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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When does the Fourier transform of a measure decay?

Let $\mu$ be a Borel measure on $\Bbb R^d$. It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies $$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$ However if ...
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Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by $$ I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t $$ where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
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What is the "best" good kernel?

A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied: $\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$, $...
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Why complex conjugate in definition of the Fourier transform?

Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be $$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$ ...
Daniel Loughran's user avatar
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Inequality for Fourier transform of a power exponential function

Let $$ f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }}, x \in \mathbb{R}, 0<\alpha<2, $$ where $\phi_1(\alpha)=\frac{\alpha}{2} \left\{{\{\Gamma(3/\alpha)\}^{1/...
Tanya Vladi's user avatar
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Positive definiteness with nonnegative weights

Is there a simple criterion to certify if some function $f: \mathbb{R} \to \mathbb{R}$ satisfies that $\sum_{i,j=1}^n c_ic_jf(x_i-x_j) \ge 0$ for all $x_i \in \mathbb{R}$ and $c_i \ge 0$? Note that if ...
Yanjun Han's user avatar
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$\Lambda f\ge 0\iff f\ge 0$ if $\Lambda$ is a Gaussian convolution kernel?

Consider $\Lambda f(x)=\int_{\mathbb R} f(x-y) e^{-y^2} dy$. Suppose that for a bounded function $f$, $\Lambda f(x)\ge 0$ for all $x\in\mathbb R$. Does it imply that $f\ge 0$ almost surely?
Ribhu's user avatar
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Inequality for sums of sines with similar frequency

Let $c>0$ be a very small constant and $N \in \mathbb N$ very large. Assume we have a function $f(x)$ for $x \in S^1$ defined as $$ f(x) = \sum_{k=\lfloor N/(1+c) \rfloor}^{N} c_k \sin(kx+b_k) $$ ...
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Direct analytic proof of positive definiteness of stable characteristic functions

Is there a direct analytic proof that the function $$ f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \...
tsnao's user avatar
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3 answers
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If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?

Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that $$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$ Do we have $\|f\...
Feng's user avatar
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Systems of parabolic equations -- Petrovskii's condition

Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$. Given a matrix field $A:Q_T\rightarrow\text{M}...
Ayman Moussa's user avatar
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Littlewood-Paley characterisation of Hölder regularity

I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
Tham's user avatar
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The asymptotic behaviour of a singular integral

Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$. I am trying to determine the asymptotic behaviour of $$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...
Medo's user avatar
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Is this formula for 2D Fourier integral of diffraction kernel correct?

Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
VojtaK's user avatar
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Building a smooth function from a rapidly decreasing sequence

Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function? More precisely: Let $\lbrace c_k\rbrace _{k \...
Peg Leg Jonathan's user avatar
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Eigenfunction expansion theorem for general manifold for smooth functions

The question starts with the well known facts that: if $f$ is a smooth function on $S^1$, then its Fourier series converges to it in smooth topology. This must be true in more general setting. I have ...
Hao Yin's user avatar
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Calculating the Fourier dimension of a real interval $\left[a, b\right]$

(Preliminaries:) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$. 2.) Define the ...
Epsilon Away's user avatar
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Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?

For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
H A Helfgott's user avatar
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Approximation of Hölder functions by Fourier series

Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$. Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
user500030's user avatar
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2 answers
211 views

Well-defined distribution and its singular support

Let $f$ be a smooth function on $X$, an open subset of $\mathbb{R}^n$, with $Im(f) \geq 0$. Let us fix an $\epsilon > 0$. Let $T_{\epsilon} := \frac{1}{f(x)+i\epsilon} $ in $D’(X)$. Now if we ...
zarathustra's user avatar
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Localize functions in the Hardy space $\mathcal H^1(\mathbb R^n)$

Let $f$ belong to the Hardy space $\mathcal H^1(\mathbb R^n)$, $B\subset \mathbb R^n$ be the unit ball. Does there exist a $\bar f\in \mathcal H^1(\mathbb R^n)$ with compact support such that $\bar f=...
Tian LAN's user avatar
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3 answers
239 views

Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?

Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let $$ F_a(t) = \sum_{k \in \mathbb Z} f(t+ak) $$ be the ...
user975628's user avatar
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Real-analytic analogue of Schwartz functions

Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
Zislu R.'s user avatar
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107 views

Recovering phase function using Fourier decomposition

I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function $$f = e^{i \phi(x)}. $$ I am interested in the following problem. If I know the function/distribution $...
VojtaK's user avatar
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Question on the existence of a certain decomposition method for real square matrices

I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
Kanghun Kim's user avatar
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238 views

Main ideas behind the proof of the Carleson theorem

I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
Basj's user avatar
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2 answers
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When I know self convolution of the complex function can I recover function itself or its modulus?

I have a function $A : \mathbb{R} \to \mathbb{C}$. I know there exists unknown function $u: \mathbb{R} \to \mathbb{C}$, such that $A$ is convolution of $u$ and its complex conjugate $A = u * u^*$. I ...
VojtaK's user avatar
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Detailed examples of induction on scale

I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...
Simplyorange's user avatar
1 vote
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A convergence problem in the space of tempered distributions

Let $K(x):=|x|^{-\alpha}$ be a function on $\mathbb{R}^{n}\setminus\{0\}$ with $0<\alpha<n$. Suppose $\phi$ is a $C^{\infty}_{c}(\mathbb{R}^n)$ function such that $$\text{(i)}\quad \text{supp}\...
Medo's user avatar
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1 answer
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Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator $...
Medo's user avatar
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Fourier series of Eisenstein series — elegant and very good approximation

I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...
Marcus's user avatar
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1 answer
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Fourier series of an arbitrary function of a cosine function

Is there a general expression for the Fourier series of the function $f(a\cos(\omega t))$ in terms of the derivatives of $f$? Obviously, the function can be expressed as a Maclaurin series $f(0)+af'(0)...
Jinyang Li's user avatar
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0 answers
85 views

Closed formula for iterated Fourier series

I'm trying to obtain a closed formula for the following integral. \begin{align} I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt] & {}+ b_{1,...
Marco's user avatar
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6 votes
1 answer
255 views

A geometric interpretation of the fractional Fourier transform

I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18): Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
Luke's user avatar
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1 answer
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How to compute the asymptotics of this oscillatory integral?

I posted this on Stackexchange but got no responses or comments. Consider the following integral, for $\epsilon\ne 0:$ $$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...
Josh Lackman's user avatar
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2 votes
1 answer
112 views

Vertical Fourier decomposition for skew-Hermitian 1-forms

In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein). Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...
Florian R's user avatar
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7 votes
2 answers
376 views

$L^p-L^q$ boundedness of this simple singular oscillatory integral operator

Let $0<\alpha<1$ and define $$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$ The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of $Hf(x):=\int \frac{...
Medo's user avatar
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1 vote
1 answer
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Uniqueness of Fourier–Stieltjes transform for finite complex valued measures

Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform $$ \hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t) $$ Question: Does $\...
Boby's user avatar
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11 votes
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Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?

Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality $$ \...
Carlos Esparza's user avatar
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1 answer
160 views

When some Fourier coefficients are fixed, can we control the extremals of the function?

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold? 1- $-n\lneqq f_{\min}$ (where $f_{\...
ABB's user avatar
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Computing sums with linear conditions quickly

Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
H A Helfgott's user avatar
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4 votes
1 answer
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Exponential sum vs. exponential integral via Poisson summation

When we want to estimate an exponential sum $$ \sum_{M<m\le M'}e(f(m)) \quad\text{with}\quad 1\le M\le M'\le 2M \quad\text{and}\quad e(x):=\exp(2\pi ix) $$ where $e(x):=\exp(2\pi ix)$ and the phase ...
snufkin26's user avatar
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3 votes
0 answers
160 views

The essential norm where some Fourier coefficients are fixed

Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$. Q. Let $\phi\in C_{2\pi}$. Is the following statement valid? $$\|\phi\|_2=\inf_{g\in C_{2\...
ABB's user avatar
  • 3,898
2 votes
1 answer
243 views

Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space

Problem Statement Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
Jacob Helwig's user avatar
1 vote
1 answer
100 views

Support of Fourier transform of random characteristic function

Let $\chi_X:\{-1,1\}^n\to \{0,1\}$ be the characteristic function of a subset $X\subseteq \{-1,1\}^n$, which is randomly drawn from all subsets with exactly $k$ elements. Is the support of the Fourier ...
BGJ's user avatar
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1 answer
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Relation between inverse Fourier transform of dyadic bump function and its regularity

I posted on MathStackExchange but there are no answers. So, I arrived at MathOverflow in order to have some comments or any idea. I am interested in oscillatory integral which has restricted range by ...
Idkwhat's user avatar
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1 vote
0 answers
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Is there an approximate formula for this summation function?

Consider the function $$\sum_{n=1}^\infty \frac{\cos(nx)}{n^r},$$ where $r\in\mathbb{N}$. Is there any approximate formula (closed form possibly avoiding this type of summation) for this function? I ...
user102868's user avatar
4 votes
0 answers
127 views

Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$

I have asked the same question on MathSE. I was thinking about the following problem. Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
Lorenzo Pompili's user avatar
4 votes
0 answers
97 views

L_infinity norm of signed sums of Fourier characters and discrepancy of Fourier matrices

Consider signed sums $\displaystyle A_f(x) =\sum_{\chi} (-1)^{f(\chi)} \chi(x)$ for some set $S$ of characters of an abelian group $G$, and signing $f$ of the characters. For a fixed set $S$ what is ...
fourier_discrepancy's user avatar
1 vote
0 answers
166 views

Fourier transform of functions mapping manifolds, is there a definition?

$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form $$ f : \mathbb{R} \to \SO(3)^n $$ Since $\SO(3)$ is a compact group so is $\SO(3)^n$. Now if ...
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