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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Invariant continuous linear maps on group algebras

Let $G$ be a locally compact group and $M(G)=C^*_0(G)$. Assume that $\Gamma$ consists of all linear maps $T$ on $M(G)$ with $T\mu=\mu*\nu$ for some $\nu\in M(G)$. Q. Any characterization of $\Gamma$ ...
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Basic doubt related to property of Fourier transform

Let $f\in L^1$ such that $\hat{f}\in L^1.$ $\hat{f}\in L^1\implies f$ is continuous. Define $g(x)=\begin{cases} 0 , \text{ if } x\in \mathbb{Q}\\ f(x), \text{ if } x\in \mathbb{Q}^c \end{cases} . $...
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The limit of the convergent Fourier series of $f$ is $f$

I have to try and solve the following question: Let $f$ be a continuous function on $[-\pi,\pi].$ Suppose that the Fourier series converges uniformly. Then its limit must be $f.$ MY ATTEMPT Say $\...
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Find Fourier series

I'm trying to show that the Fourier series of $f(\theta) = 0 $ if $|\theta| > \delta$ and $f(\theta) = 1- |\theta|/\delta$ if $|\theta| \leq \delta$ The Fourier coefficients is given by $a_n = 1/2\...
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Troubles finding the Fourier series of a sawtooth function plus a cosine function

I have a function $y(x) = y_1(x) + y_2(x)$ consisting of two other waveforms, where $ y_1(x) = \cos{\left(\dfrac{16 \pi}{5} x \right)}; \, y_2(x) = \displaystyle \sum_{k=-\infty}^{\infty} y_3(x - k); \...
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Distributional Fourier transform of $\arctan$

I understand that for a Schwartz function $f:\mathbb{R}\to\mathbb{C}$, its Fourier transform is another Schwartz function, while if $f$ was a bounded continuous function, then in general its Fourier ...
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Robust / continuous definition of the fundamental of a periodic signal

In a nutshell: Consider the space of functions from $I \mapsto \mathbb{R}$ where $I \subseteq \mathbb{R}$ with the $L2$ norm. I am looking for a function that maps those functions to probability ...
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Fourier transform of $\frac{1}{|x|^{d+a}}$ [closed]

I am interested in the solution of the following Fourier transformation $$\int \frac{1}{|x|^{d+a}}e^{-ixk}d^dx ,$$ considering a general $d$-dimensional system with $a\in\mathbb{R}_+$. How is this ...
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Complex conjugate of a complex function [closed]

Does just replacing the $i$ ( $=\sqrt{-1}$ ) by $-i$ everywhere give the complex conjugate of any complex number of a function? Will that be the same as changing the sign of imaginary part of the ...
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Fourier Transform FT

Two questions: Are there more than one definitions of the Fourier transform? Called $g(\omega)$ the FT of a real function $f(t)$, is $g(\omega) = g^{*}(- \omega)$? Thanks
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Is there a way to write down an expression for this 1d function, given the specified integration and convolution relationships?

I have two functions $n(x)$ and $g(x,y)$, and I would like to know if there is a way to express $p(x)$, given that the following relationships are true: $n(x)$ is the result of vertically integrating ...
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$2\text{D}$ Fourier Transform of Laplacian in polar coordinates

Consider a typical function written in standard $2\text{D}$ polar form: \begin{equation} f(\underline{r})=f(r,\theta)=\sum_{n=-\infty}^{\infty} f_n(r) e^{in\theta} \end{equation} executing the ...
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153 views

Can a Fourier Series converge absolutely to a non-continuous function?

I'm trying to prove for a specific function that its Fourier series does not converge absolutely for any $x$. But the Fourier series is rather messy and I was wondering if there's a way to prove that ...
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Will this heart rate waveform create an artifact in the frequency domain?

Below you will see a measurement of a human heart beat interval as a function of time. I take the Fourier transform of this data (but a much longer measurement, obviously). However, notice that the ...
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A bound of Cesàro mean of the Fourier series of bounded function

Let $f: \mathbb{T} \to \mathbb{R}$ with $m \leq f(x) \leq M$ for some $m, M \in \mathbb{R}$ and all $x \in \mathbb{T}$. We have, the $k$-th Cesàro mean of the Fourier series $f$: $$\sigma_k[f](x) = \...
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Fourier coefficients of $1_{[0,\pi]}$ and where the points converge to

Let $f(x) = 1_{[0,\pi]}$ where $f$ is defined on $(-\pi, pi]$. First, I'd like to compute the Fourier series of $f$. We have $$f(x) \sim \sum_{n \in \mathbb{Z}}\hat{f}(n)e^{inx}.$$ Then the Fourier ...
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Precise mathematical version of Shannon-Nyquist sampling theorem?

On Wikipedia, it gives the following statement for the theorem: If a function $x(t)$ contains no frequencies higher than B hertz, it is completely determined by giving its ordinates at a series of ...
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Asymptotic behavior of Fourier cosine transform $f(x)=\int_{0}^{\infty}\beta(t)\sin(\frac{t\pi}{2})\cos(tx)dt$

Suppose I have a function $f(x)$ : $$f(x)=\int_{0}^{\infty}\beta(t)\sin(\frac{t\pi}{2})\cos(tx)dt.$$ Assuming $\beta(t)\sim\displaystyle\frac{1}{t^{3/2}}$ for $t\to+\infty$ is it true that for $x\to0$...
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How to prove $\mathcal{F}(u\ast v)=\mathcal{F}(u)\mathcal{F}(v)$?

If $u,v$ are a couples of distributions with compact supports,namely $u,v\in S^{'} $,then How to prove that $$\mathcal{F}(u\ast v)=\mathcal{F}(u)\mathcal{F}(v)$$ $\mathcal{F} $ denote the Fourier ...
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How to prove that $ \lVert \Delta_q^{'}u\rVert_{L^p}\le C\lVert u\rVert_{L^p}$ is not true?

For $q\in \mathbb{Z}$,denote $$\Delta_q^{'} y:=1_{2^q\le |\xi|\le 2^{q+1}}\,(D)u$$ Prove that the inquality $$ \lVert \Delta_q^{'}u\rVert_{L^p}\le C\lVert u\rVert_{L^p}$$ for some constant $C$ ...
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Fourier dimension of the fat Cantor set

Let $C$ denote the fat Cantor set, which has positive Lebesgue measure but does not have an interior point. https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set What is its Fourier ...
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Proving Parseval's Theorem

My goal is to prove Parseval's Thoerem, that is: $$\frac{1}{T_0} \int \limits_{-T_0/2}^{T_0/2} |x(t)|^2~dt = \sum \limits_{n=-\infty}^{\infty} |c_n|^2$$ To do this, let's say I have a periodic signal $...
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STFT tensor product identity

Good evening, I'm trying to fill the proof of this lemma I found on the book "Modulation Spaces" by Bényi and Okoudjou: (Lemma 2.10) Let $F(x_1,x_2)=f_1(x_1)f_2(x_2)$ with $f_1,f_2\in S'(R^...
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Effects of Setting a Lower Bound on the Fourier Transform of a Function

We are concerned with a function in 2-D Euclidean space. $\theta: \mathbb{R}^2 \rightarrow \mathbb{R} \ $, $\theta \in L^1(\mathbb{R}^2) \cap L^p(\mathbb{R}^2) $, $p\in (2,\infty)$. As per the title, ...
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calculate Dirichlet integral get two different answer

When I calculate Dirichlet integral. $$\int_0^{+\infty}\frac{\sin x}{x}\text{d}x$$ this integral is converge. and Dirichlet Kernel:$$D_N(x)=\sum_{n=-N}^{N}e^{inx}=\frac{\sin\left(\left(N+\frac{1}{2}\...
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Fourier coefficient of $f(x) = e^{imx}$ and $g(x) = \cos(mx)$

Let $f,g$ be the functions that can be seen on the title. Let $m \in \mathbb{N}$. I need to $$S_k[f](x) = \sum_{n=-k}^{k}\hat{f}(n)e^{inx} \ \ \text{and} \ \ S_k[g](x) = \sum_{n=-k}^{k}\hat{g}(n)e^{...
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Why are the coefficients of the Fourier series called the frequency domain?

I'm trying to learn signals and systems on my own and the book I'm using refers to the coefficients of the Fourier series as the "frequency domain." So suppose we have a signal in time $x_1(...
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Is the $p$-norm of the fourier transform of a function greatest when its phase is constant?

Suppose $f\in L^2$ and define $g\in L^2$ by $g(x) = \lvert f(x)\rvert$. Based on numerical experiments I believe that $$ \Big\lVert\hat f\Big\rVert_p\leq\Big\lVert\hat g\Big\rVert_p$$ whenever $p\geq ...
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Uniform estimate on Schwartz functions away from support of Fourier transform

This question is a follow-up to this post, but hopefully it's a better attempt at formulating the same idea. Roughly speaking, I would like to obtain a uniform estimate on the size of a certain class ...
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63 views

How to prove $\sum_n e^{ik n}$ is a periodic sum of delta functions?

I want to show that $\sum_n e^{ik n}$ is an infinite periodic sum of delta functions, where $n$ were integers from $-\infty$ to $\infty$. I tried to manipulate the expression $\delta(x-a)=\frac{1}{2\...
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Fourier transform of $|t| \exp{(−|t|)}$

How can I calculate the Fourier transform of $|t| \exp{(−|t|)}$. Can somebody show me the way?
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Uniform estimate on Schwartz functions with compactly supported Fourier transform

Let $\mathcal{C}$ be the class of all even Schwartz functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the following conditions: The Fourier transform $\hat{f}$ is compactly supported; $f$ is non-...
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Lower Bound for Fourier Transform Expression

Let $f,g \in L_2(\mathbb{R}) $, and let $\hat{f}$ and $\hat{g}$ denote their Fourier Transforms. I'm trying to find a lower bound for this sum: $$ \left(\int_{-\infty}^{\infty} x^2 \, f^2(x) \, dx \...
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How to fix this argument that the periodic functions on $[-L/2,L/2]$ generate a dense subspace of $L^2[-L/2,L/2]$?

In V. Moretti's "Spectral Theory and Quantum Mechanics," example 3.32 (1), he attempts to show that the functions $$f_n(x)=\frac{1}{\sqrt{L}}e^{i\frac{2\pi n}{L}x},$$ where $n\in\mathbb Z,x\...
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Why can't we measure a monochromatic wave's frequency perfectly over a finite range?

In Fourier analysis it is said that the only way to have 0 uncertainty in a monochromatic wave's frequency is if you FT it over an infinite domain (otherwise it does not produce a delta function). ...
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Fourier transform on graphs

I would like to know by some (rather simple) real examples of graphs, how the Fourier transform processes the graph. Indeed, what exactly information is extracted (figured out) by computation of ...
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Is the partial sum of a Fourier series considered a orthogonal projection?

So my lecture notes claims that you can think of the partial Fourier series as a orthogonal projection of f onto a subspace spanned by the orthonormal basis vectors $e_{j}$ $f_{N}=\sum_{j-n}^{n}\left \...
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Convergence of the derivative in Schwartz space

Reading Grafakos' book "Classical Fourier Analysis", I got stuck in the Exercise $2.3.5.$ (a) which states that if $f ∈ \mathscr{S}(\mathbb{R}^n)$ then $$(\tau^{−he_j} f − f )/h\rightarrow \...
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Fourier shift theorem for cyclic permutation

Let $h:R^{d}\to R$. I am quite confused how to apply the Fourier shift theorem to $h(Px)$ where $P\in M^{d\times d}$ is a cyclic permutation acting on $x$: $\mathcal{F}(h(Px))(k)=?$ I guess some phase ...
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Sup Norm using FT

For some proper signal $x(t)$ the Fourier transform is given $$X(\omega)\equiv\frac{1}{\sqrt{2\pi}}\int dt x(t) e^{-i\omega t}$$ Is there a $simple$ way to write down the sup norm of the signal in ...
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Proof the bound for the Fourier coefficients: $|\hat{f}(n)| \leq Me^{-\delta|n|}$

There are some uncertainty regarding how to get this bound for Fourier coefficients. These are given: Let $\delta > 0$ and $A_{\delta} = A(e^{-\delta}, e^{\delta}) = \{z \in \mathbb{C} : e^{-\delta}...
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convergence involving Fourier transforms

Suppose we have a sequence of probability densities $\{\rho_n\}_{n\geq 0}$ on $\mathbb{R}^d$, whose corresponding sequence of Fourier transform $\{\hat{\rho}_n\}_{n\geq 0}$ (where $\hat{\rho}(\xi) := \...
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relation between fourier transforms of N series and the fourier transform of their combination

Assume there are $N$ series $X_{1},X_{2},X_{3}...X_{N}$ (Each series would be of the form $X_{k} = [(t_{1},p_{1}),(t_{2},p_{2})...,(t_{m},p_{m})]$ .Series could be non-uniformly spaced) and their ...
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55 views

Polynomial solutions to the heat equation

The well-known heat equation is given by $$ u_t - \Delta u = 0. $$ I was curious to know about polynomial solutions to this equation (of degree greater than or equal to 3). I've read the various ...
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Logarithmic integral of analytic function in upper half plane.

Let $f:[0,\infty)\to \mathbb{C}$ be a integrable function. For any $a\geq0,b\in (a,\infty],$ let us define for $z\in \mathbb{H}^+=\{Im(z)>0\}$ $$F(z)=\int_a^be^{itz}f(t)dt.$$ Then $F$ is analytic ...
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Lp Convergence of Fourier Integrals Using Hilbert Transform

I am reading about Hilbert transform and its application on Fourier analyisis, and I am triying to prove a statement given by Terence Tao in his notes on Fourier Analysis. He says that if $\varphi\in\...
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42 views

What can be said about functions that have Fourier coefficients $c_n=0$ after a certain $n$?

Let $e_n$ be an orthogonal basis of $L^2[a,b]$. Let $f:[a,b] \rightarrow \mathbb R$ be $L^1[a,b]$ such that all its Fourier coefficients in $e_n$ exist. The Fourier coefficients of $f$ in that basis ...
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What is wrong with this definition of Fourier coefficients?

something is wrong with what I am doing. I'm either: Using the wrong definition of something(but from Wikipedia, I don't think I am). Did some mistake in my calculations (also don't think so) So I ...
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Fourier coefficient of the dilation $f_m := f(mt)$

Let $f \in L^1(\mathbb{T})$ and $m \in \mathbb{N}$. Furthermore, define the dilation $f_m$ by $f_m(t) := f(mt)$ for any $t \in \mathbb{T}$. I would like to show that $$ \widehat{f_m}(n) = \begin{cases}...
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43 views

Proving $L^q(\mathbb{T}) \subseteq L^p(\mathbb{T})$ for $1 \leq p < q \leq \infty$

First, I'd like to address that this question is similar to " $L^p$ and $L^q$ space inclusion" or "Proving that $L^p \subset L^q$ when $1 \le q \le p$". But I was wondering if we ...

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