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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6
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1answer
378 views

Brownian motion and Fourier series

Let $(B_t)_{t \in [0, \infty)}$ be a Brownian motion. Can you prove me why it can be written as $$B_t= Z_0 \cdot t + \sum_{k=1}^{\infty} Z_k \frac{\sqrt{2} \cdot \sin(k \pi t)}{k \pi}$$ for some ...
2
votes
1answer
724 views

Fourier sine series: quarter range expansion

I know how to do (a). I know the sine expansion of $\phi(x)$ on $(0,l)$: $\phi(x)=\sum_{n=1}^\infty B_n \sin \frac{n\pi x}{l}$, but could not get the desired form. Through the formula I mentioned ...
2
votes
1answer
4k views

Using Fourier series to calculate an infinite sum

Given the Fourier series of the $2\pi$-periodic function defined for $$-\pi\leqslant x \leqslant \pi$$ by $$f(x) = |x|$$ is $$ \frac{\pi}{2} -\frac{4}{\pi} \sum_{k\geq 1, k\ odd}^{\infty} ...
0
votes
1answer
1k views

Deriving fourier series using complex numbers - introduction

So this is the follow up thread to the one I asked before but you don't need to read the other one for this to make sense. If you want to, read PZZ's answer: link to the thread. So I know that there ...
2
votes
2answers
451 views

The link between vectors spaces ($L^2(-\pi, \pi$) and fourier series

So in my PDE course we started with a review of complex numbers and vector spaces to introduce us to fourier series. I have a few questions about this. I know 'big ell 2' and 'little el 2' are ...
3
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2answers
1k views

Lipschitz Continuity and Hölder Continuity helps Fourier series to converge

Let $f$ satisfies $$|f(x+u) - f(x)|\leq L|u|^{\alpha}$$ for some constants $L$ and $\alpha$. If $\alpha = 1$ then $f$ is called Lipschitz continuous, and if $0 < \alpha < 1$ ...
9
votes
3answers
306 views

Meaning/Justification for Describing Functions as 'Orthogonal'

When introducing Fourier series, my lecturer stated that 2 periodic functions, $f$ and $g$, with period $2L$ are orthogonal iff $$\int^{L}_{-L}{f(x)g(x)}\mathrm dx=0$$ Wikipedia agrees, even defining ...
1
vote
1answer
195 views

Accuracy of Fourier series at discontinuities

What could I say when asked to "comment on the accuracy of Fourier series at discontinuities"? It is very vague, though I reckon it alludes to the W-G phenomenon. I have read the wiki page on Gibbs ...
2
votes
2answers
601 views

Low pass filter

A wave defined by $f(t)=a$ for $t\in (0,T)$ and $f(t)=-a$ for $t\in (-T,0)$ (the wave is $2T$ periodic) is input into a system that transmits angular frequencies $<\omega$ and absorbs those ...
6
votes
1answer
722 views

Term by term differentiation

If a function $f(x)$ is expressed as a Fourier series and we know $f'(x)$. Is it then true that if we differentiate the Fourier expression we must get $f'(x)$? E.g. if $f(x)=x^2$ for $x\in ...
1
vote
1answer
2k views

Just checking: sine series for $x^2$

Is the Fourier sine series for $x^2$ equal to $\sum {2\pi\over 2m+1}-{8\over (2m+1)^3\pi} \sin ((2m+1)x)$? (just want to check that those multiple steps of intergation by parts did not slip me up). ...
5
votes
1answer
261 views

Fourier coefficients

I hope I have understood this coreectly: A Fourier series has coefficients of order $O(n^{d+1})$ for a d times differentiable function. But what if the function is infinitely differentiable? The ...
0
votes
0answers
78 views

Discontinuities and Fourier series

Again, regarding FT, why is it true that if the $d$-th derivative is the lowest derivative that contains $\geq 1$ discontinuity then the Fourier coefficients are $O(n^{-(d+1)})$ ? Thanks.
1
vote
1answer
168 views

Fourier series — dual problem

Regarding the Fourier series expansion of a function, why are The two representations -- Knowing the function in physical space at a finite number of points Knowing a finite number of ...
2
votes
1answer
65 views

bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2

Are there any results about the bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2? More precisely, let $k$ be a positive integer and $m=4/k$. Write \begin{equation*} ...
3
votes
1answer
309 views

A question on convergence of Fourier series and the derivative of the function

Consider a function $f_p : \mathbb{R} \to \mathbb{R}$ which is continuously differentiable in $(0,2\pi)$ except at two points $x = x_c$ and $x = x_o$. At $x = x_c$, $f_p(x)$ has a jump discontinuity. ...
2
votes
1answer
131 views

Fourier transform of a wavefunction

It would be great if someone would help me check that I have done this right. I have an infinite square well, s.t. $V(x) = 0$ for $x\in(0,a)$ and $V(x) = \infty$ otherwise. Given that $\psi(x,0) = ...
3
votes
0answers
160 views

Uniform convergence of a series

This problem came from the Krantz text ($2^{nd}$ ed. ch. 9, prob. 17): Prove that the series $\displaystyle\sum_{j=1}^{\infty }{\frac{\sin{(jx)}}{j}}$ converges uniformly on compact intervals that do ...
10
votes
3answers
2k views

The mathematics of music - why sine waves?

Of course, the Fourier transform is an extremely elegant mathematical method of overwhelming simplicity, and this straight away puts sine waves (or complex exponentials) on a high pedestal. But what ...
1
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2answers
206 views

Are there any applications of Fourier series/analysis in General relativity

I'd like to know if there are any applications of Fourier analysis / Fourier series expansion in General relativity ? I mean how Fourier transform has applications in Quantum mechanics.
3
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1answer
324 views

A question on limit of a Dirichlet's integral

I want to know the idea/intuition behind the Dirichlet integral. For example consider $$I(\alpha) = \int_0^{\delta} g(t) \frac{\sin(\alpha t)}{t} dt$$ . Why would $I(\alpha)$ tend to some constant ...
1
vote
1answer
1k views

Euler's Formula Conversion with coefficients

If I have an equation such as $x(t) = \displaystyle \sum_{n=1}^N \left( a_n \cos(\omega_nt) + b_n \sin(\omega_n t) \right)$, how do I convert it to a sum of complex exponentials? In other words what ...
2
votes
1answer
301 views

Asymptotic error of Fourier series partial sum of sawtooth function

In Iwaniec's book, Topics in Classical Automorphic Forms, pg. 4, he gives the statement: $$\{x\}=\frac{1}{2}-\sum_{n=1}^N\frac{\sin 2\pi nx}{\pi n}+O((1+||x||N)^{-1})$$ where $\{x\}$ denotes the ...
3
votes
2answers
239 views

What is the reason for these jiggles when truncating infinite series?

Plotting the series $$\displaystyle y = \sum_{k} \frac{\sin kx }{k}$$ In the limit it would look like Taking a finite number of terms, I want to understand what is the reason for the jiggling at ...
1
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0answers
116 views

Does number theory have any role in the proof of convergence of Fourier series for certain functions?

Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original ...
0
votes
1answer
14k 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 ...
4
votes
4answers
617 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 ...
7
votes
4answers
329 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 ...
3
votes
1answer
437 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 ...
4
votes
1answer
545 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 ...
0
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0answers
173 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, ...
1
vote
1answer
329 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 ...
6
votes
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 ...
4
votes
2answers
1k 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 ...
5
votes
1answer
506 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 ...
2
votes
1answer
162 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 ...
3
votes
3answers
703 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, ...
2
votes
2answers
152 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 ...
1
vote
1answer
957 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 ...
0
votes
2answers
273 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} ...
1
vote
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 ...
2
votes
1answer
163 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 $f : \mathbb{R} \to \mathbb{C}$. ...
1
vote
1answer
305 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 ...
3
votes
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) = ...
0
votes
1answer
414 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 ...
0
votes
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 ...
14
votes
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 ...
0
votes
0answers
65 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) = ...
3
votes
1answer
192 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
votes
1answer
274 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 ...