The fourier-analysis tag has no usage guidance.

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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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112 views

### 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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144 views

### 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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238 views

### 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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1k views

### 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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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 ...

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269 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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196 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 ...

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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 ...

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**1**answer

292 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 ...

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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 ...

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580 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 ...

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182 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)| < ...

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198 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:
...

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147 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 ...

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617 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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296 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. ...

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578 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 ...

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

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104 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) ...

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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}}.
...

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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) = ...

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78 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)}; ...

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180 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 ...

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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 ...

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

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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, ...

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

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516 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 ...

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298 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 ...

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753 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 ...

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471 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 ...

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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 ...

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107 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 ...

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207 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 ...

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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 ...

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401 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) - ...

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158 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 ...

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249 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})}
...

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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 ...

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749 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 ...

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492 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}) ...

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398 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 ...

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175 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 ...

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5k 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 ...

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**1**answer

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 ...

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208 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 ...

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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 ...

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290 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 ...

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321 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, ...