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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1answer
196 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 ...
0
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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 ...
2
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1answer
273 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 ...
3
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0answers
204 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 ...
5
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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 ...
0
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1answer
412 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?
2
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1answer
938 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
653 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
333 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
192 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
363 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
385 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
718 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
589 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
262 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
468 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, ...
4
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2answers
593 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
380 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
298 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 ...
2
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1answer
699 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
255 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 ...
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2answers
5k 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.
5
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1answer
213 views

For a polygon on complex plane, when are the vertex 'Fourier coefficients' non-zero

Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. ...
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2answers
783 views

Gibbs Phenomenon

Can someone explain in complete detail with the appropriate convergence arguments of the Gibbs Phenomenon for Fourier Series? I know that the overshoot near a jump does not die out as the frequency ...
5
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1answer
564 views

Isoperimetric inequality implies Wirtinger's inequality

Let $C: x=x(t), y=y(t), a\le t\le b$ be a $C^1$ closed curve (not necessarily simple).The isoperimetric inequality says that $$ A\le \frac{\ell^2}{4\pi},$$ where $$A=\left|\int_C y(t)x'(t) ...
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1answer
358 views

How to express the whole part $\lfloor x \rfloor$ as analytical function or Taylor/Fourier series?

And how to express $\{ x \} = x - \lfloor x \rfloor$ as function of $sin(x)$ and $sign(x)$?
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3answers
3k views

an example of a continuous function whose Fourier series diverges at a dense set of points

Please give me a link to a reference for an example of a continuous function whose Fourier series diverges at a dense set of points. (given by Du Bois-Reymond). I couldn't find this in Wikipedia.
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2answers
2k views

Fourier cosine transform

Find Fourier cosine transform of $e^{-a^2 x^2}$ and hense evaluate Fourier sine transform of $x\cdot e^{-a^2x^2}$. I can solve this question only if there is $x$ instead of $x^2$ in the exponential ...
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5answers
5k views

Fourier series for $\sin^2(x)$

I was asked to compute the Fourier series for $\sin^2(x)$ on $[0,\pi]$. Now this is what I did and I'd like to know if I'm right. $\sin^2(x)=\frac12-\frac12\cos(2x)$ . I got the right hand side using ...
3
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1answer
123 views

A limit related to the Gibbs phenomenon

Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$\left|\int_{x(N)}^{\pi/N} D_N(t)\mathrm dt ...
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4answers
561 views

Fourier series analog of a formula in Fourier transform

Every Fourier transform formula that I know of has a corresponding Fourier series analog, except the multiplication formula $$\int_{-\infty}^\infty f(x)\hat{g}(x) dx=\int_{-\infty}^\infty ...
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1answer
841 views

Accessible proof of Carleson's $L^2$ theorem

Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$). Some time ago I ...
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2answers
1k views

What are the conditions for existence of fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$

What are the conditions for existence of fourier series expansion of a function $f\colon\mathbb{R}\to\mathbb{R}$
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1answer
668 views

Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
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4answers
863 views

Convergence of a Fourier series

Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$. I am looking for a general theorem that implies that the Fourier series of $f$ ...
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1answer
2k views

Continuity and discontinuity in fourier series?

Can somebody please explain continuity and discontinuity in fourier series?
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1answer
165 views

What else can the elliptic integral count?

I just read this document - Jacobi's Four Square Theorem. It shows how to count the number of representations of a number as the sum of four squares. I can follow the proof but currently it just ...
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2answers
452 views

Using complex exponentials as solution of ODE

I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or $A ...
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5answers
889 views

Geometric intuition behind convergence of Fourier series

I've been trying to work out the best way to understand why Fourier series converge, and it's a little embarrassing but I don't even know a rigorous proof. Can someone please help put me on the right ...
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2answers
645 views

Convergence of $\sum_{n=1}^{\infty} \frac{\sin (n x)}{2^n}$

I can't seem to solve this one. I've tried Dirichlet's test but to no avail. Any help is greatly appreciated.
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1answer
188 views

Is there a (deep) relationship between these various applications of the exponential function?

Here is a list of some applications of the exponential function. 1) The exponential mapping in Lie theory. I put this first because my intuition tells me that this must be the most fundamental, or ...
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1answer
520 views

Relation between “harmonic form” and fourier series?

I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare ...
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1answer
245 views

Lie algebra of the bounded continuous functions

I can think of the set of bounded, continuous functions from $\mathbb R \to \mathbb R$ as a group, with composition as addition of functions. In other words, this group has the rule that the ...
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3answers
732 views

Hint on how to prove $\zeta ( 2) =\pi ^{2}/6$ using the complex Fourier series of $f(x)=x$

I know how to prove $\zeta (2)=\pi ^{2}/6$ by using the trigonometric Fourier series expansion of $x^{2}/4$. How can one prove the same result using the complex Fourier series of $f(x)=x$ for $0\leq ...
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6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...