The fourier-analysis tag has no wiki summary.

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### Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution )
$$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...

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### Express measurable entropy in terms of Fourier coefficients of the measure

Let $S^1$ be the unit circle , $\mu$ be a Borel probability measure on $S^1$ and
$T:S^1\to S^1$ is a measure-preserving map (not necessarily invertible) with respect to $\mu$. The Fourier ...

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### How Fourier transform behaves if we kills the oscillation?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put
$$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$
and
...

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### Understanding Bruhat's notion of Schwartz function

I am trying to understand Bruhat's generalized Schwartz functions over (Hausforff) locally compact Abelian groups [1], following this paper [2] by Osborne. There, the Schwarz-Bruhat space ...

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50 views

### Fourier coefficients of a non-atomic measure on the unit cirle

Let $0<\lambda<1$ and $\pi:\Pi_{\mathbb{N}} \{0,1\}\to S^1$ be the continuous map from the product space of $\{0,1\}$ over the set of natural numbers $\mathbb{N}$ onto the unit circle $S^1$. Let ...

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### How Fourier-Lebsgue spaces operates functions?

Let $a, b \in \mathbb R$ such that $ab> 1$ ; put
$$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$
and
...

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### Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?

Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...

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256 views

### Ultrafilter-based Fourier-Walsh-like Functions

Here is a (little wild) question about Boolean functions with countably many variables and a wild analog for Fourier-Walsh functions and analysis based on them.
Let $x_1,x_2,\dots,x_n,\dots$ be ...

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86 views

### Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...

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### Fourier Transform of compactly supported $L^1$ functions

Background Given a (translation bounded) positive definite measure $\gamma$ lets say on $\mathbb R^d$, its Fourier transform as a tempered distribution is a positive measure $\widehat{\gamma}$.
I am ...

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158 views

### Problem with understanding an equation

I have read the article Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets and I don't know how to interpret Equation (10) on
page 630, because this equation ...

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107 views

### Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...

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### Correlation of Fourier transforms of characteristic functions

Let $A$ and $B$ ($A\subset B$) be subsets of a finite abelian group $G$. (For the sake of argument, you can take $G$ to be $\mathbb{Z}/p\mathbb{Z}$ for large $p$, say.) Write $1_S$ for the ...

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### Are the two functions equal? If not, what relation are they?

We have
$$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$
$$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi ...

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### Let $f \in M^{1,1} (\mathbb R)$ (Feichtinger's algebra /Modulation Space). Can we say $Fof\in M^{1,1}(\mathbb R)$; $F$ is an entire function?

The Modulation space ( Feichtinger's algebra),
$$S_{0} (\mathbb R) = M^{1, 1}(\mathbb R): = \{ f\in L^{2}(\mathbb R) : V_{g}(f) \in L^{1}(\mathbb R^{2}) \};$$
where $V_{g}f (x, w)$ is the short- ...

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### Let $f \in S(\mathbb R)$. Can we say $\widehat{|f|} \in L^{1}(\mathbb R)$?

Let $f\in L^{1} (\mathbb R) := \{f:\mathbb R \rightarrow \mathbb C \ \text {measurable functions} : \int_{\mathbb R} | f(x)| dx < \infty \}$ and the Fourier transform of $f$,
$\hat{f} (y) : = ...

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### convolution algebra on a compact surface in $\mathbb{R}^3$

Can one construct for all/any compact Surfaces in $\mathbb{R}^3$ a Schwartz space like object in a way so that the space can be given a structure analogous to the convolution algebra structure it is ...

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### Sets $E$ in $\mathbb{Z}$ such that any $l^2$ function with support on $E$ comes from Fourier of a continuous function

Are there infinite sets $E\subset\mathbb{Z}$ such that any $f\in l^2(\mathbb{Z})$ with support on $E$ comes from the Fourier transform of a continuous function on $\mathbb{T}$ ? If yes, is there a ...

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### Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$
where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...

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### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable
results, such as a proof of the Bieberbach conjecture in 1985, which is now
known as de Branges' theorem. Initially, his ...

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### 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 ...

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261 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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)| < ...

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### 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:
...

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### 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 ...

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498 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 ...

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### Is Modulation space is close under absolute?

It is clear that, for $1\leq p \leq \infty$, $f\in L^{p}$ iff $| f| \in L^{p}.$
By the usual modulation space $M^{p, q}(\mathbb R^{d})$, $1\leq p, q \leq \infty, d\in \mathbb N $, we mean the ...

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### 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. ...

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### $L^p$-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 ...

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### 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: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...

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### 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) ...

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### 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}}.
...

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### 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) = ...

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### 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)}; ...

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### 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 ...

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### 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 ...

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### 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 $f(x)=O(e^{-\alpha^2|x|^2})$, ...

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### 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, ...

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### 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 $Mf(\sigma-it)$ ...

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### 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 ...

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### decay rate or bound of short-time Fourier transform of nonlinear waves

Suppose we have a nonlinear wave $f(x)=e^{2\pi i N(x+\epsilon \sin(x))}$ with positive $\epsilon$ small enough. Let $w(x)$ be a smooth window function supported in a unit ball. Define the short-time ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...