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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25 views

Every vector in a Hilbert space has a Fourier representation wrt an orthonormal sequences?

I'm reading Kreyszig's text, and there is a Theorem in section 3.5 stating: Theorem: Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$. Then 1) If $\sum_{k=1}^\infty \alpha_k e_k$ ...
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26 views

Which comes first, Fourier Series or Fourier Transform/Analysis?

I am starting to learn Fourier and Laplace series, transform, derivation, etc. What should I start with? Which comes first, Fourier Series or Fourier Transform?
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16 views

Square Integrable Functions Formula

The book reads as follows: "Let $a(x)$ and $b(x)$ be square integrable functions defined on [$a , b$]. First we note that it follows from the elementary inequality $|ab| \le 1/2 (a^2 + b^2)$ ...
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22 views

What do Fourier Series for Other Symmetric Operators Look Like?

I understand that Fourier analysis works (up to constant multiples) by considering the inner-product space $E$ of smooth functions $[-\pi,\pi] \to \mathbb C$ with inner product. . . $\displaystyle ...
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1answer
44 views

Epicycles as precursors of Fourier series

How convincing an argument can be formulated to claim that the Ptolemaic epicycles were actually an early precursor of Fourier series? Ptolemy lived ~200AD, and so well pre-dates Fourier ~1800.
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1answer
36 views

How to use the 3rd and 4th boundary conditions in this?

I was solving $$ \frac{\partial^2 u}{\partial t^2}=\frac{\partial^2 u}{\partial x^2}$$ All the boundary conditions are as follows:- $$u(0,t)=0 \\ u(\pi ,t)=0 \\ u(x,0)=\sin x \\ u_t(x,0)=x^2$$ ...
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4answers
66 views

Why cannot $A\sin\alpha x +B\cos \alpha x$ be zero?

I was going through solving wave equations using fourier and I came across a note saying $A\sin\alpha x +B\cos \alpha x \neq 0$ I believe this applies to $\alpha ,A,B\neq 0$ I was solving $$ ...
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1answer
126 views

Convergence of Fourier sine and cosine series

Discuss whether or not it is possible to have a Fourier series $$a_0+\sum_{k=1}^\infty[a_k\cos(kx)+b_k\sin(kx)]$$ converge for all $x$ without either $$a_0+\sum_{k=1}^\infty a_k\cos(kx) ...
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1answer
56 views

Finding Fourier series constant and integral

I have been studying Griffith's Intro to Electrodynamics. I am studying differential equations and Fourier series. I am studying the problem discussed here: Why is this allowed? ("Fourier's ...
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1answer
32 views

Trigonometric integrals and limits

Show $$\lim_{N\to\infty}g_N(\theta_N)=2\int^\pi_0\frac{\sin x}{x}dx-\pi,$$ where $$g_N(\theta_N)=\int_0^{\theta_N}\frac{\sin[(N+1/2)x]}{\sin(x/2)}dx-\pi,$$ $$\theta_N=\frac{\pi}{N+1/2},$$ and ...
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1answer
15 views

Writing a function $f : [-\pi,\pi) \to \mathbb{R}$ as $\sum c_k e^{ikx}$ where $c_k$ is to be found

I have a function on $[-\pi, \pi)$ defined as: $$ f(x) = \begin{cases} -1 & \mbox{if} \;x \in [-\pi,0) \\ 1 & \mbox{if} \;x \in [0,\pi) \\ \end{cases} $$ And I have to write it in the form ...
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2answers
47 views

'Obtain' the Fourier transform

If $g(t) = e^{-a|t|}$ and a is a real positive constant, obtain the fourier transform. I'm a bit unsure what this is asking. I can write out the expression for the fourier transform. Should I stop ...
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0answers
15 views

Using the Fourier Series in Variational Optimization Problems

Say I have a functional $L(f)$ which takes as input the function $f:\mathbb{R}\to\mathbb{R}$, and I want to find the function that optimizes $L$. Unfortunately, there's no way to define a functional ...
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1answer
53 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
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0answers
37 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
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1answer
61 views

Challenge in trignometry and integration [closed]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
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1answer
35 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
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33 views

Fourier sine and cosine: reconstruction depends on 'noise data' outside signal

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
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33 views

Sawtooth wave as a sum of sines

Wikipedia gives the equation for a sawtooth waveform composed as a sum of sines as: $$ x_\mathrm{sawtooth}(t) = \frac{A}{2}-\frac {A}{\pi}\sum_{k=1}^{\infty}\frac {\sin (2\pi kft)}{k} $$ Where $A$ ...
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1answer
441 views

Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers ...
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2answers
37 views

Fourier series and convolution

Let $f$ and $g$ be $2\pi$-periodic, piece-wise smooth functions having Fourier series $f(x)=\sum_n\alpha_ne^{inx}$ and $g(x)=\sum_n\beta_ne^{inx}$, and define the convolution of $f$ and $g$ to be ...
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1answer
20 views

Is the complex form of the Fourier series of a real function supposed to be real?

The question said to plot the $2\pi$ periodic extension of $f(x)=e^{-x/3}$, and find the complex form of the Fourier series for $f$. My work: ...
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1answer
34 views

Fourier cosine series giving nonsense answer

I'm currently trying to find the cosine Fourier series of $f(x) = \left | \sin \frac{\pi n }{L} x\right |$ on the interval $0 < x < L$. I first started by calculating the first term of the ...
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109 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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1answer
23 views

Problem calcualting Fourier coeff. of tent function.

Consider the tent-function on $[-\pi,\pi]$ depending on some $\delta$. I.e $(1-\frac{\mid x \mid}{\delta})$, $x$ is zero when larger then $\delta$ When I compute ...
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1answer
31 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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0answers
56 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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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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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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“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
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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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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
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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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0answers
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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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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
40 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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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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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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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
63 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
26 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
42 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 ...