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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51 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 ...
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0answers
34 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 ...
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20 views

Nyquist limit explanation

Kindly explain Nyquist in easy words. The actual question is as follows. We can attempt to display sampled data by simply plotting the points and letting the human visual system merge the points into ...
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0answers
22 views

Is there addition theorem for Fourier Harmonics?

We know that in spherical harmonic expansion we have addition theorem, and we can expand a function which depends on $x,x'$ and the angle between thesis two vectors $\cos(\theta_{x,x'})$ by spherical ...
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1answer
20 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: ...
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74 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} ...
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1answer
29 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 $ ...
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1answer
46 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 ...
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1answer
56 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 ...
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1answer
42 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 ...
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32 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 ...
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1answer
30 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 ...
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1answer
23 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 ...
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0answers
12 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 ...
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1answer
50 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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35 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 ...
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33 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 ...
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0answers
13 views

Conditions of coefficients that make argument increasing

Suppose a complex function $z(t)= \sum_{i=\infty}^{\infty} c_k e^{ikt}$, which equals its Fourier series. I would like to know if there is any simple condition that guarantees that the argument of $z$ ...
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18 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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27 views

Spectral interpolation - Rotation equivalent to translation properties of Fourier transform?

I am using a spectral code for flow simulations. My aim is to obtain flow field data from points which do not coincide with the simulation grid without using inaccurate interpolation schemes in real ...
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1answer
54 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 ...
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1answer
63 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 $$ \sum_{k \in \mathbb{Z}} c_k \sin(k t). $$ ...
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103 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 ...
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1answer
73 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 ...
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0answers
37 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
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1answer
86 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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45 views

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

Theorem 2 of de Bruin'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 ...
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2answers
63 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 $y$ ...
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1answer
91 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 ...
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2answers
42 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 ...
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1answer
19 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} ...
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24 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 ...
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1answer
38 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
123 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 ...
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1answer
28 views

Hilbert space (nonseparable): 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
35 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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82 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 ...
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1answer
23 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 ...
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2answers
49 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 ...
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1answer
21 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 ...
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2answers
42 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 ...
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2answers
35 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 ...
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1answer
32 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: ...
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1answer
44 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 = ...
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1answer
28 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 ...
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0answers
22 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 ...
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1answer
21 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 ...
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3answers
90 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
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1answer
42 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 ...
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0answers
18 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} ...