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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2
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1answer
36 views

Evaluate the series $\sum_{n = 0}^\infty \frac{1}{(2n + 1)^6}$ by examining the real Fourier series of the function $f(x) := x(\pi - |x|)$

The following is a question from a past exam in my university in a course called "Mathematical Methods for Statistics". It consists of two subquestions that may or may not be related (there is a high ...
0
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0answers
104 views

Relation between permutations and fourier transform?

i dont know if this is already addressed somewhere (searching around did not find sth). The motivation is to find a way to generate or produce permutations using concepts from continuous mathematics ...
0
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2answers
67 views

$\int_0^\pi\sin(2t)e^{-in2t}dt$ complex number integral for integer values of n

$$\int_0^\pi\sin(2t)e^{-in2t} \, dt$$ wolfram alpha say the answer is $$\frac{1-e^{-2 i n π}}{2-2 n^2}$$ although using the integral trig identity $$\int ...
1
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0answers
77 views

Finding Fourier series of $x(a-x)$ by integrating the Fourier series of the delta function.

I want to find the Fourier series for $$f(x)=x(a-x).$$ Of course I could do integration by parts and find the coefficients that way, but I'm given a hint to integrate the Fourier series expansion of ...
0
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1answer
49 views

N-point FFT and 2-radix FFT

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$) For example a is a sequence: ...
4
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0answers
155 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} ...
2
votes
1answer
33 views

$ U_{xx}+U_{yy}=0$ with rectangular boundary conditions

When solving $ U_{xx}+U_{yy}=0$ with $u(0,y)=u(a,y)=u(x,b)=0,u(x,0)=f(x)$. $0<=x<=a$ , $0<=y<=b$ by the method of separation of variables I have $-X''(x)-\lambda X(x)=0 $ ...
1
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1answer
93 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
1
vote
1answer
76 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
0
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1answer
55 views

Inequality between Fourier coefficients implies inequality for $L^p$ norms on the circle

Given two functions from $L_p [-\pi,\pi]$, where $p\geq 2$, $p$ is an even integer, and $f_n>|g_n|$ for every $n$ (where $f_n$ is the $n$th Fourier coefficient), I need to prove that ...
2
votes
2answers
57 views

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
1
vote
1answer
44 views

Is this fourier even?

$$ f(x) = \begin{cases} \frac{\pi}{4}-\frac{x}{2} & [0,\pi] \\ -\frac{3\pi}{4}+\frac{x}{2}, & (\pi,2\pi) \end{cases} $$ Is it right to compute only $a_n \text{ and } a_0$ coefficient for ...
12
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7answers
513 views

Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?

Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$. Presumably, this can ...
0
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1answer
550 views

Fourier series of coshx using fourier of $e^{x}$.

I have to find the Fourier series of $coshx$ on $(-l,l)$.What I did was I found the Fourier series of $e^{x}=\sum _{n=-\infty}^{\infty }{(-1)^n (\ell^2+in\pi)\over{l^2+n^2\pi^2}}\sinh(\ell)e^{{in\pi ...
1
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0answers
23 views

Fourier transform at infinity

I have a function $f(u)$ satisfying the following properties $$ \lim_{u\to\pm\infty} f(u) = f^\pm,~~ \lim_{u\to\pm\infty} f'(u) \sim {\cal O} \left( |u|^{-3/2} \right) = 0 $$ The function $f(u)$ can ...
2
votes
1answer
3k views

Showing that complex exponentials of the Fourier Series are an orthonormal basis

I am revisiting the Fourier transform and I found great lecture notes by Professor Osgood from Standford (pdf ~30MB). On page 30 and 31 he show that the complex exponentials form an orthonormal ...
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0answers
48 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
3
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0answers
110 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 ...
0
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0answers
33 views

Construct a Fourier series that diverges almost everywhere.

Andrey N. Kolmogorov was one of the greatest mathematicians and polymaths of the 20th century. One of his first achievements was to construct a Fourier series that diverges almost everywhere. How ...
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1answer
69 views

Questions about the Fourier series

$$f(x)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} (a_n \cos{(\frac{2 n \pi x}{L})}+b_n \sin{(\frac{2 n \pi x}{L})}) \ \ \ \ \ (*)$$ The symbol $\sim$ has the following meaning: We know that the right ...
4
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1answer
74 views

Question regarding Fourier coefficients

I would like to express the product $$ \left( \sum_{k \in \mathbb{Z}} a_k \sin(k t) \right) \left( \sum_{k \in \mathbb{Z}} b_k \cos(k t) \right) $$ as $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
2
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0answers
156 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 ...
4
votes
1answer
90 views

how to compute this integral for fourier series

I am trying to find the Fourier sine and cosine series of $\frac{1}{(1+x^2)}$ from $0$ to $2$, and do not know where to even begin to evaluate this integral: $\int \frac{sin(nx)}{(1+x^2)} dx$ (and ...
2
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0answers
86 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
1answer
257 views

Fourier series of $\sin x$ using series of $e^{ix}$

I have to find the Fourier series of $\sin x$ . Assume that $\ell$ is not an integer multiple of $\pi$.(Hint: First find the Fourier series for $e^{ix}$) This is how I did it: Complex Fourier series ...
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0answers
82 views

A question about theorem 2 in de Bruijn's 1950 paper “The roots of trigonometric integrals”

Theorem 2 of de Bruijn's paper titled "The roots of trigonometric integrals" (Duke Math. J., 17 (1950)) is given by: What does it mean by "the function $q(x)$ be regular in the sector...? Does it ...
0
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2answers
93 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
4
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1answer
254 views

An integral that might be related to the modified Bessel function of second kind

It is known that the modified Bessel Function $K_z(a)$ ($a>0$)can be expressed as a Fourier transform $$K_z(a)=\frac{1}{2}\int_{-\infty}^{\infty}\exp(-a\cosh t)\cosh(zt){\rm d}t=K_{-z}(a)$$ Can ...
1
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2answers
120 views

Fourier series with respect to orthonormal sequence

Let $H$ be the space of piecewise continuous $2 \pi$-periodic functions on the real line. For $f$ and $g$ in $H$, consider the inner product $<f,g>=\frac{1}{2\pi}\int_{- \pi}^{\pi}f(x)\overline ...
0
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1answer
29 views

Rank of the harmonics in a Fourier series expansion

Let $\boldsymbol{A}(t)$ be a $T$-periodic matrix with rank $r$, and $\boldsymbol{A}_n$ the harmonics of its Fourier series expansion, so that $$ \boldsymbol{A}(t) = \sum_{n=-\infty}^{+\infty} ...
2
votes
1answer
79 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 ...
1
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1answer
53 views

Theoretical question about Fourier Series, I'm confused!

If I have a function f(x) defined on $[0,L)$, said to be periodic of period $L$ and such that $f(0)\neq0$, how should I get the Fourier coefficients? I'm hesitating between taking the even extension ...
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1answer
675 views

Parseval Identity and Fourier Series Question on function $f(x)=|x|$

Trying to compute the fourier series for $f(x)=|x|$ for $f$ on $[- \pi, \pi]$ using the trig method. I have a question as to the absolute value function. I'm using the definition of absolute value ...
1
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1answer
50 views

Nonseperable Hilbert space: Explicit ONB?

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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2answers
43 views

Using a family of functions to find fourier series

I'm given a family of functions $$T= \left \{\frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt \pi} \cos n\pi, \frac{1}{\sqrt \pi} \sin n \pi: n=1,2,3,\ldots \right \} , $$ on the interval $[-\pi, \pi]$ ...
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0answers
125 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
1
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1answer
73 views

Conditions for Uniform Convergence of Fourier Series

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a 2$\pi$ periodic function such that $\exists$ $C>0$ and $\epsilon>0$ with $|f(x)-f(y)|\leq C|x-y|^{.5+\epsilon}$. Show that the the Fourier series ...
2
votes
2answers
70 views

Let f be a continuous real valued function on R, and prove that $f(x)$ is constant using the fourier series.

I missed my class where we went over the fourier series and am having extreme issues with this homework question. $f(x) = f(x + 1) = f(x + \sqrt{2})$ Is there anyone who could be kind enough as to ...
0
votes
1answer
65 views

Inverse Fourier Transform

I need help solving the following Fourier transform question. Given, $$ X_s(f) = \frac{1}{\Delta T} \sum_{n = -\infty}^{\infty} X\left(f - \frac{n}{\Delta T} \right) $$ $$ H(f) = \begin{cases} 1 ...
3
votes
2answers
75 views

What function does the Fourier series $\pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {k^2} \cos(kx) $ converge against?

What function does the Fourier series $$\pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {k^2} \cos(kx) = \pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {2k^2} (e^{ikx} + e^{-ikx})$$ converge against ? I've ...
1
vote
2answers
129 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
0
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1answer
53 views

fourier series representation

Find the Fourier series with period $2$ of $$f(x) = -x,\qquad-1<x<1$$ so I find that $a_0$ and $a_n$ both are $0$ since odd functions so the Fourier series is on the form: ...
2
votes
1answer
60 views

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$. I have computed $$c_n = \frac 1 {2\pi} \int^{\pi}_{-\pi} |\sin(x)|e^{-inx} dx = ...
0
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1answer
35 views

Redundant assumption in an exercise concerning fourier series?

So here is my problem, I have to solve the following exercise, Let $\phi\in L^1[0,1)$ and $\psi\in L^{\infty}[0,1)$, both of period 1 and $\int_0^1\psi(t)dt=0$. Show that $$\lim\limits_{n\rightarrow ...
2
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0answers
26 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 ...
1
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1answer
28 views

Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
1
vote
3answers
1k views

Calculating own dft via matlab?

We are asked to code our own dft function from the formula : If everything is done correctly it should give the same result with matlab's own dft function, in the end I'm comparing them but they ...
1
vote
1answer
134 views

Uniform convergence of the Fourier Series using Bessel's inequality

Consider the Fourier series of $f$, $$ \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx) $$ Let $$f_n(x)= a_n \cos(nx) + b_n \sin(nx)$$ Then to show that $f_n(x)$ is uniformly ...
0
votes
0answers
22 views

Complex form fourier series of a sum of e

The heart of the problem is finding a fourier series in its complex form for: $\displaystyle\sum _{k=-\infty }^{\infty } e^{-4|t-k|}$ The form I know of is $\displaystyle\sum_{k=-\infty}^{\infty} ...
1
vote
1answer
56 views

Find Fourier series coefficients of $f(x)$.

$T=2$ $$f(x) = \begin{cases} 1, & \text{$-\frac12\le x \le\frac12$} \\[2ex] |2x|, & \text{$\frac12 < x \le1\frac12$} \\ \end{cases}$$ The image: I found that $a_0=\frac12$. Since ...