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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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How to choose some $h$ so its Fourier transform supported in some set?

Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$ Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
 Analyst 's user avatar
1 vote
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A parametrix construction for heat boundary value problem using Fourier transformation

Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3 $ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
L19's user avatar
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Sobolev estimates on domain with boundary

Could someone point me to a reference for the proof of the following Sobolev estimate $$ \|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)}) $$ for ...
L19's user avatar
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Can the best constants in harmonic analysis be approximated in principle?

Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
Simplyorange's user avatar
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1 answer
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$\|\hat{f}\|_{L^q}< \infty \implies \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty $

Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $ \int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier ...
 Analyst 's user avatar
3 votes
0 answers
79 views

Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive ...
Alexey S's user avatar
1 vote
0 answers
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Question about stationary phase with Hessian close to $0$

Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define $$ I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
Johnny T.'s user avatar
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Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?

Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions $$ \chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1) $$ where $\chi(t)$ ...
Mark's user avatar
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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 ...
Guy Fsone's user avatar
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1 answer
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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 ...
Grandes Jorasses's user avatar
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0 answers
121 views

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: $\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$, $...
Dr. Pi's user avatar
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1 vote
1 answer
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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
4 votes
1 answer
334 views

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
3 votes
0 answers
67 views

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
2 votes
1 answer
58 views

$\Lambda f\ge 0\iff f\ge 0$ 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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1 vote
1 answer
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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 $$ f(x) = \sum_{k=\lfloor N/(1+c) \rfloor}^{N} c_k \sin(kx+b_k) $$ ...
HHN's user avatar
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3 votes
0 answers
128 views

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
  • 109
5 votes
3 answers
348 views

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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4 votes
0 answers
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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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0 votes
1 answer
111 views

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
  • 103
0 votes
1 answer
125 views

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
  • 391
0 votes
0 answers
20 views

Bounding $\widehat{G_{m+1}}\ast\widehat{H_m}(0)$ when $\frac{1}{2}\leq\widehat{H_m}(0)\leq\frac{3}{2}$ and $H_m,G_{m+1}$ are smooth over $\mathbb{T}$

To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that $$\left|\widehat{H_m}(k)\right|\leq C\log(\left|...
Epsilon Away's user avatar
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0 answers
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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
  • 141
-1 votes
1 answer
112 views

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
3 votes
0 answers
47 views

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
  • 31
0 votes
2 answers
257 views

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
5 votes
0 answers
205 views

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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0 votes
0 answers
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Existence of a specific family of functions on an abelian group with vanishing properties on rank 2 subgroups

Fix a prime $p$, and let $W_0\subset W$ be an inclusion of a codimension one $\mathbb{F}_p$ vector spaces. Let $W_e$ denote a fixed nontrivial coset of $W_0$ in $W$. The question is whether there ...
Chris H's user avatar
  • 1,607
2 votes
1 answer
188 views

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
0 votes
2 answers
170 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
1 vote
0 answers
42 views

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
6 votes
3 answers
199 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
5 votes
1 answer
189 views

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
1 vote
0 answers
98 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
  • 141
1 vote
0 answers
100 views

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
0 votes
0 answers
181 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
  • 577
0 votes
2 answers
166 views

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
  • 141
6 votes
0 answers
153 views

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
0 answers
47 views

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
  • 391
1 vote
1 answer
108 views

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
  • 391
2 votes
1 answer
256 views

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
  • 386
0 votes
1 answer
123 views

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
0 votes
0 answers
80 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
  • 253
5 votes
1 answer
174 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 ...
Waiganjo's user avatar
4 votes
1 answer
159 views

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
2 votes
1 answer
106 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
  • 215
7 votes
2 answers
336 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
  • 391
1 vote
1 answer
106 views

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
  • 611
11 votes
2 answers
666 views

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
0 votes
1 answer
148 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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