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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15k views

Use WolframAlpha to compute the real Fourier series of a function

How can I use Wolfram|Alpha to compute the Fourier series (with real coefficients $a_0, a_n$ and $b_n$)? (The 'Fourier series' command seems to summon the complex series) I.e. $f(x) = x + \pi$ for ...
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4answers
622 views

Application to Fourier series

I have seen the following problem in a test, and there are some elementary solutions to it. I am curious if there is a solution involving Fourier series. Here it is: Let $(a_n),(b_n)$ be two ...
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4answers
335 views

The leap to infinite dimensions

Extending this question, page 447 of Gilbert Strang's Algebra book says What does it mean for a vector to have infinitely many components? There are two different answers, both good: 1) The ...
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1answer
440 views

Find the Complex Fourier coefficients

This is a revision question I've been working on. Show that if a $2\pi$-periodic function $f$ has the complex Fourier coefficients $c_{k}$ and $g(t)=f(t+a)$, where $a$ is a constant, the the Fourier ...
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1answer
553 views

Exponential Decay of Laplace Coefficients

Laplace coefficients are Fourier coefficients used in Celestial mechanics calculations $$ b^n_s (\alpha) \equiv {1 \over \pi} \int_0^{2\pi} {\cos n \phi \over (1 - 2 \alpha \cos \phi + \alpha^2)^s} d ...
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0answers
176 views

d-dimensional Riesz-Fischer theorem

In $\mathbb R$, the Riesz-Fischer theorem states that any square-summable series is the Fourier series of a square integrable function. Is this true in $\mathbb R^d$? Thanks a lot in advance, ...
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1answer
332 views

Fourier transform on a simple smooth 1-manifold

Assume a very simple smooth 1-manifold, with a single chart covering, What I'd like to know is, can we use and Fourier transform for functions on this manifold just as we did for the case of ...
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1answer
228 views

About $2$-periodic continuous solutions of $f(x)+f(x+1)=f(2x+1)$

Suppose I want to find all the continuous solutions to the functional equation $$f(x)+f(x+1)=f(2x+1),\tag{E1}$$where $f$ is a continuous and $2$-periodic function defined on the dyadic rationals. I ...
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2answers
2k views

Identifying the product of two Fourier series with a third?

Given the product of two functions defined explicitly through their Fourier coefficients (of unknown undeveloped form): $\sum_k{c_k e^{i k t}} \cdot \sum_k{c'_k e^{i k t}}$ Is there any way to ...
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1answer
523 views

Fourier series of almost periodic functions and regularity

Let $f$ a $2\pi$-periodic function represented by its Fourier series $\displaystyle\sum_{k=-\infty}^{+\infty}c_ke^{ikx}$. We know that $f$ is smooth if we have $\displaystyle\lim_{|n|\to ...
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1answer
167 views

How large are the second, third, fourth, etc. ringing artifacts in Gibbs phenomenon?

I've read that in the Gibbs phenomenon, partial Fourier series will over- or underestimate a function's value in neighborhoods of jump discontinuities. Specifically, the maximum error will converge to ...
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3answers
745 views

Fourier series at discontinuities

I was reading about Fourier series and have a doubt concerning it. The book I am reading from does not seem to help. As I understand, ...
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2answers
160 views

Generate a Monte Carlo sample from a PDF defined by a Fourier Series

I have a probability distribution (PDF) defined by a Fourier series.. actually it's a purely cosine series over a known range. The PDF quite smooth, so most of the power is in the low 5 or so ...
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1answer
1k views

Fourier transform from limit of fourier series [duplicate]

Possible Duplicate: Derivation of Fourier Transform? How is the Fourier transform obtained by taking the limit of the Fourier series as the period goes to infinity? In particular I am ...
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2answers
290 views

Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc?

Consider any continuous $2\pi$ periodic function (of bounded variation) $f : \mathbb{R} \to \mathbb{R}$ and its fourier series given as $f(\theta) = \frac{a_o}{2} + \sum\limits_{n = 1}^{\infty} ...
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1answer
175 views

Integral representation of a partial sum of a Fourier series using complex exponentials

Consider a periodic function $f: \mathbb{R} \to \mathbb{C}$ of period $T = 2\pi/\omega_0$. What I'd like to know is the integral representation of a partial sum of its Fourier series using complex ...
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1answer
164 views

reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
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1answer
308 views

expression for Dirichlet's kernel like sum

It is given in the book that the Dirichlet's kernel $D_n(t) = 1/2 + \sum\limits_{k=1}^{n} \cos(kt)$ is given as $\frac{\sin(n+1/2)t}{2\sin(t/2)}$. I'd like to know if there is any such expression for ...
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1answer
164 views

Fourier coefficients in oscillation problem with viscosity

On the oscillation problem of a rope with fixed extremities, $$\left\{\begin{matrix} \left.\begin{matrix}\left.\begin{matrix} u_{tt}(t,x) = a^2u_{xx}(t,x)\\ u(0,x) = \varphi(x)\\ u_t(0,x) = ...
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1answer
430 views

Proving that the fourier coefficients for a pretty smooth function are pretty small

Let $f:[0,2\pi] \rightarrow \mathbb{R}$ be $C^k$ for some $k >0$. Prove that $|\widehat{f}(n)|n|^k|$ is bounded above by some constant independent of $n$. To do this, we've been ...
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1answer
4k views

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft. Greetings I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do ...
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1answer
1k views

Making use of Fourier series to evaluate an infinite sum

Show that $$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k \sin(ax)}{a^{2}+k^{2}}=\frac{\pi}{2}\frac{\sinh(ax)}{\sinh(\pi a)}, \;\ x\in (-\pi,\pi)$$ It appears to me this series is crying out for the use of ...
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0answers
66 views

Dimension of space of band-limited, periodic, real functions

Dear all, I'm interested in the space of functions of $d$ variables which can be put in the following form $$f(x_1, \ldots, x_d) = ...
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1answer
202 views

Terminology for multidimensional Fourier series

Dear All, I'm computing multidimensional Fourier series of a function $f$ defined on $(0, L_1)\times(0, L_2)\times\cdots\times(0,L_d)$. The series reads $f(\vec x)=\sum_{\vec k}\hat f(\vec ...
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1answer
277 views

Substituting Periodic Fourier series expansion equation with standing wave equation

Substituting Periodic Fourier series expansion equation with standing wave equation Greetings All I can re-create a periodic signal using Fourier series expansion using sin and cos waves. But how ...
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1answer
275 views

Convergence in the mean of Fourier series

I need to do some research on fourier series for my analysis class so I'm trying to find info (preferably a book or paper with the proof) on this: "If $f$ is Riemann integrable on $[-l,l]$ then its ...
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0answers
205 views

Fourier Series generated by a function and periodic with $2\pi$

Consider the Fourier series (in exponential form) generated by a function $f$ which is continuous on $[0,2\pi]$ and periodic with period $2\pi$ , say $$f(x)\sim\sum_{n = - \infty }^{+ \infty }\alpha ...
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1answer
303 views

Cusp forms' Fourier coefficients sign changes

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N ...
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1answer
414 views

The Wiener algebra question

I want to show that the maximal ideal space of the Wiener algebra $W$ is $ \{ M_z : z \in \mathbb{T} \}$ where $M_z = \{ g \in W : g(z)=0 \}$ Could you please help me?
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1answer
953 views

Fourier series odd and even functions

I'am a little confused. In my text book it is written that all odd function can be described by a sine series. I have this following equation from an exercise: $$A_{0}+\sum\limits_{n=1}^\infty ...
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1answer
669 views

How can I increase/decrease (frequency/pitch) and phase using fft/ifft

How can I increase/decrease (frequency/pitch) and phase using fft/ifft I think I have the basic code but I’m not sure what to do next PS: It's done in Octave/matlab code Example I have a signal that ...
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1answer
338 views

One more question about decay of Fourier coefficients

Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$ Does it imply that $f_s=O(s^{-(p+\psi)})$ for some ...
2
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1answer
195 views

FFT and changing frequency and vectorizing FOR loop

I can increase and decrease the frequency of a signal using the combination of fft and a Fourier series expansion FOR loop in the code below but if the signal/array is to large it becomes extremely ...
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1answer
364 views

Fourier series coefficients

Suppose I have a function $S_n(t) = \sum_{k=1}^n a_k \sin((k-1/2)\pi t)$, with unknown coefficients $a_k$. $S_n$ is perodic with period 4. If I can observe $S_n$ over the interval $[0,4]$, I can ...
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1answer
391 views

Fourier transform without Gibbs oscillations

Is there a way to obtain the Fourier transform of a function (e.g., sinc) without the Gibbs phenomenon?
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1answer
1k views

Differentiability and decay of magnitude of fourier series coefficients

I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a ...
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2answers
723 views

Fourier transform of $\mathrm{sinc}(4t)$

I'm preparing for an exam in the signals and systems class I'm taking. One of the practice exams has a problem that requires you to take the Fourier transform of $\text{sinc}(4t)$. From a table of ...
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1answer
602 views

rapidly decaying sequence and fourier series coefficients of a compactly supported smooth function

In this question the term rapidly decaying sequence is used. What is the definition of a rapidly decaying sequence.(in cases of terms being purely real or complex.) How to prove that the sequence of ...
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1answer
267 views

Bounds for Fourier series

Fourier series of function f: $$f(x)=\sum_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$$ Suppose $f_{s}\sim\frac{1}{s^{p}}$. What can we say about $f(x)$? Can we find some bounds for $f(x)$ like ...
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1answer
476 views

Aliasing in DFT: mathematical expression

The Fourier transform of a sinc function is the top hat function. So, if $\{y_k\};\ k\in\{0,1,...,n-1\}$ are samples of the sinc function, sampled $T$ apart, the discrete Fourier transform is ...
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1answer
94 views

Why is the following sum true?

Why is the following true? $$\sum_{j=0}^{n-1}w(2\pi j/n)\left[\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi ik(j-m)/n}\right]=w(2\pi m/n)$$
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1answer
107 views

Reconstruct elements of sequence from inverse transform

Let $v(x)\in L^1\cap L^2$ be a function, and $w(y)\in L^2$ be its Fourier transform. Let $\{w_k\}$ be an infinite sequence of samples from $w(y)$, sampled $T$ apart. Now, ...
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2answers
618 views

Inverse Fourier transform relation for $L^2$ function

If $f$ is an $L^2$ function and $\hat{f}$, its Fourier transform, also in $L^2$, can the Fourier transform and its inverse be written as $$\hat{f}(\omega)=\int_{-\infty}^\infty f(x) e^{i\omega ...
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1answer
384 views

Bound for Fourier series

Let $\{a_k\}$ be an infinite sequence, with $\sum_{k=\infty}^\infty \vert a_k\vert^2<\infty$. Let $f(\omega)=\sum_{k=-\infty}^\infty a_k e^{ik\omega}$ be its Fourier series. By Plancheral's ...
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1answer
305 views

Existence of Fourier transform

Let $\{x_k\}$ be an infinite series. It is known that $\sum_{k=-\infty}^\infty \vert x_k\vert^2<\infty$ and $\sum_{k=-\infty}^\infty \vert x_k\vert$ does not converge. How do I prove that the ...
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1answer
714 views

Is it possible to link the eigenvalues of a matrix to the Fourier transform of the matrix?

I'm trying to get insight into the eigenvalue spectrum of a square matrix (large N, symmetric,positive semi definite matrix) using Fourier transforms (I've tried transforming a bunch of thigngs: the ...
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2answers
302 views

reference request for proof of Gibbs phenomenon at jump discontinuities

Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
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2answers
468 views

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function - a reference request

Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to ...
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3answers
256 views

The fourier series $\sum_{m\neq n} \frac{1}{n^2 - m^2} \cos \frac{m\pi x}{2a}$

A Fourier series arising in perturbation theory in quantum mechanics is $$\sum_{m\neq n} \frac{1}{n^2 - m^2} \cos \frac{m\pi x}{2a} \, .$$ where $n$ is an odd positive integer and $n$ runs through ...
4
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2answers
6k views

Derivation of Fourier Series?

Can someone point me to the full derivation of the Fourier Series? I'm having problems understanding how the a's and b's coeffients are worked out.