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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Nyquist Sampling Theorem

I'm working on the proof of the Nyquist sampling theorem. Mainly I'm wondering about the regularity conditions. In particular, suppose $f$ is continuous on $\mathbb{R}$ and $\hat{f}(k)=0$ unless ...
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24 views

Expanding unity in terms of orthogonal functions cos( alpha(i) * y)

It is written in the book I am reading without proof that if we expand unity in terms of orthogonal functions cos( alpha(i) * y), we get: (Please check this link) ...
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1answer
46 views

Proof of Parseval's Theorem for Fourier Series

Ok so I want to prove the above expression, I substituted the complex fourier series for f and using the fact f may be complex-valued, carried on by representing $|f(x)|^2$ as $f(x)f(x)^\ast$ where ...
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Is there a specific name for Fourier cosine series divided by its input?

I am thinking about a univariate model $$ y=a_{0}x^{-1}+\sum_{k=1}^{n}a_{k}\cos(kx)x^{-1}. $$ It seems that this form looks like a Fourier cosine series w.r.t. $x$ divided by $x$. Could you tell me ...
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1answer
38 views

Termwise Integration of Fourier Series

This is a question from Edwards and Penney 4th edition Differential Equations and Boundary value problems from section 9.3. Suppose that $f(t)$ is a piecewise continuous period $2L$ funtion. ...
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20 views

Express the fourier coefficients of a autocorrelation function (complex form)

Here are some hints from the instructor: "Just plug the expression for the autocorrelation function into the formula for the Fourier coefficients, you get a double integral, and then smuggle in an ...
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1answer
67 views

Poisson summation formula clarification regarding Fejer kernel

Define $$\mathbf{F}_R(t) = \begin{cases} R \left(\dfrac{\sin(\pi R t)}{\pi R t}\right)^2 & t \neq 0\\[10pt] R & t = 0 \end{cases} $$ A problem in Stein's Fourier Analysis asks ...
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This $\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$ integral: does a closed form exist?

$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{in x}dx}{1+\tan^m(x)}$$ Does a closed form for the above exist, ideally for $n,m\in\mathbb{C}$ (most bounds probably removed at one point using ...
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35 views

Prove $\frac1T \int_0^T\left(\sum_{k=-\infty}^{\infty}c_ke^{j{\frac{2\pi kt}{T}}}\right)^2dt= \sum_{k=-\infty}^{\infty}|c_k|^2$

This question relate to fourier series in electrical engineering but I post it here as it's only mathematical concern. I cannot prove this $$\frac1T ...
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1answer
27 views

Contradiction between $a_0$ and $a_k$ for Fourier Series

I need to calculate the Fourier Series for the function $f(x) = |x| \; f:[-\pi,\pi] \to \mathbb{R}$ When calculating $a_k = {1 \over \pi} \int_{-\pi}^{\pi} f(x) \cos{(kx)} dx \; (k \in \mathbb{N_0})$ ...
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26 views

Hölder Condition for Fourier Series

So I'm trying to prove that the function (as represented by a Fourier series) $ f(x) = \sum_{k=0}^\infty 2^{-k\alpha}e^{i2^kx}$ satisfies the Hölder Condition: $|f(x+h)-f(x)| \le C|h|^\alpha$, with $0 ...
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1answer
41 views

Fourier decomposition of the Mandelbrot set

It is not clear that the boundary of the Mandelbrot set is an analytic curve, even though it is connected. Nevertheless, we can approximate the boundary with a curve by iterating a finite number of ...
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13 views

Nyquist Frequency, filter, phase/amplitude

Problem The problem seems quite simple and I believe it is though I have no idea how to approach it. I have tried googling 'Nyquist frequency' but have not had any luck with problems similar to this. ...
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1answer
68 views

Fourier Series/Parseval's Theorem

I have pretty much completed this question and have found the Fourier representation to be; $$ f(x) =\frac A2 +\sum_{n=0}^\infty 2A\frac{\cos(((2n-1)(\pi x))/2f_o)}{\pi(2n-1)} $$ Now I don't ...
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1answer
42 views

Calculating Fourier expansion using Legendre Polynomials

I'm trying to write any function of the type $t^m$ using Legendre polynomials $P_n(t)$ . That means: $$t^m=\sum_{n=0}^\infty\langle P_n,t_m\rangle P_n =\sum_{n=0}^\infty a_{mn}P_n$$ Where I have to ...
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47 views

Fourier expansion of the complexified Gram series

Consider the Riemann's R function, also known as the Gram series: $$\text{R}(x)=1+\sum_{k=1}^{\infty}\frac{\left(\log x\right)^{k}}{kk!\zeta(k+1)}$$ Now consider the form: $$\text{R}\left(e^{2\pi ix} ...
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1answer
24 views

Discontinuous functions with finite Fourier series approximation?

Yesterday I posted a question regarding the computation of complex Fourier coefficients for the functions $$f(t) = \sin(2 \pi t)$$ $$f(t) = |\sin(2 \pi t)|$$ where $0 \leq t \leq 1$. The first ...
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50 views

Finding complex Fourier coefficients

This is probably an easy question, but I'm a little bit stuck, so any help will be appreciated. PROBLEM Find the complex Fourier coefficients of: $$f(t) = \sin(2\pi t)$$ and $$f(t) = |\sin(2\pi ...
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2answers
118 views

Summing a series (from a physics problem)

How might we show that $$\sum_{k = 0}^{\infty}\frac{2}{2k + 1}e^{-(2k + 1)\pi x/a}\sin\left( \frac{(2k + 1)\pi y}{a}\right) = \tan^{-1}\left( \frac{\sin(\pi y/a)}{\sinh(\pi x/a)} \right) $$ where $x, ...
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32 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
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33 views

Finding the number of derivatives for series problems

I have the following problem: How smooth are the following functions? That is, how many derivatives can you guarantee them to have? $$a)\;\;\;\;\; ...
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54 views

Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
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46 views

Convergence of Fourier series in finite number of terms

Let $f(t)$ be a continuous function that is periodic on some interval $[0,p]$ on its domain. Let $\omega = 2\pi/p$ and observe that the Fourier series of $f(t)$ is given by $$ f(t) = ...
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1answer
48 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
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1answer
162 views

Using the Parseval Identity to compute $ \sum_{n=1}^{+ \infty} \frac{1}{(4n^2-1)^2}$

Parseval's Identity: For continuous $f: [- \pi , \pi] \to \mathbb{R}$ $$ \sum_{n=- \infty}^{+ \infty} |c_n|^2 = \frac{1}{2 \pi} \int_{ - \pi}^{ \pi} |f(x)|^2dx, \text{ where } c_n = ...
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2answers
104 views

Trigonometric series problem

I have the following problem from my Fourier analysis book I would need some guidance with. I have tried it, but apparently I made some mistakes...here is my problem: We have: $$\sin \theta ...
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2answers
69 views

Fourier Series - Integration

Could someone explain where I am going wrong with the following fourier series calculation please? I'm trying to compute the $A_{0}$ and $A_{n}$ coefficients for the fourier series: \begin{align} ...
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1answer
24 views

A relation for Fourier series

For $f$ and $f'$ in $L^2(0,1)$, define $e_k(x)=e^{2\pi ikx}$, $k \in \mathbb{Z}$. And define the Fourier series: $f=\sum _{k \in \mathbb{Z}}c_ke_k$, where $c_k=\left \langle f,e_k \right ...
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40 views

Creating a function which satisfies a given set of points

I have been tasked to write a program in matlab which will approximate a function $f(t)$ as a sum of sines and cosines given that it is defined in the domain $0 - 2\pi$. I have a set of points that ...
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2answers
33 views

How does one derive the complex form of the Fourier series?

Specifically, I have gone from the Fourier Series in this form: $$\sum\limits_{n=1}^{\infty} a_n\cos(nx) +b_n\sin(nx)$$ and I have taken it to this form: $$\sum\limits_{n=1}^{\infty} \frac{(ib_n - ...
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1answer
51 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
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How can we derive $\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)]$?

When I was reading digital communication theory, I couldn't derive following equation $$\frac{1}{2j}\mathbf{F}^{-1}[Y_b(f-f_c)-Y_b^*(-f-f_c)]=\frac{\sqrt{2}}{2j}\mathbf{F}^{-1}[Y(f)u(f)-Y(f)u(-f)] $$ ...
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53 views

$f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$

The problem I am considering is: For $f$ a $2\pi$-periodic and Riemann integrable function, show that $f$ is real valued iff $\overline{ \hat{f}(n) } = \hat{f}(-n)$. Here $\hat{f}(n)$ represents the ...
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1answer
23 views

Fourier Coefficients of a Sequence of Functions

Let $f_k$ be a sequence of Riemann integrable functions over $[0,2\pi]$ such that $$\lim_{k\rightarrow\infty}\int_0^{2\pi}|f_k-f|=0$$ for some function $f$. Let $\hat{g}(n)$ denote the $n$th Fourier ...
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47 views

How to find the value of this sum?

The sum below numerically (to 13th digit at least) is the same as $\ln 2$. So there should be a way to prove it analytically, but I haven't succeeded. Any suggestions? ...
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Expanding a piecewise defined function, what will the series converge to at $x=-1,0,1$? [closed]

If we expand $$f(x)=\begin{cases} (x+1) & -1<x<0; \\ -x & 0<x<1 \end{cases}$$ what will the series converge to at $x=-1$, $x=0$, and at $x=1$? Hey I tried to work this out on ...
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1answer
32 views

Is $f(t-N t_0)=\sin(2\pi f_0t)\cos(2\pi f_1t)$ always true?

Is it true that multiplying two sinusoidal functions, always result in some periodic waveform. i-e $$f(t-N t_0)=\sin(2\pi f_0t) \cos(2\pi f_1t)$$ If so, then how can we calculate the period ( ...
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1answer
83 views

Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
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36 views

Is it possible to solve a system of equations comprising FFTs?

Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in the Real domain. Also consider $F(x)$ as the *Fast Fourier Transform function* (the ...
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51 views

How did Fourier find the formula for the fourier series coefficients?

The modern proof use the dot product but did he use that ?
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The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
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PSD Rational function to power series

I have a Power Spectral Density given as a Rational function that is: \begin{equation} \phi(e^{i\omega}) = 1/(1-1.7464e^{-i\omega}+1.2602e^{-2i\omega}-0.4366e^{-3i\omega}+0.625e^{-4i\omega})^2 ...
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Decay of Fourier coefficients of $\frac{1}{f}$

Let $\alpha > 0$ and define \begin{equation*} \mathbb{H}^{\alpha}\left[-\pi,\pi\right] = \left\{f:\left[-\pi,\pi\right]\mapsto\mathbb{R} \;s.t.\; \sum\limits_{n\in\mathbb{Z}} \left\lvert ...
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856 views

Why is this allowed? (“Fourier's Trick”; finding the coefficients in a Fourier Series)

In my textbook (Introduction to Electrodynamics, D. Griffiths), we derive the equation for some strange potential function. Eventually, we get to this (for $n \in \mathbb{Z}^+$): $$ V_0(y) = ...
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1answer
19 views

Fourier coefficients.

I don't quite see how the following hold and would appreciate an explanation: (1) The Fourier coefficients of $cos(\frac{6\pi n}{N})$ are $\delta[k-3]+\delta[k+3]$ (2) ...
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94 views

Proof of Wirtinger inequality

Quoting from Ana Cannas da Silva's book on Symplectic Geometry: "As an exercise in Fourier series, show the Wirtinger inequality: for $f\in C^1([a,b])$, with $f(a)=f(b)=0$ we have $$ ...
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4answers
144 views

Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
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27 views

Discrete Fourier Transform of the infinite series

I am reading this book and having hard time understanding how to get to eq(2) from eq(1) $$P(k,t) = e^{-\alpha t} \sum\limits_{l,m=-\infty}^\infty (-i)^m e^{ik(l+m)} I_l(\alpha t) I_m(i\beta t) ...
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13 views

Is it true that $\theta_{1,1}^{4N} \in J_{2N,2N}(2N)$?

I need examples of Jacobi forms for full congruence subgroups $\Gamma(N) $ of $SL(2,Z)$. As a particular case, take the theta function $\theta_{1,1}(t,z) := \sum_{n\in\mathbb{Z}} exp(\pi it(n + ...
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44 views

About Fourier coefficient definitions

I'm studying Fourier analysis and my book gives the following definitions for the Fourier series and Fourier coefficients: Fourier series of $2\pi$-periodic function $f(\theta)$ is defined as: ...