Summation formula

Under the condition f continuous, integrable and: $|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0) we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
2
votes
1answer
294 views

Fourier transforms of finitely additive bounded measures

Given a finitely additive positive regular bounded measure $\mu$ on ${\mathbb R}^n$ (i.e. a positive linear functional on $C_b({\mathbb R}^n)$), I wonder what can be said about its Fourier transform. ...
4
votes
2answers
564 views

$L^p$-norm of Fourier series in terms of coefficients, $p \neq 2$

It is known that the $L^2$-norm of a Fourier series equals the $l^2$-norm of the coefficients. Are there similar results in the case of $L^p$-norm for $p\neq 2$? Can it be expressed explicitly in ...
1
vote
2answers
268 views

Bandwidth approximation for a nonlinear problem

Can anyone please help me with this problem. I must let you know from the beginning that it's not an easy one. "Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ , ...
4
votes
1answer
101 views

A Laplacian semi-group estimation

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
1
vote
0answers
81 views

The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion: \begin{equation*} B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}. ...
-17
votes
1answer
2k views

Can Poisson Summation formula break?

The Poisson summation formula states if $f: \mathbb{R} \to \mathbb{R}$ then $\displaystyle \sum_{n \in \mathbb{Z}} f(n) = \sum_{n \in \mathbb{Z}} \hat{f}(n) $ where $$\hat{f}(\xi) = ...
0
votes
1answer
77 views

The influence of eigendecomposition on the periodicity of a (rank 2) Hermitian matrix (of functions)

Let $\boldsymbol{R}(u,v);~u,v\in\mathbb{R}$ be a Hermitian matrix (of Hermitian functions) with entries \begin{equation} r_{ij}(u,v) = 1 + Ae^{-2\pi i \phi_{ij}(ul_0 + vm_0)}; ...
0
votes
0answers
178 views

Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is ...
0
votes
0answers
90 views

Two distribution spaces ${\mathcal S}'/{\mathcal P}$ and ${\mathcal S}_\infty'$

Let ${\mathcal S}'$ be the set of all distributions. Denote by ${\mathcal P}$ the set of all polynomials, which is embedded into ${\mathcal S}'$ as a closed subspace. Equip ${\mathcal S'}/{\mathcal ...
4
votes
1answer
288 views

Uniqueness theorems related to Hardy Uncertainty Principle

Uncertainty Principles state that a function and its Fourier transform cannot be simultaneously sharply localised. A well known result due to G.H.Hardy says that if $f(x)=O(e^{-\alpha^2|x|^2})$, ...
1
vote
0answers
98 views

A Fourier analysis-type identity arising in potential theory

I have recently been learning about potential theory in the complex plane, and I would like to understand why the following proposition is true: Suppose $f: \mathbb C \to \mathbb R$ is smooth, ...
17
votes
1answer
512 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
15
votes
3answers
510 views

An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
1
vote
0answers
294 views

Obtaining a pointwise bound on the convolution of two singular measures

I am confused about a passage in the paper by T. Tao A sharp bilinear restriction estimate for paraboloids. We are in Section 7, near equation (34) (pag.16 of the arxiv). Notations and ...
5
votes
1answer
751 views

Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
20
votes
3answers
468 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
0answers
62 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
1answer
105 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
2answers
206 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
0answers
66 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
1answer
396 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
1answer
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
1answer
246 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
0answers
208 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
2answers
748 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
2answers
487 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
2answers
395 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
0answers
169 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
5answers
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
1answer
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
0answers
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
1answer
261 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
0answers
286 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
2answers
315 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, ...
2
votes
0answers
166 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
1answer
316 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 ...
2
votes
2answers
550 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 ...
17
votes
2answers
756 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
1answer
256 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 $L^2(0,1)$?