Skip to main content

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).

Filter by
Sorted by
Tagged with
2 votes
1 answer
25 views

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 ...
daㅤ's user avatar
  • 3,294
0 votes
1 answer
41 views

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 ...
Diffusion's user avatar
  • 5,591
0 votes
0 answers
8 views

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 ...
Duomo Feng's user avatar
0 votes
1 answer
42 views

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 ...
RunningMeatball's user avatar
4 votes
1 answer
64 views

Is it true that $\widehat{(\delta_{x_{0}}\otimes T)} = \hat{\delta}_{x_{0}}\otimes \hat{T}$?

For a fixed $x_{0} \in \mathbb{R}$ consider the Dirac delta distribution $\delta_{x_{0}}$. Its Fourier transform is given by $\hat{\delta}_{x_{0}}(p) = e^{-px_{0}}$, in the sense that $\hat{\delta}_{...
InMathweTrust's user avatar
-1 votes
0 answers
44 views

Integral representation of Dirac delta? [closed]

Is there any integral representation of the Dirac delta distribution as follows: $$\delta(x-x_{0}) = \lim_{n\to \infty}\int dx f_{n}(x-x_{0})\frac{1}{\sqrt{|x|^{2}+m^{2}}}$$ for a fixed parameter $m &...
Idontgetit's user avatar
  • 1,929
0 votes
0 answers
79 views

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 ...
CBBAM's user avatar
  • 6,275
-1 votes
0 answers
35 views

Discrete Fourier Transform: choice of basis [closed]

I have two sets of N real numbers $\{E_m\}_m$ and $\{t_j\}_j$. I impose the following conditions: $\frac{1}{N} \sum_{m=1}^N e^{-i\,E_m(t_j-t_k)}=\delta_{jk} \hspace{1cm} \forall j, k$. $\frac{1}{N} \...
BlockSlicer's user avatar
1 vote
0 answers
47 views

Fourier transform of real exponential

I'm currently reading Zworski's semiclassical analysis and have some question regarding the following example: In the second equality, I verified the computations by expanding the inner products, and ...
mtcicero's user avatar
  • 529
3 votes
0 answers
74 views

$L^2$ vs $L^\infty$ projection

Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is $$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$ where $d_\...
Davide Maran's user avatar
  • 1,199
0 votes
0 answers
46 views

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 ...
Martha's user avatar
  • 1
2 votes
0 answers
131 views

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 ...
Siegfriedenberghofen's user avatar
3 votes
0 answers
45 views

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 $C_n$...
Jackson Walters's user avatar
1 vote
0 answers
26 views

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. ...
Dr Potato's user avatar
  • 812
0 votes
0 answers
13 views

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 ...
MathArt's user avatar
  • 1,329
4 votes
1 answer
73 views

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 ...
math123's user avatar
  • 21
1 vote
1 answer
68 views

Calculating $\int_{-\infty}^{\infty}e^{ixt}\cos(at)e^{-\frac{t^2}{2}}\,dt$ and $\int_{-\infty}^{\infty}\cos(at)e^{-\frac{t^2}{2}}\,dt$ [closed]

Problem Statement: Derive the formulas: $$\begin{align*} \mathcal{F}\{\cos(at)f(t)\} &= \frac{1}{2} \left( F(x + a) + F(x - a) \right)\\ \ \mathcal{F}\{\sin(at)f(t)\} &= \frac{1}{2i} \left( F(...
user1718's user avatar
2 votes
0 answers
18 views

Sum of Dirichlet kernel for angle differences over $n$ angles on unit circle

Let $$D(\theta_i-\theta_j):= \frac{\sin((n+1/2)(\theta_i-\theta_j))}{2\sin\frac{\theta_i-\theta_j}{2}}$$ being the Dirichlet type of kernel of angle difference between $\theta_i$ and $\theta_j$ where $...
chloe's user avatar
  • 1,052
1 vote
0 answers
30 views

Fourier Transform of product of unit vectors

I have a general integral of the form $$\int_0^{2\pi} d\phi e^{-i F \cos(\phi)} \left(\cos(\phi) \hat{x}+ \sin(\phi) \hat{y}\right)^m$$ That is, in the end, I get bunch of tensors with chains of $\hat{...
Quantization's user avatar
2 votes
0 answers
28 views

Definite integral of Modified Bessel function, exponential and trigonometric functions

I am trying to solve the following integral; $$ \int_{0}^{\frac{\pi}{2}} e^{\gamma \cos\theta} I_{1}(\epsilon\sin\theta)d\theta,$$ where $\gamma\in\mathbb{R},\epsilon\in\mathbb{R}^{+},$ and $I_{1}$ is ...
Nelly Clark's user avatar
4 votes
1 answer
91 views

Is the Short-Time Fourier Transform an Isometry in $L_2(\mathbb{R}^d)$?

I have been studying the Fourier transform and came across an interesting property that it acts as an isometry in $L_2(\mathbb{R})$. Specifically, given a function $f \in L_1(\mathbb{R})$, the Fourier ...
Mark's user avatar
  • 7,880
1 vote
1 answer
81 views

Estimate of Fourier coefficients of $x^{-1/4}$

I'm studying whether the function $f(x)=x^{-1/4}$ on $[0,1]$ has $p$-summable Fourier coefficients for some $1<p<2$, i.e., $(\widehat{f}(n))_{n\in \mathbb{Z}}\in \ell^p(\mathbb{Z}).$ Apparently, ...
Roddick Yu's user avatar
2 votes
0 answers
50 views

Rate of Uniform Convergence of Fourier Series to a Smooth Function?

I'm wondering if there are any known results on the rate of uniform convergence of a Fourier partial sum to a smooth function ?. More specifically, I am wondering ...
user avatar
0 votes
0 answers
16 views

FFT for the Estimation of Power Spectra (Welch's Method) - DFT Definition

I was reading Peter Welch's famous paper "The Use of Fast Fourier Transform for the Estimation of Power Spectra: A Method Based on Time Aver. aging Over Short, Modified Periodograms" the ...
Rouben's user avatar
  • 1
1 vote
0 answers
53 views

About a calculation in Grafakos' Classical Fourier Analysis.

I'm reading Grafakos' book on Fourier Analysis and at some point he says "There is an analogous calculation when $g$ is the characteristic function of the unit disk $B(0, 1)$ in $\mathbb{R}^2$. A ...
huh's user avatar
  • 464
2 votes
1 answer
88 views

If we know the even fourier series are we able to find the odd version?

If we have a sequence, call it $a_{n}$ where $n>0$, and we take it as the fourier coeficents of an even function... $$ F_{e}(x) = \sum^{\infty}_{n=1}a_{n}\cos(2\pi nx) $$ ... and we know the form ...
Aidan R.S.'s user avatar
4 votes
1 answer
172 views

Bound of an integral function

Let $$f(r):=\int_{\mathbb{R}^d}\left|\int_{\mathbb{R}^d}e^{2\pi i \langle x,y\rangle}e^{-|y|^4+r^{1/2}|y|^2}dy\right|dx,$$ for $r\geq 0$. Is it possible to uniformly bound $f$ on $\mathbb{R}_+$? i.e. $...
mathex's user avatar
  • 616
0 votes
1 answer
37 views

Proving that sine series is complete [closed]

Let's consider the following family of functions $$ S = \{\frac{2}{\pi} \sin nx \}_{n \in \mathbb{N}}. $$ Of course for each $n \in \mathbb{N}$ we have $s_n \in L^2(0, \pi)$ for $s_n \in S.$ I would ...
Hendrra's user avatar
  • 3,036
0 votes
0 answers
21 views

Use a Riemann sum to approximate the integral $\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot x}-1)dx$ in 1d

Consider the function $f:[-1,1]\setminus\{0\}\to \mathbb{R}$ given by $f(x)=\frac{1}{|x|^{2.5}}.$ For dimension $d=1,$ Consider the integral below: $$\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot ...
Chang's user avatar
  • 329
1 vote
0 answers
24 views

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 ...
S.C.'s user avatar
  • 5,064
1 vote
0 answers
64 views

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 ...
Aristo's user avatar
  • 61
2 votes
1 answer
69 views

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 \begin{equation} I(\lambda) = ...
stoic-santiago's user avatar
-1 votes
0 answers
18 views

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 ...
zoran's user avatar
  • 127
0 votes
0 answers
14 views

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 ...
travelingbones's user avatar
0 votes
0 answers
38 views

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 ...
eraldcoil's user avatar
  • 3,650
0 votes
1 answer
55 views

Is convolution theorem on $l^2(\mathbb{Z})$ valid?

I have a doubt about Fourier transform $F:L^2([0,2\pi])\to l^2(\mathbb{Z})$. If $f,g\in l^2(\mathbb{Z})$ then $f*g\in l^2(\mathbb{Z})$, then, $\mathcal{F}^{-1}(f*g)\in L^2([0,2\pi])$. Question $\...
eraldcoil's user avatar
  • 3,650
1 vote
0 answers
31 views

Small Question from the Proof of Poisson Summation

Let $f$ be a Schwartz function on $\mathbb R$. I want to prove that $$\sum_{n \in \mathbb Z}f(n) = \sum_{n \in \mathbb Z}\widehat{f}(n)$$ The beginning of my proof goes like this: the decay property ...
Johnny Apple's user avatar
  • 4,429
0 votes
0 answers
39 views

Fourier analysis of sine: magnitude result clarification

I want to calculate the Fourier transform of $sine(2 \pi 60 t)$ and I have difficulty understanding/calculating the magnitude result. It is already known that the only frequency on this sine function ...
Christianidis Vasilis's user avatar
2 votes
2 answers
78 views

Exercise 1.3 in Anton Deitmar‘s "A first course in Harmonic Anaylsis" , 2nd Edition

I am learning the Fourier Analysis, but I don't how to solve the following problem: Let $C(\mathbb{R}/\mathbb{Z})$ contains all periodic, continuous, complex-valued functions on $[0,1)$ Let $f \in C(\...
Leaves's user avatar
  • 83
1 vote
0 answers
36 views

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
math_boy's user avatar
0 votes
0 answers
27 views

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 $f(u)$ ...
Nidish Narayanaa's user avatar
0 votes
0 answers
80 views

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 ...
unknown's user avatar
  • 391
0 votes
0 answers
29 views

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 ...
propriofede's user avatar
1 vote
0 answers
48 views

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 ...
H-a-y-K's user avatar
  • 729
1 vote
1 answer
39 views

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 ...
Adam Teixidó Bonfill's user avatar
0 votes
0 answers
32 views

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 ...
myfakeaccount's user avatar
2 votes
1 answer
96 views

Convert the Laplace transform of the Bessel function to a Fourier transform

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, ...
Lucas Bitencourt's user avatar
0 votes
0 answers
31 views

Using complex phases to do linear regression

The following is non-standard but was interested to see if there is value in following this path. Consider a linear regression problem without intercept, so simply, $y=ax$ and some data is provided $(...
play's user avatar
  • 307
1 vote
0 answers
29 views

How to prove that sum of infinite complex harmonics makes a continuous time impulse function?

I was trying to substitute complex fourier series coefficients into complex fourier series formula and I came up with this identity in order to get the same function back. How can I prove that this ...
U.AL's user avatar
  • 57
1 vote
1 answer
72 views

A special kind of Poisson summation formula

There is a Poisson summation formula as follows: Let $V$ be a smooth function with compact support on $\mathbb{R}$. For $X > 1$ and $q > 1$, we have $$\sum_{n\equiv a~\mathrm{ mod }~q}{V}\left( ...
XUSEN's user avatar
  • 39

1
2 3 4 5
211