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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fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
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9 views

When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
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16 views

fourier transform extrapolation - formula for one frequency [on hold]

Sorry guys if that's a simple question. I really can't find the answer. What is a formula for a function generating a sinus of selected frequency based on data available from results of fourier ...
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10 views

What is the formula for single frequency generation function obtained from FFT?

What is the correct formula of a function that generates specific tone from fourier transform? I thought that having: transformata - an array with FFT of a source sample. v = transformata[freq] - ...
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15 views

Fejer's Theorem in relation to the Fourier Transform

I have this question that relates the Fejer theorem with the Fourier Transform. Any help would be appreciated. If $f$ is of moderate decrease then $$\int_{-R}^{R}\left(1-\frac{|\xi|}{R}\right) ...
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1answer
38 views

Why do Fourier Series only work from $- π$ to $π$?

Take Euler's famous example: $$\dfrac{1}{2} x= \sin x-\dfrac{1}{2} \sin 2x+ \dfrac{1}{3} \sin 3x- \dfrac{1}{4} \sin 4x+\cdots $$ What is the reason this only works on $[-π,π]$?
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30 views

Proof that these are Fourier coefficients

I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$ In Murphy's book ...
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15 views

WHY?The Squared Euclidean Imbalance are equal to Fourier coefficients

I'm reading the classical paper about distinguishing attack, How Far Can We Go Beyond Linear Cryptanalysis ,Thomas Baign`eres, Pascal Junod, and Serge Vaudenay. The only proposition I don't ...
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13 views

Obtain the complex Fourier Series of the following function:

$$f(t)=t^3 \;\;\;\;\;\;\;\;\;\;\;\; -3/2<t\leq 3/2 $$ $$f(t)=f(t+3)$$ I've tried setting up an integral for $C_n$ coefficients using the formula $$C_n = \frac{1}{L} \int^{L/2}_{-L/2} f(t) ...
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12 views

Amplitude Spectrum, Nyquist Frequency, mixed/min/max wavelets

The problem is here. Now I know the definition of mixed/max/min phase wavelets, whether the roots lie within the unit circle or not. Starting from n = 1, let $$ x_t = ( 5, 6) $$ $$ X(z) = 5 + 6z $$ ...
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1answer
29 views

Why is the zeroth coefficient in a Fourier series divided by 2?

I just learned that $a_0$ is basically the average of a function $f(x)$ on the interval $[-\pi, \pi]$, and that a Fourier series is given by $$ f(x) = a_0 + \sum_{n=1}^\infty (a_n \sin(nx) + ...
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1answer
32 views

How can I find this integral for a fourier series?

I have to calculate the following integral $$ b_n = \dfrac{1}{\pi} \int_{-\pi}^{\pi} \dfrac{1}{2}x \sin nx dx$$ The correct answer is apparently $$\dfrac{(-1)^{n-1}}{n}$$ But I have no idea how I ...
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1answer
30 views

Evaluate the following Dirac delta integrals:

a) $ \int^{+\infty}_{-\infty} \delta'(t-\pi)e^{-t^2} \; dt$ b) $ \int^{+\infty}_{-\infty} \delta(-3t)(\frac{e^{-t^2}}{\ln(t^2 + 3)}) \; dt $ c) $ \int^{+\infty}_{-\infty} \delta(4t)\sinh{t^2} ...
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Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
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19 views

Value of $L$ in Fourier series when $-\frac{\pi}{2}<x<\frac{\pi}{2}$ for $|\sin{\pi x}|$

The period of $|\sin{\pi x}|$ is $1$. So $T = 1$ and $L$ is $\frac12$ (from calculation of $a_0$,$a_n$,$b_n$ in Fourier series). Now, when we define $-\frac{\pi}{2}<x<\frac{\pi}{2}$ for ...
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2answers
33 views

Fourier transform of 1 cycle of sine wave

Consider the signal: $\begin{align*} f(t) &= \sin(\omega t) \tag{$0 \leq t \leq 2\pi/\omega$}\\ &= 0 \tag{elsewhere} \end{align*}$ How to compute the Fourier transform of $f(t)$? I ...
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1answer
20 views

Fourier series with a weighted mean square norm

I am interested in Fourier series with a non-uniformly weighted error norm. What I mean by this is that the usual Fourier series of a periodic function is a minimizer of the mean squared error: $$ J_N ...
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16 views

Convergence of Fourier coefficients of a periodic function

Given a function $g: [0,\pi] \to \mathbb{R}$, if for example $g(0) = g(\pi) = 0$ and we write the odd and periodic extension of $g$ as a Fourier series $$ g(x) = \sum_{m=1}^{\infty} {\hat{g}_m \sin{m ...
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1answer
21 views

Please help with this Discrete fourier transform question

Consider the ODE $\frac {d^2u}{dx^2} + 2\pi\frac {du}{dx} + \frac 54\pi^2u = g(x)$ where g is a periodic fuction with period 1 given by $g(x) = e^{\pi x}$ , $ 0 \le x \lt 1$. It is desired to find ...
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25 views

Triangular signal Fourier transform. Please help me [closed]

Please help me. I don't know how to solve part b) (the triangular signal fourier transform).
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1answer
25 views

Is the Fourier Series correct?

Could you tell me if the following Fourier series of the function $f(x)=x^2, -\frac{L}{2} \leq x \leq \frac{L}{2}$ is correct?? $$$$ $$a_0=\frac{2}{L} ...
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40 views

Show sum involving sines is non-negative

I want to show that \begin{equation} \sum_{\substack{k \geq 1 \\ k \text{ odd}}} k e^{-k^2 a} \sin(kx) \geq 0 \qquad \text{for all } x \in [0,\pi], \, a > 0. \end{equation} How should I start? I ...
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8 views

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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1answer
18 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
25 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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7 views

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
36 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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10 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
39 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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2answers
26 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
25 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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1answer
18 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
29 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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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
50 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
30 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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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
18 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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2answers
41 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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When has Discrete Fourier Transform unitary absolute value?

I have a vector $\mathbf{v}$ with $N$ complex components. Only the first $N_1$ components are not null, i.e. $v_i \ne 0$ only for $i \in \{0, \dots,N_1-1\}$. Now, I compute the Discrete Fourier ...
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2answers
114 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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26 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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30 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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32 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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44 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
41 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
106 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
90 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 ...