Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or 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
79 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
5
votes
0answers
50 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
135 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 ...
5
votes
0answers
183 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
317 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
273 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
52 views

How to find Green's function using Fourier-Bessel expansion

The Green's function satisfies the non homogeneous Bessel equation can be written as $xg''+g'+\left(k^2x-\frac{m^2}{x}\right)g=-\delta(x-\xi)$ where $m\geq0$ and an integer. The boundary conditions ...
4
votes
0answers
57 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
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0answers
976 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
votes
0answers
18 views

On converting from real to complex Fourier series

Let a real-valued function $f$ be defined as following: $$f\left ( x \right )=\left\{\begin{matrix} 2k-x, x\in\left [ 2k-1,2k \right ) & \\ x-2x , x \in \left[ 2k,2k+1\right )& ...
3
votes
0answers
69 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
3
votes
0answers
65 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
3
votes
0answers
38 views

For which algebras Taylor series and the Fourier series can be generalized?

I'm not a professional mathematician. The question is in the title. But most of all I'd like to know about this for quaternions algebra with non commutative multiplication. I'd like to know about ...
3
votes
0answers
30 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
3
votes
0answers
27 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
34 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
67 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
117 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
71 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
40 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
102 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
27 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
votes
0answers
99 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
3
votes
0answers
63 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
92 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
68 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
473 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
61 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
33 views

Characterize a large class of shapes using a finite number of parameters

I am doing some numerical computations searching for an optimal shape for a certain functional. In my particular case, the shape $\Omega$ is a 2 dimensional star shaped domain by the origin, which ...
3
votes
0answers
257 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
267 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
399 views

Is this sum equal to the Möbius function?

In the wikipedia page Uses of trigonometry under the section Number theory and in the page for the Möbius function there is an explanation for how to calculate the Möbius function from the GCD=1 ...
3
votes
0answers
166 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
215 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
19 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
2
votes
0answers
26 views

Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
2
votes
0answers
39 views

Computing Fourier Series coefficients

Hello I have to calculate the Fourier series coefficients for the following function: $$f(t)=\sum_{n=-\infty}^{+\infty} \Pi(\dfrac{t-nT_o}{T_o/2})$$ where "$\Pi$" indicates the rectangular function. ...
2
votes
0answers
20 views

Drawing a continuous function with divergent Fourier series at $x=0$…

Does anyone know how the graph looks like for a continuous function with Fourier series diverging at $x=0$ ? The example due to Fejer (a variation of the du Bois-Reymond construction), is explicitly ...
2
votes
0answers
18 views

half range fourier series, even and odd extension

Hello, I have some problems understanding what is above on the image. Firstly, he defines an "odd extension" of any function. I don't really understand what this means, how is it an "odd extension" ...
2
votes
0answers
29 views

Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
2
votes
0answers
41 views

Fourier Series of the batman equation

I want to represent the batman equation as a Fourier Series. (I got the equation here : Is this Batman equation for real?) But a part of it is an ellipse and when I tried to calculate an the integral ...
2
votes
0answers
24 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
2
votes
0answers
39 views

Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge(fourier series coefficients)?

Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$ $S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f. Where ...
2
votes
0answers
56 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
2
votes
0answers
27 views

Prove $\hat{f}(\omega)\neq 0$ if $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete

Let $f\in L^1(\mathbb{R})$. s.t $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete. Prove that $\hat{f}(\omega)\neq 0$ for all $\omega\in\mathbb{R}$ Suppose the system is complete for any $g\in ...
2
votes
0answers
52 views

Fourier series using Bessel function

so Im stuck on the following problem; Use the identity $\exp(ix\sin\theta) = \sum\limits_{k=-\infty}^\infty J_k(x)\exp(ik\theta)$ to find the Fourier series of $\cos(\theta + 4\sin\theta)$, where ...
2
votes
0answers
45 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
2
votes
0answers
35 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
2
votes
0answers
65 views

Functions over a finite domain that cannot be represented by Fourier series

Given a double Fourier series for some $f:[0,L]\times [0,R]\to \mathbb{R}$ of the following form $$\sum_{k,l=0}^{\infty}a_{kl}\cos\left(\frac{2k\pi ...