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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$f$ is smooth and periodic function,$\exists \lambda$ such that $f^{(4)}=\lambda f$ prove:$\exists \lambda$ such that $\lambda= (\frac{2\pi n}{T})^4$

Given $f:\mathbb{R} \rightarrow \mathbb{C}$ is smooth and periodic with period $T>0$ and exists $\lambda \in \mathbb{C}$ such that $f^{(4)}(x)=\lambda f(x)$ for any $x \in \mathbb{R}$. prove: ...
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0answers
51 views

Generalized Fourier series in $L^2$ that do not converge pointwise a.e.

For a Hilbert space $L^2$ we have the notion of an orthonormal basis $\{f_j\}$ being a sequence of orthonormal elements such that any element $f$ in $L^2$ can be approximated by partial sums in terms ...
2
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1answer
44 views

Identity of $\coth $ using Fourier series

The exercise wants me to prove the identity $$\pi \coth \pi a= \frac{1}{a}+ \sum_{n=1}^{\infty}\frac{2a}{n^2+a^2}$$ using the Fourier series of $\cosh ax, \; x \in [-\pi, \pi], \; a \neq 0$. ...
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1answer
41 views

Can a sine wave be expressed as a sum of square waves.

The opposite is possible, throught simple fourier analysis. For this question, suppose we have the periodic functions square wave functions: $f(x) = \begin{cases} 1 & 0\leq x < 1 \\-1 & ...
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2answers
27 views

Prove that orthonormalsystem is an orthonormalbasis

We have an orthonormalsystem in $L^2(0, 2\pi)$: $\{e^{ikx} : k \in \mathbb{Z}\}$. Now I want to show that it's also an orthonormalbasis. I thought the easiest way to do that would be to show that ...
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2answers
50 views

How is the Fourier transform a geneeralization the the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message the the latter is a genaralization of the first. I know that the idea that the Fourier ...
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0answers
31 views

Fourier series & Fourier transformation [on hold]

Tell me where we use Fourier series & transform in real life? Please mention an example problem which helps me to understand easily about Fourier series &Fourier transformation?
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3answers
51 views

Using complex variables to find sums of Fourier series

Use complex variables to find the sum of the Fourier Series: $$\sin(\theta) + \frac{\sin(2\theta)}{2^{2}} + \frac{\sin(3\theta)}{2^{3}}+\cdots$$ where $\theta$ is a real variable.
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1answer
32 views

Using Hölder condition to find upper bound on Fourier coefficients

First I want to stress that I don't want an answer, perhaps a hint. Let $f(x)$ have period $2\pi$ and let $|f(x) -f(y)| \leq c|x-y|^{\alpha}$, for some constants $c$ and $\alpha$ for all $x$ and $y$. ...
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1answer
30 views

Fourier transform of a pyramid

Has anyone calculated the Fourier coefficients for a pyramid function? Let us define the pyramid function as, z = f(x,y). We are looking at 5 planes making up the pyramid. The 4 base points and apex ...
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1answer
35 views

Fourier series of constant on $2\pi$ intervals

I want to find a fourier expansion of only sines representing $g(x) = 1$ on the interval $[0, \pi]$. So I extend the function on $[-\pi, \pi]$ such that it is odd, and calculate $$b_k = \frac 1\pi ...
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4answers
380 views

Sum function of a series

Does anyone know what is the sum function $f(x)$ of the series $\displaystyle\sum_{n=1}^\infty \frac{\cos(nx)}{n^2}$? I have no idea how to find a sum function... Any help would be appreciated.
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2answers
119 views

Why is periodic harmonic analysis only possible with sines?

This paper shows that if we consider odd functions on $(-\pi,\pi)$ in $L_2$, then the only $2\pi$-periodic function $f$ for which $f(nx)$ is a complete orthogonal system is the sine function. I'll ...
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1answer
61 views

Convergence of the series $\sum_{n=0}^\infty \sin(n! \pi m \sin(1))$

In this exercise I was asked to prove the convergence of the following infinite sum: $$\sum_{n=0}^\infty \sin(n! \pi m \sin(1)),$$ where $m$ denotes any integer. I don't have any idea on how to ...
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1answer
31 views

Inverting a differential operator on $C^\infty$ functions using Fourier series

I am working on the following problem: Let $S^1 := \mathbb{R}/2\pi \mathbb{Z}$ and suppose $p(x) = a_0 + a_1x + \cdots + a_kx^k$ is a polynomial such that for all $n \in \mathbb{Z}$ we have $p(in) ...
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3answers
86 views

Is there a direct method for evaluating this integral: $\int_{0}^{2\pi}\ln^2(2\sin(\frac{x}{2}))dx$?

I stumbled upon this integral while attempting to evaluate $\sum_{n=1}^{\infty}\frac{\cos(n\theta)}{n}$. I started with the series $-\ln(1-z)=\sum_{n=1}^{\infty}\frac{z^n}{n}$, replaced z with ...
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2answers
43 views

how to solve this complex exponential integration ??

During exercising and example of Fourier Series , I encountered with an integration : $$ \frac{E\omega_o}{4\pi j}\int_{0}^{\frac{\pi}{\omega_o}}\Big[e^{-j\omega_o (n-1)t}-e^{-j\omega_o ...
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0answers
20 views

The size of the set of continuous function of periode T

I have a naive question. The Fourier series give an injection between continuous function of periode $T$ and the set of real valued sequences. But, don't we expect the set of continuous function of ...
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0answers
33 views

How to find the Fourier coefficient of this function

How can we find the Fourier coefficient of the following function, f(x)=0 if -$\pi$$\leq$x<0 f(x)=1, if 0$\leq$x<$\pi$ and f(x)=f(x+2$\pi$) 1)how can we figure out that f(x) is integrable ...
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1answer
21 views

Problem in computing complex integrals for fourier transform

This is from a problem set of open course 8.02 by MIT OCW. I am not able to understand how the integral was solved. I have basic knowledge of Fourier transformation, and the Dirac delta function ...
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1answer
85 views

Simplifying an expression with Fourier transform

Can anyone simplify the following expression? I guess something from Fourier transform can help: $$f(\omega) = \lim_\limits{R \to \infty} \frac{1}{R^2} \int_{r=0}^{R}{re^{i \omega r^{-\gamma}}} ...
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0answers
29 views

Solution procedure for poisson equation

Consider the Poisson equation in the rectangle $Q=\{(x,y):0<x<1,0<y<1\}$, $$u_{xx}+u_{yy}=F(x,y)$$ $$u(0,y)=0,\,u(1,y)=0$$ $$u(x,0)=\phi(x),\,\,u(x,1)=\psi(x)$$ My Question: Is ...
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1answer
39 views

Are the real parts of the vectors constituting the discrete Fourier transform matrix linearly independent?

Let W denote the n- dimensional symmetric discrete Fourier transform matrix and $W_{i}$ denote its column vectors. Then, is the set { Re($W_{i}$) | i= 1... n } linearly independent? Or similarly, find ...
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1answer
36 views

How does this conjecture correspond to Carleson's theorem for the case of $d=1$?

From this source, on page 36 (bottom) there is a conjecture stated and it was said that the case of $d = 1$ corresponds to Carleson's theorem. A picture included here : But when I look at wiki ...
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1answer
30 views

Turn a function into an odd function?

I was asked to let: $$ f(x) = \begin{cases} 1-x, \text{if $0 \le x < 1$} \\ 0, \text{if $ 1 \le x < 2$} \end{cases}$$ Let fodd be the $4$ periodic odd extension of $f(x)$. Find the ...
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2answers
67 views

Do they have a mistake in this heat equation?

I need to know if there is a mistake in these notes: In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a ...
2
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0answers
51 views

Finite Ramanujan expansions over a finite field

I'm wondering if we could have an analogy in finite fields. After all, the Discrete Fourier Transform (DFT) has been generalized to finite fields as well (with essentially identical properties as in ...
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2answers
137 views

Infinite sum of elements in a finite field

This is a bit of a curiosity that intrigues me. Let $p$ be a prime and consider the sum of reciprocals of squares divisible by $p$. This is just $$ \dfrac{1}{p^2}\sum_{n=1}^\infty \dfrac{1}{n^2} = ...
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Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq ...
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2answers
68 views

What could be the mathematical equation of the given signal?

We know that Fourier series for periodic signal $y(t)$ is given by $$ y(t) = \sum\limits_{m=0}^{+\infty} a_m \cos(w_m t) + \sum\limits_{m=0}^{+\infty}b_m \sin(w_m t). \quad (2)$$ Now,I want to find ...
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1answer
30 views

Fourier transform with initial limit starting from 0

For a function $f(p)$, the Fourier transform is $\hat{f}(p)=\int_{-\infty} ^{\infty}e^{-i xp}f(x)dx$. What are the conditions that i write it as; $\hat{f}(p)=2\int_{0} ^{\infty}e^{-ixp}f(x)dx$? I ...
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1answer
60 views

How does one define the Fourier transform of a probability distribution?

Say $p_X$ and $p_Y$ are two probability distributions on a $m$ element set. Then I see an equality written as, $$\sqrt{m} \vert \vert p_X - p_Y \vert \vert _2 = \sqrt{ \sum_{k=0}^{m-1} \vert ...
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0answers
27 views

Fourier Cosine Expansion of Piecewise Continuous Function

Hi I am trying to represent this following function: $$f(x)=\begin{cases} 35.6236 + 0.161087e^{59.9842x},0\leq x < 0.1 \\ 35.6236 + 0.161087e^{59.9842 (-x + 0.2)},0.1\leq x \leq 0.2 \\ ...
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1answer
31 views

Question about the limits of definite integrals

Let me take an example that I've come across while studying Fourier series, We all know that $$\int_{-a}^{a} \sin \left( \frac{n\pi x}{a} \right) dx = 2 \int_{0}^{a} \sin \left(\frac{n \pi x}{a} ...
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2answers
76 views

Role of i in Fourier transform

I've seen several derivations of the Fourier transform, but most don't cover the conversion to the form $$ S(f) = \int_{\infty}^{-\infty} s(t)e^{-i2\pi ft} \;\mathrm{d}t $$ What is the role of ...
2
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1answer
34 views

What can we say about the transform of a function on a truncated domain, with respect to the transform on the full domain?

Let $f$ be a function on R and $\hat f$ its Fourier transform. Consider a truncated version of $f$ called $\bar f$ whose value outside an interval is $0$. Formally, $\bar f(x) = f(x) * 1_{x \in I}$ ...
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4answers
123 views

Closed form of $\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$ for $\theta\in (-\pi,\pi)$

We have to find the closed form of the following series $$\sum_{n=1}^\infty (-1)^n\frac{\sin(n\theta)}{n^3}$$ for $\theta\in (-\pi,\pi)$. We tried to use the following form of the sine ...
2
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1answer
57 views

A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Preliminaries: Given ${s \geq 0}$, let ${\mathrm{H}_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm \begin{equation*} \| u ...
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1answer
53 views

Fourier series converges

Suppose $S_N(x)$ is the Fourier series of $f(x)$, a continuous function. Now, I've understood that if $S_N(x)$ converges uniformly to some $g(x)$ then is must be that $f\equiv g$. What about the ...
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1answer
19 views

Find a specific intersection point of line and Fourier series using Newton–Raphson method when graphs have more then one intersection points

I need to find an intersection point of two graphs in polar coordinates. First is defined by a simple line $y=kx+b$. Second — by Fourier series $$ r=r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] ...
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2answers
35 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
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1answer
43 views

Fourier transform of periodic function

Is it possible to Fourier transform a periodic function f(x) = f(x+L) with period L, numerically, only over the range x = 0 to L and use periodic boundary conditions to enforce the periodicity of the ...
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2answers
52 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
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2answers
116 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
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0answers
47 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
2
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1answer
39 views

Intuitive reason for Fourier Series Convergence

I read that Fourier Series Converges to average of left side and right side limits at Jump Discontinuities. What is the intuitive explanation for it? Is it something regarding Energy minimization?
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1answer
31 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
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3answers
41 views

Are the coefficients on the Fourier transform arbitrary?

I was just wondering the other day about a convention I'd always taken for granted. I've seen the Fourier transform written a lot of ways. The first way (which, for reference, I'll call Scheme 1) is: ...
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0answers
48 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
0
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1answer
32 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...