# Questions tagged [fourier-transform]

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### Comparison principle for porous medium equation in Fourier variables

Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u)$. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
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### Existence of nonzero entire function with restrictions of growth

Question. Is there an entire function $F$ satisfying first two or all three of the following assertions: $F(z)\neq 0$ for all $z\in \mathbb{C}$; $1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...
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I am interested in the following question. So let suppose we have finite number of point particles on plane $\mathbb{R}^2$. We can assume that every $j$ point is represented by Dirac delta function $\... 0 votes 0 answers 18 views ### How to calculate the power transformation of a spectral density function There is a problem I have been trying to solve for a while. Let$X_t$be a stationary (univariate) time series. The spectral density of the moving average process $$X_t=\sum^{\infty}_{j=-\infty}a_je_{... 1 vote 1 answer 75 views ### Eigenvalues of a circulant: DFT or Inverse DFT Convention? Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ... 1 vote 1 answer 106 views ### Uniqueness of Fourier–Stieltjes transform for finite complex valued measures Let \mu be a finite complex valued measure on \mathbb{R} and let \hat{\mu} be it's Fourier–Stieltjes transform$$ \hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t) $$Question: Does \... 0 votes 1 answer 148 views ### When some Fourier coefficients are fixed, can we control the extremals of the function? Let n be a odd number. Does there exist any 2\pi-periodic continuous function f :\mathbb{R}\to \mathbb{R} such that the following points simultaneously hold? 1- -n\lneqq f_{\min} (where f_{\... 0 votes 0 answers 51 views ### Fourier transform of an exponential function with radical argument divided by a radical I have f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}} where t_0 and A are constant. I need to take the Fourier transform of f(t). I made few substitutions to take it to a form ... 0 votes 0 answers 34 views ### Are standing waves equivalent to travelling waves as a modelling tool to solve wave equation? I am learning Fourier Analysis from Elias Stein's excellent textbook. He starts off by explaining difference between standing waves and travelling waves and then demonstrating how you can either to ... 3 votes 0 answers 157 views ### The essential norm where some Fourier coefficients are fixed Let us denote C_{2\pi} by the set of all 2\pi-periodic continuous functions f:\mathbb{R}\to \mathbb{R}. Q. Let \phi\in C_{2\pi}. Is the following statement valid?$$\|\phi\|_2=\inf_{g\in C_{2\... 2 votes 1 answer 89 views ### Is the Fourier multiplier$\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2)\hat \psi(p)$justified for any real function$G$? I am confused about a claim asserted in the paper "Higher Order Schrodinger Equations" published in IOP Science. The authors claim that a Fourier multiplier identity $$\mathcal F(G(-\hbar^2 ... 3 votes 1 answer 177 views ### The proof of the invertibility of \Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2} Suppose that n is even. Any suggestion/appraoch to prove that S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2} is invertible? 0 votes 0 answers 115 views ### Question on possibility of uniquely defining the FRFT via certain properties I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ... 0 votes 0 answers 104 views ### Decay of the Fourier transform I have read this post (and similar ones): Decay of the Fourier transform of a non-differentiable function and I have the following doubt: If f\in C^{n}(\mathbb{R}) f^{(n+1)} is piecewise and of ... 2 votes 1 answer 208 views ### Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space Problem Statement Let g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\} be an integrable function (assumption I1). Suppose \mathcal T is a rotation, and f:\mathbb R^d\to\mathbb C (assumption C) is an ... 1 vote 0 answers 34 views ### Does the constrained Wasserstein barycenter admit a blue noise property? Let (E,d) be a metric space and \nu be a probability measure on \mathcal B(E). In this paper, it is mentioned that sampling from \mu can be described as choosing n\in\mathbb N, x_1,\ldots,... 3 votes 0 answers 99 views ### A new arranging of discrete sine transform Let n be even and consider the discrete sine transform of type 5 which is the matrix$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$Let us denote by s_{-,l} the l^{\text{... 0 votes 0 answers 103 views ### Fourier transform and rotations in 3d Let f\in \mathcal{S}(\mathbb{R}^3) be a Schwartz function invariant under rotations in \mathbb{R}^3 and let \hat f\in \mathcal{S}(\mathbb{R}^3) be its Fourier transform, i.e. \hat f(p)=\int_{\... 9 votes 2 answers 587 views ### How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up? In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ... 3 votes 0 answers 323 views ### Is this FFT algorithm known? Recently I've been thinking about alternatives to the usual Cooley-Tukey FFT for multiplying polynomials. I think I've come up with a pretty nifty algorithm for multiplying polynomials. So my question ... 5 votes 1 answer 470 views ### Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential decay Let f:\mathbb{R}\to \mathbb{R} be a function in L^2 satisfying |f(x)|\ll e^{-a_1 x}, a_1>0, for x\to \infty. (Variant: assume as well that |f(x)|\ll e^{a_2 x}, a_2>0, for x\to -\... 9 votes 2 answers 434 views ### Distribution f such that (a) \widehat{f} has compact support, (b) \mathbb{E}(|X|) is minimal? (What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem) Let f:\mathbb{R}\to [0,\infty) be such that (a) \int_{\mathbb{R}} f(x) dx = 1, (b) \... 6 votes 1 answer 615 views ### Fourier optimization problem related to the Prime Number Theorem Let \kappa>0 be given. What is the function f:\mathbb{R}\to [0,\infty) with \int_\mathbb{R} f(x) dx = 1 such that$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{... 6 votes 2 answers 208 views ### Does the (distributional) support of the Fourier transform of an$L^p$-function with$p<\infty$have positive measure? Suppose that$f \in L^p(\mathbb R^n)$such that$1\leq p < \infty$. Let$\hat f$be the Fourier transform of$f$. Clearly, if$p=1$or$p=2$then the support of$\hat f$has positive Lebesgue ... 3 votes 0 answers 152 views ### Green-Tao's "Polylogarithmic bound for$r_4(N)$" On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of$\psi\big(\frac{k}{N}\big)$for all$k$, $$K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\... 4 votes 1 answer 284 views ### When does the sine transform result in a positive function? For each x>0 is,$$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ... 2 votes 0 answers 74 views ### For \Phi a majorant of 1_{[-1/2,1/2]}, how small can the total variation of \widehat\Phi be? Let \Phi:\mathbb{R}\to \mathbb{R} be a real-valued, symmetric, non-negative function such that \Phi(t)\geq 1 for |t|\leq 1/2. Assume furthermore that \Phi and \widehat\Phi are both in L^1\... 5 votes 2 answers 229 views ### An optimization problem: \Phi(0), \widehat{\Phi}(0), \Phi a majorant (This is a problem that arose from my own answer to Mean value theorem for Dirichlet series - optimize? ) Let \Phi:\mathbb{R}\to \mathbb{R} be a real-valued, symmetric, non-negative function such ... 2 votes 0 answers 99 views ### Evaluate action of f(\frac{d}{dx}) using the Fourier/Laplace transform Consider a function f(x) that is numerically defined in -1 \leq x \leq 1 interval (assume N samples). I am trying to compute the action of f(d/dx) on a function g(x) using the Fourier ... 1 vote 0 answers 75 views ### Kernel representation of a power of (pseudo-)differential operator Let \mathcal{T} be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ... 4 votes 1 answer 392 views ### The decay of Fourier coefficients and the continuity of functions Let f be a function on \mathbb{T}=[0,1] ( 1 -periodic) with bounded variation. Prove that if \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) , then f\in C(\mathbb{T}) . I do not ... 6 votes 1 answer 354 views ### Harmonic analysis for a beginner I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ... 3 votes 0 answers 193 views ### A generalization of Weierstrass transform As stated in this article, the Weierstrass transform of f(x) is defined as: \begin{equation} W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy \end{equation} which can be ... 0 votes 1 answer 159 views ### Find an integral kernel for the solution of a partial differential equation: an initial value problem Consider the following partial differential equation with an initial condition u(x,0)=f(x): \begin{equation} \frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\... 1 vote 1 answer 219 views ### Why we have f=0 Define the Fourier transform for a suitable function f\in L^1(\Bbb R) by \widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx. Assume the condition$$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^... 4 votes 2 answers 436 views ### A proof of Bernstein's inequality I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below: "The function$\frac{\xi^\beta}{|\xi|...
Let $f\in L^1(\mathbb{R})$ such that $$\int_0^\infty e^{-yt}e^{ixt}\widehat{f}(t)dt=0,……(i)$$ $$\int_{-\infty}^0 e^{yt}e^{ixt}\widehat{f}(t)dt=0, ……(ii)$$ for all $x\in \mathbb{R}, y>0.$ Questions: ...