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.

learn more… | top users | synonyms

5
votes
3answers
251 views

A question related to Wave Equation

Let $L>0$. Suppose $f, g$ are $C^2$ functions on $\mathbb{R}$ such that $$f(t)+f(-t)+\int_{-t}^t g(s)\,ds=0$$ and $$f(L+t)+f(L-t)+\int_{L-t}^{L+t} g(s)\,ds=0$$ for all $t\in \mathbb{R}.$ Does it ...
1
vote
0answers
136 views

Eigenfunction of the $n$-torus

Let $\phi$ be an eigenfunction of the Laplacian $\Delta$ on the $n$-torus $T^n$, with eigenvalue $-\lambda$, i.e. $\Delta \phi + \lambda \phi =0$, then : $$ \phi (x)= \sum_{|n^2|=\lambda} \hat{\phi} ...
1
vote
0answers
235 views

Fourier Series on a 2-Torus

Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a ...
0
votes
1answer
55 views

Fourier series inequality with polynomial

I have the following question: Let f be in $\mathbf{L}_{\mathbf{R}}^2([-\pi;\pi])$. Show that $$\left({\int_{-\pi}^\pi |x^nf(x)|\,\mathrm{d}\lambda(x)} \right ) \leq \frac{2*\pi^{2n+1}}{2n+1} ...
3
votes
1answer
49 views

Analysis: Show that $f(x)$ converges, pointwise and uniformly on $\mathbb{R}$ to a differentiable function $f$ that satisfies:

Show that $$f(x) = \sum_{k=1}^\infty \frac{\sin kx}{k^3} \tag1$$ converges, pointwise and uniformly on $\mathbb R$, to a differentiable function that satisfies $$\int_0^{\pi/2} f(x)\,dx = ...
2
votes
4answers
436 views

Type of convergence of Fourier series of a function

For given function, $$f(x) = \begin{cases} 1, & |x|\leq\frac{\pi}{2} \\ 0, & \pi\geq|x|>\frac{\pi}{2} \\ \end{cases}$$ The calculated Fourier series is: $$\begin{align} a_0 &= ...
1
vote
1answer
88 views

Approximation using a Fourier transform with low pass filter

I need to approximate a function f, but I cannot do so with frequencies that exceed 1kHz What is the best approximation I can get? Is taking the Fourier transform then zeroing any term above 1kHz the ...
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 ...
2
votes
0answers
62 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
7
votes
2answers
201 views

Does $\sum_{n=0}^\infty\frac{\sin(2n+1)}{2n+1}=0$?

I've come to a bit of a sticking point in my answer to problem 14A given here http://www.maths.cam.ac.uk/undergrad/pastpapers/2011/ib/List_IB.pdf (note that this is a past paper that I am trying for ...
1
vote
1answer
84 views

$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0 }\sum_{n=-\infty}^{\infty}\delta f(\delta n)$?

Assume that $f$ is continuous and moderate decrease, show that $$\int_{-\infty}^{\infty}f(\xi)d\xi = \lim_{\delta \to 0,~ \delta>0}\sum_{n=-\infty}^{\infty}\delta f(\delta n)$$ From the ...
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 $$ ...
3
votes
1answer
818 views

Fourier and Legendre series

Find the Fourier sin series for the function $f(x) = x^3$ on the interval $0\leq x \leq L$. the Legendre series for the same function. One representation involves an infinite number of terms, ...
4
votes
1answer
301 views

Fourier Series and Inner Product

When working with Fourier series, the inner product is defined as $$\int_{-L}^L f(x)g(x)dx$$ I see this definition everywhere and we know that $\rm{sin}\big(\frac{n\pi x}{L}\big)$ and ...
1
vote
2answers
394 views

Trigonometric Identities and Fourier Series

I have the series: $$2+\sum_{m=1}^n 4(-1)^m\cos(m\pi x)$$ Here, $x\in (-1,1)$. I need to show that this equals some fraction with only cosine terms and $n$ (no $m$). Just looking for some ...
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?
11
votes
3answers
722 views

Fourier series of function $f(x)=0$ if $-\pi<x<0$ and $f(x)=\sin(x)$ if $0<x<\pi$

$$f(x) = \begin{cases}0 & \text{if }-\pi<x<0, \\ \sin(x) & \text{if }0<x<\pi. \end{cases}$$ My attempt: I went the route of expanding this function with a complex Fourier series. ...
2
votes
1answer
774 views

Fourier series representation for $\sin(x/2)$

Is there a faster approach for finding the Fourier series of $$\sin(x/2)~,\cos(x/2)~~~\text{etc}$$ other than the usual approach?
7
votes
5answers
2k views

The Fourier series $\sum_{n=1}^\infty (1/n)\cos nx$

The series $$\sum_{n=1}^\infty \frac{\sin nx}{n}$$ is the Fourier series of the odd $2\pi$-periodic extension of $(\pi-x)/2, 0<x<\pi$. My question is : $$\sum_{n=1}^\infty \frac{\cos nx}{n}$$ ...
1
vote
1answer
1k views

Odd harmonics only in Fourier transform

If a function is even or odd, it implies that there are respectively only cos and sine terms in its Fourier expansion. But is there a condition for a function to have an expansion with only odd or ...
1
vote
0answers
84 views

Prove complex Fourier Series in 2D

Prove the complex form of Fourier Series in 2Dimension from periodic function (period $2\pi$) in $x$ and $y$, defined in region $\Omega\subset\mathbb{R^2}$ $$f(x,y)\sim\sum_{-\infty}^{\infty} ...
1
vote
2answers
143 views

Series of Fourier coefficients

Let $\mathscr B=\{\phi_n\}_{n\in\mathbb N}$ be an orthonormal basis of the real Hilbert space $L^2([0,1])$, and given $f,g \in L^2([0,1])$ let $\langle f,g\rangle$ denote the usual scalar product ...
2
votes
1answer
727 views

Relationship between DFT index values, frequency in a Fourier series and Hz.

I have a sound file recorded at 44.1 K samples per sec, and some FFT and IFFT algorithms. The sound file is a vector with about $ 2^{17} $ elements. My objective is to find which of the index values ...
6
votes
2answers
494 views

How many ways to calculate: $\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}$ where $u \not \in \Bbb{Z}$

Today I have encounter a series: $$\sum_{n=-\infty}^{+\infty}\frac{1}{(u+n)^2}=\frac{\pi^2}{(\sin \pi u)^2}$$ where $u \not \in \Bbb{Z}$ . I have known a method to computer it (by Residue formula): ...
2
votes
2answers
283 views

Show that $f_n = \sin(n\pi x)$ for $n\ge 1$ forms an orthonormal set in $C[0,1]$ with respect to the $L^2[0,1]$ norm.

Show that $f_n = \sin(n\pi x)$ for $n\ge 1$ forms an orthonormal set in $C[0,1]$ with respect to the $L^2[0,1]$ norm. My attempt, $$\langle \sin(m\pi x), \sin(n \pi x) \rangle^2 = \int^1_0 ...
4
votes
1answer
113 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
1
vote
0answers
36 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
1
vote
1answer
52 views

Determining if $x\left(t\right)$ is real with Fourier series?

The question states that $x\left(t\right)$ has Fourier coefficients $a_k=\{x, k=0; j\left(\frac 1 2\right)^{|k|},k\neq0$. I am to determine whether $x\left(t\right)$ is real. Here is what I've done ...
0
votes
2answers
135 views

Help with some Fourier series questions

I need some help with this question about Fourier Series. 1) If $f\in{L_{1}(T)}$ (that's $f$ periodic with period $2\pi$ and $|f|\in{L_{1}([-\pi,\pi]}$)) with Fourier series ...
2
votes
0answers
138 views

question on fourier series

Given a function ${f}(x)$ , which is continuous in the region $M_1<x<M_2$. Let ${F}(x)$ be the representation of ${f}(x)$ in Fourier series , such that ${f}(x)\approx{F}(x)$ = ...
2
votes
1answer
445 views

FFT Algorithm for an interpolating polynomial

I'm trying to use the Fast Fourier transform algorithm to determine the trigonometric interpolating polynomial of degree $16$ for $f(x) = x^2\cos(x)$ on $[-\pi,\pi]$ I see a computer result in my ...
6
votes
2answers
539 views

How to visualise Fourier Transform of a function?

I solved many problems on Fourier series,transforms and inverse fourier transforms as part of my academics. And i am aware that FT converts a time domain signal to frequency domain and IFT is vice ...
3
votes
2answers
57 views

Fourier sums in cosine and sine and Borel resummation

is there a method to evaluate the fourier sums ?? $$ \sum_{n=0}^\infty t^n \sin(nx)= F(x,t) $$ $$ \sum_{n=0}^\infty t^n \cos(nx)= G(x,t) $$ my idea is that i need to use these sums to apply Borel ...
1
vote
1answer
60 views

Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?

In a section on fourier transforms, my textbook contains these steps for an example: $$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$ $$= ...
1
vote
2answers
72 views

Calculate Integral Using Fourier Series?

I got this integral that I have been asked to calculate: $\int_{0}^{2\pi} |3+4e^{10ix}+5e^{100ix}|^{2}dx$ I tried using Parseval's identity and tried to convert it to Fourier series. I think there is ...
1
vote
1answer
71 views

Let $\alpha$ be a real number, and let $g(\theta) = f(\theta - \alpha), \theta \in \mathbb{R}$. Find the Fourier series for g.

Suppose that $f(\theta)$ is a $2\pi$ periodic function with a known Fourier series. Let $\alpha$ be a real number, and let $g(\theta) = f(\theta - \alpha), \theta \in \mathbb{R}$. Find the Fourier ...
1
vote
1answer
73 views

Do we have a general form for this integral?

Is there a general formula or recursion for this integral? $$\int_0^1\left(\frac{\arcsin x}{x}\right)^n\text{d}x,\ \ n\in\mathbb{N}$$
1
vote
0answers
129 views

Probability density function of $A = B + C$ via Joint Characteristic function of $B$ and $C$

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
2
votes
2answers
186 views

A integral with polygamma

I was doing a integral, the last part is $$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$ I ran this on Maple, it turns into polygammas...How we evaluate this? I think there should be a way to evaluate ...
3
votes
2answers
178 views

Another integral with Catalan

Show that: $$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$ I evaluated this by some Fourier series. Is there any other method? Start with substitution of ...
0
votes
1answer
80 views

Need help with a integral

I was evaluating $$ \int_{0}^{^\pi/_2}x\ln\left(\vphantom{\large A}\cos\left(x\right)\right)\,{\rm d}x $$ I like to try with the fourier series $$ \int_{0}^{^\pi/_2} \left[\,\,\sum_{k = 1}^{\infty} ...
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 ...
4
votes
1answer
435 views

proof of Poisson formula by T. Tao

I do not understand one thing in an article on the blog of Terence Tao: For instance, restricting a function $f: G \rightarrow \mathbb{C}$ to a subgroup $H$ causes the Fourier transform $\hat f$ ...
2
votes
1answer
77 views

What is the degree of the fourier expansion

Let $ f:\{-1,1\}^3 \rightarrow \{-1,1\} $ , $f(x)= \operatorname{sgn}(x_1+x_2+x_3)$; (Majority function), then Fourier expansion of $f$ is $f(x)= \frac{1}{2} ...
2
votes
2answers
410 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
1
vote
0answers
281 views

Calculation of the Fourier series of $f(t)=e^{it\alpha}$ and $f(t)=|t|$

I have to compute the fourier series of these 2 functions: 1 . $$f(t)=e^{ita}\text{ for }-\pi < t < \pi ;\qquad a \in \mathbb{R}\backslash \mathbb{Z}$$ 2. $$f(t)=|t| \text{ for }-\pi < t ...
5
votes
2answers
283 views

Is my Fourier Series computation done correctly?

See my fourier series calculation of this function if you please! $ f(t)=\left\{\begin{array}{ll} 0, & \text{for } \ -\pi<t<0 \\ 1, & \text{for } \ 0 < t < ...
0
votes
1answer
114 views

A small question on fourier series

Why the series is divergent, but the equation holds? $$\sum\limits_{i=1}^{\infty }{\sin kx}=\frac{1}{2}\cot \left( \frac{x}{2} \right)$$
0
votes
2answers
70 views

Fourier analysis with a changing and continuous period

Traditional Fourier analysis picks a period and then describes a function as: $$f(x) = \frac{1}{2} a_0 + \sum_{k=1}^\infty\, (a_k \cos{(\omega \cdot kx)} + b_n \sin{(\omega \cdot kx)})$$ I am ...
2
votes
0answers
25 views

Diagonable Kernels over a Riemannian Surface

This question is motivated by this paper. There, they develop a stippling method which requires a kernel to be diagonal. Meaning a symmetric bilinear function $K\colon \chi\times \chi\to \mathbb{R}$ ...