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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Using Paley-Wiener Theorem to prove the decay of $G(x-y)$

This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function: \begin{eqnarray} G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
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Looking for sufficient conditions for positive Fourier transforms

I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words $$ \int_0^\infty f(x)\cos(x\omega) \, ...
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Reference on Fourier analysis on compact groups

I am looking for a reference for Fourier analysis on compact (Lie) groups. The kind of theorems I would like the book to cover/do are the Peter-Weyl theorem, define Fourier transforms and use the ...
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Does Ahlfors–David regularity of a measure imply its Fourier asymptotic behavior?

Let $\mu$ be a Borel probability measure on $R^d$. If $\mu$ satisfies $\mu(B(x,r))\le Cr^\alpha$ for any $x\in R^d$ and $r>0$, then Strichartz (Fourier asymptotics of fractal measures, J. Funct. ...
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Vector-valued interpolation for sublinear operators

Grafakos in his $\textit{Classical Fourier Analysis}$ formulates (see Exercise 4.5.2 therein) the following vector-valued version of the Riesz-Thorin interpolation theorem. $\textbf{Theorem}$ Let $1\...
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Flat function with a spectral gap

I am looking for a sequence of functions $f_n,n\geq 1$ in $L^2(\mathbb R)$ such that $f_n$ is equal to $1$ on $[-n,n]$ and $\hat{f_n}$ vanishes on $[-1,1]$. Actually, I would also like $f_n$ to be $...
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Fourier transform of a function of bounded variation

I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
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Integrability of Fourier transform of truncated fractional power

Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...
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1 vote
2 answers
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Is there a function that is not absolutely integrable in [−π,π] so that its Fourier Series Exists? [closed]

For existence of Fourier coefficients of a function f is sufficient that f is absolutely integrable in [−π,π] but, is this condition necessary? that is, is there a function that is not absolutely ...
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Integrability of a function under a condition on its Fourier transform

Let $f\in L^1(\mathbb{R})$ and continuous on $\mathbb{R}$ such that its Fourier transform $\hat f$ equals zero in a neighborhood of zero. Let $F$ be function such that $\hat F$ exists and $$\hat f(x) =...
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Sobolev extension from a discrete set of points

Let $1 > \alpha > 0$ and fix some $C > 0$. Consider $\Omega \subset \mathbb{R}^n$ a bounded domain and $Y \subset \Omega$ a discrete (finite) set of points. For $f: Y \to \mathbb{R}$ define $$...
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Fourier Transform of an even function

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$, where $f$ is an even continuous function. Denote $$ F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\...
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Parseval-Plancherel identity involving absolute value

Let $\hat{f}$ be the fourier transform of $f$. By Parseval-Plancherel identity, for suitable $f,g$, we have $$\left\|\hat{f}*\hat{h}\right\|_{L^2_{\xi}}^2=\left\|f\cdot h\right\|_{L^2_{x}}^2.$$ Let ...
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Fourier analysis and fractional calculus

Do Fourier transform properties still hold in the case of fractional derivatives ? i.e I have seen many times that some lectures define fractional derivative as : $$\frac{d^{\alpha}}{dx^{\alpha}}f=...
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Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the torus?

Consider any continuous function $f$ on an $m$-dimensional torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
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Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?

Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the ...
Rajesh D's user avatar
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3 answers
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Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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(Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
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Radon transform range theorem and radial functions

(UPDATED for rapid decay considerations + new question) In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
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Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
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eigenvectors of a graph Laplacian VS Fourier basis

Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
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Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$

It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
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Analog of the Birkhoff's ergodic theorem for the sequence of squares

Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation. Then by the classical ...
Tony419's user avatar
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10 votes
2 answers
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Fourier series of $\log(a +b\cos(x))$?

By numerical computation it seems like, if $a_0 < a_1$: $$ \begin{multline} \log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\ + \frac{a_0}{a_1}\cos(\omega t) - \frac{...
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Example of a bounded function whose mean-zero mollification diverges at a point

For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by: $$ \varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0. $$ $\...
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Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity

The classical heat semigroup on $\mathbb{R}$ is given by $$ W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0. $$ Then the Littlewood-Paley vertical square ...
Tony419's user avatar
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1 answer
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the fractional integration method of the proof of Stein-Tomas theorem?

In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...
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Strichartz estimate for the Schrödinger equation

Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP: $$i\partial_t ...
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Strict inequality in decoupling inequality

I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032. Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
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8 votes
2 answers
906 views

Lower bound on exponential sums

Let $k\geq 2$. Consider the following norm of exponenetial sum: $$ I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy. $$ Bourgain mentioned on Page 118 of https://...
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Inequality about exponential integrals

I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski. During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
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Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) [duplicate]

I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens. ...
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A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
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1 answer
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Weyl symbol of product

Are there explicit formulas for the Weyl symbol of $-f(x)D_x^2 $ where $D_x:=-i\partial_x $ and $\partial_x$ is the derivative and $f$ some sufficiently smooth function? In the standard quantization ...
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11 votes
1 answer
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Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
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4 votes
1 answer
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Idea behind Carleson's theorem modern proof "intitial reductions"

I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for. For any $f \in L^2(\mathbb{R})$, let $\...
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Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$

Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform. ...
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1 answer
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Evolution equation generated by Fourier multiplier

I am on the hunt for techniques regarding a field which I am not familiar with. More precisely, I am considering equation of the form $$ i \partial_{t} u(t,x) + p(D)u(t,x) = 0, \ \ u_{|_{t=0}}=u_0(x)...
Yilin Ma's user avatar
7 votes
1 answer
332 views

Compactly supported probability measure in high dimensions with fast Fourier decay?

For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
Sitan Chen's user avatar
1 vote
0 answers
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Higher-order inner products of an orthonormal basis

Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...
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What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
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1 answer
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Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that: $$ \lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0 $$ where $\Vert \cdot \Vert_1$ is the $L_1$ ...
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0 answers
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On Pitt's inequality (weighted Fourier inequality)

One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
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Wigner distribution

The Wigner distribution of $u\in L^2(\mathbb R)$ is defined as a function $W(u)$ on $\mathbb R^2$ given by $$ W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} ...
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Fourier dimension of radial set

In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
Manlio's user avatar
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4 votes
2 answers
661 views

Earliest use of deconvolution by Fourier transforms

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
AChem's user avatar
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5 votes
1 answer
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Sobolev convergence of Fourier series

Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...
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Existence of the inverse Fourier transform, Carr Madan

I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant. So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
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Example of periodic semidifferentiable function without absolutely convergent Fourier series

Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but with a non-absolutely convergent Fourier ...
Phil-W's user avatar
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3 votes
3 answers
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Logarithm of the Fourier transform?

I've found this paper on the logarithm of the discrete fourier transform which proves that $$ log F = 1/4 i \pi (I - (1 +i)F + F^2 - (1 - i)F^3) $$ where $F$ is the unitary discrete Fourier ...
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