# Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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### Does the fourier transform of the product of test function and bounded function belong to $L^p$?

Let $\mathcal D(\mathbb R^n)$ be the set of test functions, i.e., functions in $C^\infty(\mathbb R^n)$ with compact support. Let $f\in \mathcal D(\mathbb R^n)$, and $m:\mathbb R^n\to \mathbb C$ be a ...
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### Non-uniqueness for Fourier Interpolation

I am working through a past exam and I am asked to show that under certain conditions, Fourier interpolation with basis function $e^{inx}, n=0, \ldots N-1$ is unique and to provide an example where it ...
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### iFFT a Known Transformed Function to Get the Unknown Complex-valued Initial Function

Background: In my case, I need to get the solution of a series of 2D equations. The analytical expression of this solution ($f$) is not available but the transformed one is. Therefore, I need to ...
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### Reference Request: Hausdorff–Young inequality for the inverse Fourier seires

Let $\hat f : \mathbb Z^d \to \mathbb C$ denote a function in $\ell^p(\mathbb Z^d)$ where $p \in [1,2]$. Let $f : \mathbb T^d = (\mathbb R / 2\pi \mathbb Z)^d \to \mathbb C$ denote the inverse ...
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### Why is a lot of Fourier analysis done on an annulus?

I am studying harmonic analysis from these lecture notes and a lot of results and definitions always assume that the Fourier transform of a function has support in an annulus or a ball. The same ...
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### Fourier coefficients for $f(x) = 3 - \sin(3x)-\dfrac{1}{3}\cos(9x)$.

I am trying to find the fourier coefficients for $f(x) = 3 - \sin(3x)-\dfrac{1}{3}\cos(9x)$ I have understood that the overall period of the function is $\dfrac{2}{3}\pi$, and can due to formulas find ...
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### Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$

$\newcommand{\on}[1]{\operatorname{#1}}$ In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
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### What is the intuition for representations of the symmetric group?

What is the (physical) intuition for representations of the symmetric group? In particular, matrix coefficients of the Fourier coefficients corresponding to a representation. For the cyclic group $C_n$...
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### Recovering Fourier series coefficients from the Fourier transform of a function extended on the unit circle.

I'm working on a problem involving Fourier transforms and functions extended on the unit circle. Given a function $f(x)$, I'm considering its extension on the unit circle and its Fourier transform. ...
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### Confusion of phase spectrum definition of Fourier transform

As we know, the Fourier transform is generally in complex form say, $$X(\omega) = A + jB$$ The magnitude spectrum is without disputing $$|X(\omega)| = \sqrt{A^2+B^2}$$ But, how about the definition of ...
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### Evaluating the convolution of $e^{-at^2}$ and $e^{-bt^2}$ via Fourier transforms

Problem Statement: Use the convolution theorem on the function $f(t) = e^{-at^2}$ and $f(t) = e^{-bt^2}$, $a, b \in \mathbb{R}$. Calculate $(f \ast g)(t)$. I got a hint that I should first ...
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### Introductory questions to fourier transform: help with the visualization of integration (with respect to one axis) of a 2D function

I'm not very familiar with more advanced concepts, but I have a decent understanding of single variable calculus with real-valued functions $f: \mathbb R \to \mathbb R$ (via Spivak). I wanted to learn ...
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### Continuation of a function [closed]

if I have a function $f(t)=e^{it}g(t)$, where $t=x\cdot \xi$ for $x,\xi \in R^n$. What are the conditions for defining $e^{it}= f(t)/g(t)$. The function $g$ is zero on a set with zero measure. If you ...
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### Intuitively, why does $I(\lambda)$ decay as $\lambda \to \infty$ if $\Phi$ is not constant?

I'm quoting a few lines from Sogge's Fourier Integrals in Classical Analysis. Stationary phase is of central importance in classical analysis since integrals of the form I(\lambda) = ...
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### Spectral theory for this kind of operators

Let $a(x,\xi)$ be the symbol for a differential operator $$|\partial^{\alpha}_{x}\, \partial^{\beta}_{\xi} a(x,\xi)| \leq C (1+|x|+|\xi|)^{m-|\alpha|-|\beta|}$$ $|x|+|\xi|\geq c$. Is there a spectral ...
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### Equivalence of Fourier Transform on $\ell_2(\mathbb{Z}_+)$ and $L_2(\mathbb(R)_+)$ via equivalence of $H_p( \mathbb{D})$ and $H_p(\mathbb{C}_+)$?

Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients. We know that the Fourier Transform is an isometric ...
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### Weighted $L^2$ space on Torus.

I'm studying weighted $L^2$ spaces in the circle $[0,2\pi]$ Definition 1 A weight is a function $w\colon [0,2\pi]\to \mathbb{R}^+$ (non negative) Definition 2 The weighted $L_w^2([0,2\pi])$ is defined ...
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### Is there uncertainty principle for Fourier series?

I know there exists many types of uncertainty principle for Fourier transform. I tried to search but I couldn't find any such principle for Fourier series
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### Continuity of Fourier Transform of a function of Fourier Coefficients

Suppose that: I have a a periodic function $u(t)=u(t+2\pi)$ that can be represented by a vector of Fourier coefficients $\tilde{U}=[U_0,\,U_1,\,\dots,U_H]^T$. I have a continuous function $f(u)$ ...
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### The Fourier transform of product of derivatives

I have the task to compute the Fourier transform of the product in matlab: $$\left( \frac{\partial u(t, x)}{\partial x} \right)^2 \left( \frac{\partial^2 u(t, x)}{\partial x^2 } \right)$$ I was ...
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### Exercise about identity regarding a function in Schwartz space which Fourier coefficients are Fourier transforms of another function

The following is a simple yet challenging exercise about Fourier transforms. It is part of an older exam for my undergrad course in Elements of Functional Analysis. The professor usually comes up with ...
1 vote
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### general quadratic equation with variable coefficients

Consider an equation of the form $a(x)x^2 + b(x)x + c(x) = 0, a(x) \neq 0$. The solutions can be found from the equations $x = \dfrac{-b(x) \pm \sqrt{b^2(x) - 4a(x)c(x)}}{2a(x)}$ which doesn't ...
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### Finding a Closed Form Expression for a Distribution Defined by an Integral Involving Sine and Bessel Functions

I am seeking a closed form expression for the following distribution: $$D(t,x) = \int_0^\infty d\omega\, \omega^2 \sin(\omega t) J_0(\omega x),$$ where $J_0(x)$ is the Bessel function of the first ...
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### Prove that $F[f](y) = o(1/y)$

I'm reading by calculus lectures and there is the following task: How can we prove that if $f$ is even, strictly monotonically decreasing on $[0, \infty)$ and $f \in C^1(\mathbb{R})$ then it's Fourier ...
I want to calculate the Fourier transform of the function $f(t)$, defined as $f(t)=0$ if $t<0$ and $f(t)=J_{n}(t)$ if $t\ge0$, in which $J_{n}(t)$ is the Bessel function of the first kind. That is, ...