Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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29 views

Fourier Coefficient

I have to compute the coefficient $b_3$ of the odd Fourier Series associated with the function $y=2-x$ in the interval $(0,1)$, period $2$. By using the formula $$ b_k = \frac{1}{T}\int_{-T}^{T} ...
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55 views

Moving limits inside inside an infinite sum, a special case.

I have come over a problem where I have found that this expression is most likely equal to a square wave with period 4 and phase shift 1. $$ f(t) = \lim_{n \rightarrow \infty}\sum_{k=1}^n ...
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1answer
16 views

Compute the Fourier Series of a trig function

I want to compute the Fourier series for the following function $$ g_n(\theta) = -2nK_{n}(\theta)\sin(n\theta)$$ where $K_n(\theta)$ is the Fejer Kernel. I tried to compute the Fourier coefficients ...
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1answer
47 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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2answers
25 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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0answers
33 views

Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
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26 views

Fourier Polynomials: standardly used term?

When teaching Fourier series to students, I realized that one of my references (only one or two I know that does this) calls the $n$-th partial sum of the Fourier series of an $L^2$ function $f$, the ...
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26 views

“Fourier” subsets of a complete basis

If we have some complete basis where the basis functions have a finite bandwidth in fourier space, and we are interested in reproducing a function with a finite bandwidth, we know that there is some ...
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26 views

How to transform an even function into an odd one?

Expand $$ x(t) =\begin{cases} t,& 0 < t < \pi/2 \\ \pi - t,& \pi / 2 < t < \pi \end{cases}$$ in Fourier sine series. First, $x(t)$ needs an horizontal translation into an ...
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0answers
32 views

Fourier series - Why does $\hat f(0) \ne 0$?

Let $f\in C^1$, $2\pi$-periodic, and let's assume $\int_{-\pi}^\pi |f'|^2 \le 1$. Prove: $$\sum_{n\in\mathbb{Z}} |\hat f(n)|^2 \le \frac{1}{2\pi}$$ There's a $c\in\mathbb{C}$ such that: ...
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1answer
94 views

Show that $f_n\to f$ uniformly on $\mathbb{R}$

Let $$P_n(x) = \frac{n}{1+n^2x^2}$$. First, I had to prove that $$\int_{-\infty}^\infty P_n(x)\ dx = \pi$$ And that for any $\delta > 0$: $$\lim_{n\to\infty} \int_\delta^\infty P_n(x)\ dx = ...
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2answers
42 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
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2answers
52 views

Show that $f(x)\equiv 0$.

Let $f:[0,2\pi]\to\mathbb{R}$, which is $2\pi$ periodic and continuous. It is given that for every $n\in\mathbb{Z}$:$$\int_0^{2\pi} f(x)e^{i\left(n+\frac{1}{2}\right)x} = 0.$$ Show that $f(x)\equiv ...
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40 views

Good reference for Fourier Analysis

Would you please indicate a good reference about Fourier analysis (Fourier series, convergence theorems: pointwise, uniform convergence, $L^2-$convergence...etc)? It should concern the organisation of ...
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0answers
32 views

Finding the zeroth Fourier coefficient using limit

The $\text{n:th}$ Fourier coefficient (for the $\cos(nx)$ part) is defined by $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(\theta) \cos(n\theta)d\theta.$$ Inserting $n=0$, we get $$a_0 = \frac{1}{\pi} ...
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1answer
39 views

Which way does the Fourier Transform go?

This might be a silly question, but I'm really confused by the way Fourier Transform was taught in my algorithms class, and everything else I found on the internet. The way we defined FT is first ...
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1answer
34 views

Double integral calculation and fourier transform

I try to find the following $$\int_{\mathbb{R}}^{}\int_{\mathbb{R}}^{} e^{-y(x+z)-(x^2+z^2)} dxdz$$ and I change variables $x=r\cos(\theta)$ and $z=r\sin(\theta)$ and the integral becomes: ...
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1answer
44 views

Find a function $u(x,t)$ satisfying some initial conditions for a vibrating string of length $\pi$.

Solve the following problem for a vibrating string of length $π$: Find a function $u(x, t), 0 ≤ x ≤ π, t ≥ 0$, satisfying $∂^2u/dt^2 = ∂^2u/dx^2, 0 < x < π, t > 0$ the boundary conditions ...
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0answers
22 views

Matrix representation of nonlinear functions

Let $\tau : [0,1]\rightarrow [0,1] $ be a continuous invertible map. Then the 'extension of $\tau$ to the space of square integrable real valued functions on $[0,1]$ is defined by the linear operator ...
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0answers
19 views

Coefficients of Fourier-Bessel series for a Neumann condition

What is the expression for coefficients of Fourier-Bessel series for a Neumann condition? I know what it is for Dirichlet condition. $\frac{\partial f}{\partial x} = 0$
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22 views

Expanding a formula

We have the below formula $$\int_{-\pi}^{\pi}s_n^2 (x) dx =\int_{-\pi}^{\pi}\left[\frac{a_0}{2}+\sum _{k=1}^n a_k \cos (k x)+b_k \sin (k x)\right]^2 dx,$$ using the aforementioned formula, how (from ...
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2answers
62 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
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2answers
25 views

Fourier sine series

Compute the Fourier sine series of $f(t)=t$ over the interval $[1,3]$. The question I have is that over $[-L,L]$, the cosine series is $0$ but does this still apply over the interval $[1,3]$? So ...
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0answers
91 views

Characterize a set of functions

While computing matrix elements of the evolution operator in Quantum Field Theory for the harmonic oscillator using the path integral formalism, I came across the assumption that all physically ...
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2answers
41 views

Definition of Fourier coefficients in PMA Rudin

W.Rudin in his book "Principles of MA" defines Fourier coefficients by $(62)$ i.e. $$c_m=\dfrac{1}{2\pi}\int _{-\pi}^{\pi}f(x)e^{-imx}dx \qquad (62)$$ But in 8.10 Definiton for orthonormal system ...
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13 views

Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]

This is probably a ridiculous/simple question but I am still having trouble! Find appropriate sine and cosine terms and coefficient formulas to approximate a function on [1,3]. I understand how to ...
3
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1answer
50 views

Which initial functions can be solved by separation of variables

consider the wave equation on $(0,\pi)$, i.e. let $f,g\in L^2(0,\pi)$ be two fixed functions and consider the following problem: $$\partial_{tt}u=\partial_{xx}u, \quad x\in(0,\pi), t\in (0,\infty) \\ ...
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1answer
20 views

convergence of series in inner product space

let $V$ be some inner product space and $\lbrace {e_i\rbrace }_{i\in\mathbb{N}} \subset V$ be some countable orthonormal set. I am wondering if for any $x\in V$ the series $$\sum\limits_{i=1}^{\infty} ...
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1answer
60 views

Fourier series of: $[\log(\sin x)]^2$

What is the Fourier expansion of: $${ \left[ \log\left( \sin x \right) \right] }^{ 2 }$$ This is a well known Fourier series: $$-\log(\sin x ...
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1answer
76 views

Asymptotic expansion of $f(x)= \sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$ at the origin

The function $$f(x)= \sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$$ is odd, uniformly convergent on all intervals $[\epsilon,\pi]$ for $0 < \epsilon < \pi$. Hence $f$ is continuous on $(0,\pi]$. ...
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0answers
38 views

$1 / (2 \pi)$ factor in Fourier transform

I have been unable to see why the $1 / (2 \pi)$ appears in Fourier transform. Would you please justify it to me? Problem: Let $$ f(x) = \int_{-\infty}^{\infty} \mathrm{d} k \, e^{ikx} \tilde{f}(k) ...
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1answer
21 views

Why does this Fourier inner product equal this sum?

This is part of a derivation in a text that I am struggling to follow. It says that if we write $e_k(t) = e^{2 \pi i k t}$ then $$\langle \sum_{n=- \infty}^{\infty} \langle f, e_n \rangle ...
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0answers
13 views

Quesion about Parsevals formula for Fourier-Legendre Series

Question: A function $f(x)$ defined on $(-1,1)$ can be expanded as $$ f(x) \backsim \sum_{n=0}^{\infty} c_nP_n(x) $$ What do Parsevals formula look like for this expansion? My solution: Ok so I ...
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1answer
31 views

Deriving the coefficients during fourier analysis

I'm self-studying Fourier transforms, but I'm stuck on a basic point about integration during the derivation of an expression for the coefficients of the Fourier transform. For a function of period ...
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0answers
24 views

Complex coefficient in Fourier series

Why can $$\sum_{n=1}^N a_n \frac{e^{2 \pi i n t} + e^{-2 \pi i n t}}{2} + b_n \frac{e^{2 \pi i n t} - e^{-2 \pi i n t}}{2i} $$ be written as $$ \sum_{n=-N}^N c_n e^{2 \pi i n t}$$ for some setting of ...
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1answer
28 views

How to find the Fourier's coefficient $a_n$ of the Fourier's series of $\sin(x)$ on $(0,\pi]$, $0$on $(-\pi,0]$

Considering $g(x)$, periodical with a period of $2\pi$ defined by \begin{equation*} g(x)= \begin{cases} 0 & \text{for $x \in (-\pi;0]$} \\ \sin(x) & \text{for $x \in ...
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1answer
32 views

How to show that the Fourier's series of $f(x)=x$ uniformly converges?

How to show that the Fourier's series of $f(x)=x$ uniformly converges? After finding its coefficient, I got: $$\sum\limits_{n=1}^{+\infty}\frac{2(-1)^{n+1}}{n}\sin(nx)$$ I showed the pointwise ...
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0answers
55 views

Pointwise version of Fejer's theorem (convergence of Cesaro means)

Prove a pointwise version of Fejer's theorem: If $f\in \mathscr{R}$ and $f(x+),f(x-)$ exist for some $x$, then $$\lim \limits_{N\to \infty}\sigma_N(f;x)=\frac{f(x+)+f(x-)}{2},$$ where ...
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1answer
25 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
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1answer
56 views

How these fractions becomes this?

Today I was trying to solve an integral for a Fourier series. I looked at the solution and this was the solution: \begin{align*} C_n &= \frac{1}{2\pi} \int_0^{2\pi} x^2 e^{-inx} \,\text{d}x \\ ...
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1answer
21 views

Fourier Series of saw tooth function

I have a function $f(x) = \frac{x}{\pi} \in (-\pi , \pi]$ I googled but couldn't find a solution done using complex exponential and I tired to do it as follows. $$a_k = ...
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1answer
32 views

Dirichlet theorem and expansion of fourier series

Dirichlet's theorem says that any function $f(x)$ on the interval $[-a,+a]$ can be expanded as a Fourier series: $$f\left ( x \right )=\sum_{n=0}^{\infty}\left [ a_{n} \sin \left ( \frac{n\pi ...
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2answers
52 views

Proof of an infinite sum using Fourier Series

I was revising for my calculus exam and I came across a question that asked to find the Fourier Series of $f(x)=1+x$, on $-1<x<1$, which I did. Which I found to be: $$f(x) = ...
5
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3answers
119 views

Obtaining Fourier series of function without calculating the Fourier coefficients

In this question in one of the answers it's shown how to get from $$f\left ( x \right )=\sum_{n=1}^{\infty}\frac{\sin\left ( nx \right )}{10^{n}}$$ to $$f\left ( x \right )=\frac{10 \sin ...
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4answers
113 views

Application of Fourier Series and Stone Weierstrass Approximation Theorem

If $f \in C[0, \pi]$ and $\int_0^\pi f(x) \cos nx\, \text{d}x = 0$ , then $f = 0$ Define $ g(x) = \begin{cases} f(-x) & \text{if } -\pi \leq x < 0;\\ f(x) & ...
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1answer
25 views

Problem regarding Euler's formula and finding Fourier coefficient

I'm learning about Fourier series at the moment and there is an example in the literature that I have trouble following. (I'm sorry if I wrote a misleading title for the problem, I was not sure what ...
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1answer
34 views

Finding a limit involving Fourier series and Dirichlet's kernel

Find the limit $$\lim_{n\to\infty} \int_0^{2\pi} (x+\frac{\pi}{2})^2 \frac{\sin((n+\frac{1}{2})x + x\cos nx}{\sin\frac{x}{2}}\ dx$$ So we may define $f = (x+\frac{\pi}{2})^2$ and then look at the ...
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1answer
91 views

Representing the function $f\left ( x \right )=\frac{1}{e^{2}e^{\cos\left ( x \right )}-1}$ in terms of Fourier series

The function is periodic with main period of $2\pi$, and it is even. So only the coefficients of the cosine terms remain. Wolfram alpha gives the result for $a_{0}$ as follows: I guess it is only ...
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1answer
35 views

What is the Fourier series of $f(x)$

What is Fourier series of $$f(x) = \sum_{n=1}^\infty \frac{\cos nx}{2^n}$$ Now, it was claimed that since $f(x)$ converge uniformly and: $$f(x) = \sum_{n=1}^\infty \frac{e^{inx} + ...
3
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1answer
69 views

Fourier series of $\frac{1}{5+4 \cos x}$ using contour integration

The function $$f(x)=\frac{1}{5+4 \cos x}$$ is periodic with the main period being $T=2\pi$. The graph is easily obtained, but here is a graph from Desmos as it looks better: The function is even, ...