Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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6
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0answers
250 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
5
votes
0answers
41 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
5
votes
0answers
170 views

Expansion in Fourier series involving a complicated “argument”

I know how to expand a function $f(x)$ into a Fourier series with the period $2L$: $$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(n\pi x/L)+\sum_{n=0}^\infty b_n\sin(n\pi x/L),$$ but what if I ...
5
votes
0answers
308 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
5
votes
0answers
265 views

What are the conditions sufficient and necessary on $g(t)$ for the Dirichlet integral to be equal to $\frac{\pi}{2} g(0+)$?

Dirichlet Integral of a function $g\colon \mathbb{R} \to \mathbb{R}$ is defined as $$ DI(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ assume $\alpha \in \mathbb{N}$ For the equality ...
4
votes
0answers
50 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
4
votes
0answers
117 views

Prove a function has $k$ continuous derivatives from its Fourier series

Here is the problem. Let $k\in \mathbb{N}$. Suppose that there is a constant $C$ such that $|c_n|<\frac{C}{|n|^{k+1}}$ ($c_n$ here is the $n$th Fourier coefficient). Prove that ...
4
votes
0answers
2k views

Finding coefficients of a double Fourier series

This is the end of a PDE (heat equation in 2D) I am trying to solve with bounds from $0 < x < L$ and $0 < y < H$. It is a Newmann condition problem (i.e. all derivatives of $x$ and $y$ at ...
3
votes
0answers
21 views

Local behavior of a Fourier series and a intgral

So I have to calculate an integral that involves a Fourier series of some function. I would like to get some kind of local control of the function near zero the series is ...
3
votes
0answers
31 views

Discrete Fourier Transform of a shift of a tuple over a finite field

Let $a = a_0 a_1 \cdots a_{N-1}$ be a sequence over a finite field $\mathbb{F}_q$, where $N \mid q^n-1$ for some $n$. Let $\xi_N$ be a primitive $N$-th root of unity in the extension ...
3
votes
0answers
47 views

Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
3
votes
0answers
87 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
3
votes
0answers
63 views

Are there any new research results on approximating Riemann $\Xi(z)$ by a Fourier transformation

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$$ The functional equation for $\zeta(s)$ is equivalent ...
3
votes
0answers
33 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
3
votes
0answers
69 views

Inverse Fourier transform on infinite series

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. I have derived that $$\hat{f}(y)=\sum_{n=-\infty}^\infty f(n)1_{[-\pi,\pi]}(y)e^{-iny}$$ in $L^2$ convergence. Let ...
3
votes
0answers
26 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
3
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0answers
57 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
3
votes
0answers
86 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
3
votes
0answers
60 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
3
votes
0answers
667 views

Fourier Transform of spherical harmonics

I am trying seeking for definition (or some source) of the Fourier Transform of Spherical Harmonics (see https://en.wikipedia.org/wiki/Spherical_harmonics). Any help will be really appreciated. ...
3
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0answers
396 views

Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
3
votes
0answers
58 views

A Fourier series $\frac{1}{1+t^2}$

What is the Fourier series of the function $$ f(t) = \frac{1}{1+ a t^2}$$ over $[0,1]$, where $a >0$ is some constant? I mean, are the coefficients known?
3
votes
0answers
235 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

By analogy with Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ ...
3
votes
0answers
193 views

How is study of fractals related to fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
3
votes
0answers
161 views

Uniform convergence of a series

This problem came from the Krantz text ($2^{nd}$ ed. ch. 9, prob. 17): Prove that the series $\displaystyle\sum_{j=1}^{\infty }{\frac{\sin{(jx)}}{j}}$ converges uniformly on compact intervals that do ...
3
votes
0answers
204 views

Fourier Series generated by a function and periodic with $2\pi$

Consider the Fourier series (in exponential form) generated by a function $f$ which is continuous on $[0,2\pi]$ and periodic with period $2\pi$ , say $$f(x)\sim\sum_{n = - \infty }^{+ \infty }\alpha ...
2
votes
0answers
56 views

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB?

How to find the Total Harmonic Distortion of a Periodic Signal through MATLAB? I just need help in confirming if my way of approach to finding the THD seems valid, I'm new to MATLAB so I'm not quite ...
2
votes
0answers
31 views

Accessible textbook about basic Fourier analysis in terms of integrals wrt measures

I am looking for a basic and accessible textbook (or set of lecture notes) that discusses basic fourier analysis but in terms of measures and integrals with respect to measures. Not sure if it is done ...
2
votes
0answers
30 views

Can the Fourier Series be made “ Shorter ”?

I have tried to give only the intuitive part of my question and haven't included many specific details. Please help me frame it more precisely. I have inluded the symbol (*) where I need more details. ...
2
votes
0answers
113 views

Prove that periodic analytic function can be written as $\sum_{-\infty}^{\infty} c_n e^{2\pi inz}$

This question involves the following homework problem: PROBLEM Suppose $f$ is analytic in the upper half plane and periodic of period 1. Show that $f$ has an extension of the form ...
2
votes
0answers
46 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...
2
votes
0answers
33 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
2
votes
0answers
23 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
2
votes
0answers
22 views

Fourier Series from product of to functions

I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ...
2
votes
0answers
42 views

Fejer's Theorem in relation to the Fourier Transform

I have this question that relates the Fejer theorem with the Fourier Transform. Any help would be appreciated. If $f$ is of moderate decrease then $$\int_{-R}^{R}\left(1-\frac{|\xi|}{R}\right) ...
2
votes
0answers
45 views

Show sum involving sines is non-negative

I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation} How should I start? I ...
2
votes
0answers
33 views

This $\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$ integral: does a closed form exist?

$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$$ Does a closed form for the above exist, ideally for $n,m\in\mathbb{C}$ (most bounds probably removed at one point using ...
2
votes
0answers
17 views

Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
2
votes
0answers
23 views

Inequality on $L_1$ norms of tirgonometric polynomials generated with a smooth function

Let $\varphi\in C_0^\infty(\mathbb R)$ and for $n\ge1$ $$ f_n(x)=\sum_{k=-\infty}^\infty \varphi(k/n)e^{i k x}. $$ I seem to remember that there is an inequality $\|f_n\|_{L_1(\mathbb T)}\le C$, where ...
2
votes
0answers
43 views

Phase speed of 2D wave

I'm a little stuck with understanding the properties of 2D waves. I have the wave $e^{2\pi i(jx+ky-\omega_{j,k}t)}$=$\exp\left(2\pi i\left(\left[\begin{array}{l} j \\ k ...
2
votes
0answers
34 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

Let $K$ be a fixed integer, and $\mathcal{F}$ the set of trigonometric polynomials with at most $K$ nonzero terms. Let $(f_n)$ be a sequence in $\mathcal{F}$ converging pointwise (on $\mathbb{R}$) to ...
2
votes
0answers
18 views

Extract left- and right-going waves from FFT

I am looking for a way to get information on the direction a wave is traveling. What I have got is a real function $\Phi(x,t)$ and it's time derivative $\dot{\Phi}(x,t)$ that obeys the wave equation ...
2
votes
0answers
116 views

Explicit example of a divergent Fourier series?

I've been reading Widder's Advanced Calculus text, which says that there are some continuous functions that have divergent Fourier series, which are summable to the function (C, 1). I'd greatly ...
2
votes
0answers
35 views

Fourier series/Limit

Assume $f$ satisfies the assumptions of Dirchlet's Theorem - i.e. $f$ is a piecewise continuous complex function that has one-sided derivatives at each point in $[-\pi,\pi]$. Determine the following ...
2
votes
0answers
57 views

Fourier Series, Parseval Identity

I need to prove $$\sum_{n=1}^{+\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan(\alpha\pi)},$$ with $\alpha$ a non integer complex. I know that I have to use the Parseval's ...
2
votes
0answers
42 views

Seeking better understanding of Fourier transform?

I'm quite confused on the one part of the Fourier transform. I don't understand what is the term $\left(u*x + v*y \right)$ mean. I mean $u$ and $v$ are the axis for frequency domain and $x$, $y$ are ...
2
votes
0answers
92 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
2
votes
0answers
61 views

Need help on computing odd, even extensions of a function

OK I am going over d'Alembert solutions. And I came across the following example. $$ f(x) = \begin{cases} \frac{3}{10}x &0 \le x \le \frac{1}{3} \\ \frac{3(x-1)}{20} & \frac{1}{3} \le x \le ...
2
votes
0answers
56 views

Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
2
votes
0answers
57 views

Harmonic Oscillator and Fourier Series

I am currently studying Fourier Series (on my own). I am using a few different references/sources. Some are more trying to give an intuition about Fourier Series and others are more rigorous. ...