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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$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$: Show uniformly converges within $[\epsilon, 2\pi - \epsilon]$

$S(x)=\sum_{n=1}^{\infty}a_n \sin(nx) $, $a_n$ is monotonic decreasing $a_n\to 0$, when ${n \to \infty}$. I need to prove that for every $\epsilon >0$, the series is uniformly converges within ...
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How do I calculate Fourier series of an $f(x)$ with discontinuities inside its period?

I need to calculate Fourier series of: $$\sin(x)- \operatorname{IntegerPart}[\sin(x)]$$ This seems just a common sine function, with its value set to 0 at its max and mins, so the period is just the ...
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189 views

Heuristic\iterated construction of the Weierstrass nowhere differentiable function.

I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example ...
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Scaling property of Fourier series and Fourier Transform

This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series. The Fourier transform of $f(ax)$ is ...
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Performing a differentiation on a Fourier series

I'm tutoring a set of problem sheet to do with Fourier series and one problem is as follows: The Fourier series for a sawtooth wave is, $f(x)=x=-\sum^{\infty}_{n=1}\frac{2(-1)^n\sin(nx)}{n}$ ...
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Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...
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433 views

When convergence in mean implies uniform convergence?

With Fourier series, I'm confused about Bessel's inequality and Parseval's identity. I understand that Bessel's inequality becomes Parseval's equality if and only if both integrals $ ...
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121 views

Proving the locus of a Fourier series is a system of perpendicular lines [closed]

From "Fourier's series and integrals" by H.S. Carslaw, there is the following question: Prove the zero locus of $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^2} \sin(n x) \sin(n y) = 0$ is ...
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491 views

Analytic functions and Fourier Series

I'm taking my first real analysis course and I'm trying to get a better feel about analytic functions. My understanding is that an analytic function is one which can be written as a power series. My ...
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967 views

Prove: Fourier series of $e^{\cos x} \sin (\sin x)$ is $\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$

I'd love your help with proving that the following series $$\sum_{n=0}^{\infty}\frac{\sin (nx)}{n!}$$ is the Fourier series of $e^{\cos x} \sin (\sin x)$. I tried to find $\hat f(n)$ using ...
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How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
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200 views

$f,g$ are two continuous functions with period$=1$, are the Fourier coefficients $f*g=f(n)g(n)$?

Let $f,g$ be two continuous functions with period$=1$. Are the Fourier coefficients of $f*g$ are given by the products $f(n)g(n)$ (of the $n$-th coefficient in each series)? Thanks!
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1answer
290 views

Conditions for a finite Fourier series

Under what circumstances is the Fourier series of a function guaranteed to have a finite number of coefficients?
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176 views

How many terms in a series expansion

General: If $f \in C^1$ is a periodic function defined over some multi-dimensional space, then it should be possible to express $f$ as a FINITE fourier series. is this true of any periodic basis? is ...
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Dealing with integrals and Fourier transforms.

I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
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1answer
308 views

Sum of sinusoidals (frequency/phase of acoustic beats)

I have a function that's the sum of two sinusoidals: $ A \cos(\Theta_1 + \omega_1 t) + B \cos(\Theta_2 + \omega_2 t) $. It basically forms an acoustic beat pattern. I need to find the frequency of ...
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Intuition behind decay of Fourier coefficients

Many other posts have discussed the standard result that the smoothness of a function is related to the rate at which its Fourier coefficients decay. For example, there are proofs that show that if ...
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191 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
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160 views

Fourier series of $\operatorname{sinc}(x)$

I am wondering if the function $\mathrm{sinc}(x)=\frac{\sin x}{x}$ can be represented in terms of Fourier series? Thank you.
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393 views

Fourier series of a periodic function with infinite number of extreme points

My formula booklet in signals analysis states that a condition for the Fourier series of a periodic function $x(t)$ to exist is that one period of $x(t)$ contains a finite number of maxima and minima. ...
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Convergence of Fourier series for $|\sin{x}|$

I was solving this question I saw in a textbook. The question is : Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$. Which I had $ f(x) = \frac{a_{0}}{2} + \sum ...
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260 views

Convergence of Fourier Series

Is there an $f\in L^1(\mathbb{T})$ whose Fourier series converges a.e. on $\mathbb{T}$ but not a.e. to $f$?
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Approximating $|x|$ by a linear combination of $1, \cos x, \sin x, \cos 2x, \sin 2x$

Let $\phi(x) = |x|$ for $x \in (-\pi, \pi)$. Suppose we approximate $\phi(x)$ by a linear combination of the functions $\{1, \cos x, \sin x, \cos 2x, \sin 2x\}$. What linear combination of the form: ...
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Applying Fourier series for $|\sin x|$

Why when we apply Fourier series for $|\sin x|$ from $0 < x < \pi$ , we set $2L = 2\pi$? Shouldn't it be $2L = \pi$? In Schaum's Outline of Advanced Calculus book, there's a question that ...
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Finding coefficients of a double Fourier series

This is the end of a PDE (heat equation in 2D) I am trying to solve with bounds from $0 < x < L$ and $0 < y < H$. It is a Newmann condition problem (i.e. all derivatives of $x$ and $y$ at ...
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236 views

Obtaining Poisson's formula from a integral of summation

If $$u(r,\theta)=\frac1{\pi}\int_0^{2\pi}\Bigg[ \frac12+\sum_{n=1}^{\infty}r^n\cos n(\theta-\phi) \Bigg]f(\phi)d\phi,$$ can anyone help show me why this implies ...
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Is this sum equal to the Möbius function?

In the wikipedia page Uses of trigonometry under the section Number theory and in the page for the Möbius function there is an explanation for how to calculate the Möbius function from the GCD=1 ...
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3answers
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Fourier cosine series and sum help

I have been having some problems with the following problem: Find the Fourier cosine series of the function $\vert\sin x\vert$ in the interval $(-\pi, \pi)$. Use it to find the sums $$ \sum_{n\: =\: ...
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1answer
118 views

Using orthogonality of sines to find coefficients from a given boundary condition

I'm trying to solve Laplace's equation $\nabla^2 \phi=0$ in Cartesians on $0<x<a,\; 0<y<b,\;0<z<c$ by separation of variables. A boundary condition gives the equation ...
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Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
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For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
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solving coefficients of a fourier series

So I am just learning intro to fourier series and have a quick question regarding computation of the coefficients. Let our function be $$ f(x) = \sin{\frac{\pi x}{L}} $$ Recall that the fourier ...
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Integration of Fourier series

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be $2\pi$-periodic and integrable on $[-\pi, \pi]$. Assume that $f(x)\sim \frac{a_0}{2}+\sum_{n=1}^\infty (a_n \cos nx+b_n \sin nx )$ is its Fourier ...
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Condition for Fourier series

I read that Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series. What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a ...
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156 views

the convergence of Fourier series

Assume now we have $f(x)\in L^1([0,1])$, then we don't necessarily have the convergence of the partial sum of the Fourier series, moreover, by the theorem of kolmogorov, we can even have a.e. ...
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395 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 ...
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756 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 ...
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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} ...
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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 ...
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479 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 ...
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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 $0 < \alpha < 1$ ...
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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 ...
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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 ...
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651 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 ...
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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 ...
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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). ...
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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 ...
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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 ...
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67 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*} ...
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325 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. ...