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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0
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1answer
260 views

Given fourier series,finding functions

4.How to prove that there is a continuous periodic function $f$ (with period $2\pi$), such that $$\hat{f}(n) = \log(n)/(n^{3/2}).$$ $n\neq 0$ and $\hat{f}(0) = -1$. I know only the basics of ...
2
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1answer
361 views

What does Plancherel's (or rather Parseval's actually) formula say for this f?

Given $$f(x) = 1+\sum_{n=1}^{\infty}\frac{\sin (nx)}{3^n}$$ what is the easy way to find out the following equation's answer is odd or even? $$\begin{align*} ...
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0answers
44 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
5
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2answers
504 views

Do all analytic and $2\pi$ periodic functions have a finite Fourier series?

Consider a function $f:\mathbb{R}\to\mathbb{R}$ which is periodic with period $2\pi$. Let us impose the condition that $f$ is analytic. Now does that imply that $f$ has a finite Fourier series? PS : ...
6
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2answers
267 views

Completeness and Fourier series convergence

Consider the question: In an inner product space $V$, when does the Fourier series of $x$, $\sum\limits_{n=1}^k\langle e_n,x\rangle e_n$ converges to $x$ as $k\to\infty$? Well, certainly is converges ...
2
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0answers
129 views

Fast Fourier Transform $\mathbf{\tilde{A}}=\mathbf{A}e^{i \theta(\mathbf{k})}$

I have a vector which takes form $\mathbf{\tilde{A}}=\mathbf{A}e^{i \theta(\mathbf{k})}$, where $\mathbf{k}$ is the frequency vector ($k^2=k_x^2 +k_y^2+k_z^2$), $i$ is unitary complex number, while ...
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0answers
44 views

To show $\sum_{n=1}^{\infty}a_{n}\sin nx$ is a Fourier series of $f\in L^{\infty}$ if $na_{n}=O(1).$

I want to show that if $a_{n}\downarrow 0$ and $na_{n}=O(1)$ then $\displaystyle \sum_{n=1}^{\infty}a_{n}\sin nx$ is a Fourier series of $f\in L^{\infty}$. Can anybody help me?
0
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1answer
293 views

Fourier Coefficients Absolutely Summable?

When are the Fourier coefficients of $2\pi$-periodic functions summable? More specifically, which ones of $f(x) = (\cos x)^{100}$, $g(x) = \sin(\tan x)$, and $$ h(x) = \begin{cases} x+\pi, & : ...
0
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2answers
103 views

Fourier transform help

Find the Fourier tranform of $f(x)=x^2e^{-x^2}$ In a previous question when I found the Fourier transform of $f(x) =e^{-x^2}$, I used the formulas $F(f')=i\omega F$ and $F(xf)=iF'$. Will they be ...
0
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1answer
250 views

What function do I pick for $\sum_{n=1}^{\infty}(1/n^6)$? [duplicate]

Possible Duplicate: Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series. What function do I pick for the summation from $$\sum_{n =1}^{\infty}\frac{1}{n^6} \ ?$$ ...
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3answers
901 views

Solution of Laplace's equation in an annulus with constant Dirichlet conditions?

What's the solution to Laplace's equation $\nabla^2V=0 $ in the annulus with centre 0, inner radius 1, and outer radius 2, with boundary conditions $V=0$ on the inner boundary and $V=1$ on the outer ...
3
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2answers
134 views

Sum the series: $ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \cdots \ \text{ad inf}$

How do I sum the following series? $$ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \sin{3\alpha} + \cdots \ \text{ad ...
2
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0answers
102 views

To show $\displaystyle\sum_{n=2}^{\infty}\frac{\cos nx}{\log n}$ is a Fourier series.

I want to show $\displaystyle \sum_{n=2}^{\infty}\frac{\cos nx}{\log n}$ is a Fourier series but i am stuck how to show the given series is a Fourier series. Can anybody help me?
0
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0answers
93 views

Sufficient condition for the trigonometric series to be a Fourier series.

Question: If the trigonometric series $\displaystyle\frac{a_{0}}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx+b_{n}\sin nx)$ converges uniformly to $g(x)$ on $[-\pi,\pi]$ then $g$ is continuous on $[-\pi,\pi]$ ...
0
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4answers
327 views

Let $a_{n}\downarrow 0$ and if series $\sum a_{n}\sin nx$ is a Fourier series of function $f\in L^{1}$ then $\sum \frac{a_{n}}{n}<+\infty.$

I want to show that if $\displaystyle\sum a_{n}\sin nx (a_{n}\downarrow 0)$ is a Fourier series of $f\in L^{1}$ then $\displaystyle \sum \frac{a_{n}}{n}<+\infty.$ I know i have to use some property ...
1
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3answers
452 views

Solving $f''+f=\sin x$ with Fourier series

Does there exist a twice differentiable periodic function $f$ such that $f''(x) + f(x) =\sin(x)$ for all $x \in [-\pi, \pi]$? How to solve this differential equation using Fourier series? I ...
1
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2answers
484 views

Please check my answer to $\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$

$$\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$$ I have this answer, please let me know if there is a more beautiful proof. My answer: at first, we prove two inequalities: If $x\in ...
0
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1answer
65 views

Recovering an Operator on $L^2$ Given its Action as a Composition on the Spectrum of $f$

Let $\ell^2 = \ell^2(\mathbb{Z})$ denote the Hilbert space of square summable complex sequences on $\mathbb{Z}$ and suppose that $\sigma:\mathbb{Z} \to \mathbb{Z}$ is a function such that the linear ...
3
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1answer
375 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
1
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1answer
445 views

Differential equations, HEAT equation with insulated ends.

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f (x)$ throughout ...
2
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1answer
157 views

About $\sum_{k=1}^\infty \frac{b_k}{k}$, where $b_k$ are Fourier coefficients

This is my first post here. I have some troubles with this property of the Fourier coefficients. Indeed, let $f(x)$ be a continuous real function, with compact support $[a,b] \subset (0,2\pi)$, and ...
1
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1answer
380 views

Bounded Fourier Coefficients for monotonic functions

How to show that if $f$ is bounded and monotonic on $[a,b]$ then $|\hat{f}(n)| \leq \frac{c}{n}$, i.e the Fourier coefficients are bounded?
16
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1answer
597 views

Series which are not Fourier Series

How to show that $$ \sum_{n=2}^\infty \frac{\sin{(nx)}}{\log n} $$ not the Fourier series of any function? I have shown that the series is convergent by Dirichlet test. Let $a(n)=\frac{1}{\log ...
2
votes
2answers
138 views

Does the Series $\sum_{\substack{k=-\infty\\k\neq0}}^{\infty}\frac{e^{\frac{-\pi ik}{5}} - 1}{k}$ Converge?

I'm trying to prove that this series converges, although I'm not entirely convinced that it does: $$\sum_{\substack{k=-\infty\\k\neq0}}^{\infty}\frac{e^{\frac{-\pi ik}{5}} - 1}{k}$$ This is a ...
3
votes
1answer
113 views

Fourier Series of $\sin^k(x/2)$

I'm stuck on a seemingly simple problem: What is the fourier Series for $\sin^k(x/2)$? I've tried Mathematica with no luck. Thanks for your help!
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0answers
735 views

Fourier series /spectrum of different cosine functions

I was given the following task. b) In this task you will concatenate the seven cosines from task a) into one 7 sec long vector. To concatenate vectors in MATLAB use: x=[x1 x2 x3 x4 x5 x6 x7]; ...
2
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2answers
146 views

what's the difference between “convergent” and “reconstruct-able”?

I am reading this book: http://www.abdn.ac.uk/~mth192/html/maths-music.html There is a sentence on page 54: "However, the question of convergence of the Fourier series is not the same as the question ...
1
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3answers
570 views

Fourier Transform Equation

Can anyone help me by explaining how to answer the following: Determine the fourier transform of: $f(x) = e^{-4x^2-4x-1}$ thanks, Euden
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8answers
1k views

Conceptual/Graphical understanding of the Fourier Series.

I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
6
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1answer
1k views

Proving a family of orthogonal functions is complete over a certain interval

I'm reading Applied Partial Differential Equations by DuChateu and Zachmann, and the first couple of chapters contain quite a bit of review of Fourier series, as well as theory about L2 integrable ...
2
votes
1answer
175 views

the integral of the inverse of a Fourier series

Let $\{a_h\}$ be a double-sided complex sequence such that $\sum_{h=-\infty}^{\infty} |a_i| <\infty$ with $a_{0}\neq0$. Set $f(x) := \sum_{h=-\infty}^{\infty} a_h \exp(ixh)$ and assume that ...
6
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1answer
233 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
2
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1answer
124 views

Interval type for Fourier Analysis on $L^2( [-\pi,\pi) ) $

Why is it that in many texts the natural domain of choice is $L^2 ( [- \pi, \pi) ) $ as opposed to $L^2 ( [- \pi, \pi] ) $? I would to think of the space $L^2 ( [- \pi, \pi] ) $ as the completion of ...
3
votes
1answer
236 views

summation of a doubly infinite series

I want to show that $\sum\limits_{n=-\infty}^{\infty} \frac{(-1)^{n}}{x + \pi n} = \csc x$. If you let $f(z) = \frac{\pi \csc \pi z}{x + \pi z} $ and integrate around the square in the complex plane ...
5
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0answers
270 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 ...
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1answer
1k views

Computing Coefficients of Complex Form Fourier Series

I am having some trouble knowing how to correctly start a problem of finding the Fourier Coefficients using complex exponential form. The problem is given below: $$g_1(t)=\begin{cases} 1,~~\qquad ...
24
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3answers
1k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
2
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3answers
953 views

time-frequency domain

im confused on how these folks seems to like convert a frequency into a time function, and a time function into a frequency function. i know that time function uses amplitude that varies over time, ...
0
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1answer
76 views

Should a fourier saw wave function return values greater than 1?

For a sine function, the maximum value sin(x) returns will always be 1 (and -1) correct? Is this the same for a Fourier function? I'm writing a program that ...
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2answers
9k views

How to find inverse Fourier transform

I have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi i ...
1
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1answer
756 views

Fourier transforms of cos and sin

I have the function of time $f(t)=\cos(t10\pi) + \sin(t10\pi)$ and i wish to transform it. By using the tables, i have $\pi [\delta(w-10\pi) + \delta(w+10\pi)] + (\frac{\pi}{j})[ \delta (w-10\pi) = ...
9
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4answers
2k views

Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$

I need to compute Fourier series for the following function: $f(x)=\frac{-\pi}{4} $ for $-\pi \leq x <0$, and $\frac{\pi}{4} $ for $ 0 \leq x \leq \pi$, and then to use it and compute ...
2
votes
2answers
1k views

$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$

$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within ...
1
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1answer
109 views

How do I calculate Fourier series of an $f(x)$ with discontinuities inside its period?

I need to calculate Fourier series of: $$\sin(x)- \operatorname{IntegerPart}[\sin(x)]$$ This seems just a common sine function, with its value set to 0 at its max and mins, so the period is just the ...
4
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1answer
186 views

Heuristic\iterated construction of the Weierstrass nowhere differentiable function.

I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ...
2
votes
3answers
4k views

Scaling property of Fourier series and Fourier Transform

This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series. The Fourier transform of $f(ax)$ is ...
2
votes
4answers
2k views

Performing a differentiation on a Fourier series

I'm tutoring a set of problem sheet to do with Fourier series and one problem is as follows: The Fourier series for a sawtooth wave is, $f(x)=x=-\sum^{\infty}_{n=1}\frac{2(-1)^n\sin(nx)}{n}$ ...
3
votes
1answer
1k views

Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...
0
votes
2answers
428 views

When convergence in mean implies uniform convergence?

With Fourier series, I'm confused about Bessel's inequality and Parseval's identity. I understand that Bessel's inequality becomes Parseval's equality if and only if both integrals $ ...
0
votes
2answers
119 views

Proving the locus of a Fourier series is a system of perpendicular lines [closed]

From "Fourier's series and integrals" by H.S. Carslaw, there is the following question: Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is ...