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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5 views

Unique real sequence $(U_n)_{n\ge0}$ such that $ f_x(\theta)=U_0(x)+2\sum_{n=1}^{\infty}U_n\cos(n\theta) $

Let $f_x(\theta)=e^{x \cos(\theta)}$ I have to show that exist a unique real sequence $(U_n)_{n\ge0}$, $x\rightarrow U_n(x)$ such that for all real $x$ and $\theta$: $$ ...
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0answers
15 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
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0answers
24 views

Full series expansion of the floor function

We know if $x$ is not an integer we have $$\left \lfloor x \right \rfloor=x-\frac{1}{2}+\frac{1}{\pi }\sum_{k=1}^{\infty}\frac{\sin(2\pi kx)}{k}$$ Is there an series expansion of floor function ...
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0answers
22 views

Using Fourier Series to solve differential equations [on hold]

Given the equation: $\frac{d^{2}i}{dt^{2}} + 10\frac{di}{dt} + 10i = \frac{dv}{dt}$ in which $v(t) = 10(\pi ^{2} - t^{2})$ and $-\pi \leq t\leq \pi $, $v(t+2\pi ) = v(t)$. Find a particular solution ...
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1answer
21 views

Question about the frequency domain and the fourier transform

if you have a signal say x(t) in continuous time and you transform it using the Fourier transform for continuous time you get X(w) which is the frequency domain representation of this signal x(t). ...
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1answer
26 views

Complex Fourier Series of $|x|$

How would I write the Fourier series for $|x|$ in complex form over the interval $[-2,2]$? I have already tried writing $$|x|=\sum c_ne^{i\pi nx/2}$$ where ...
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1answer
40 views

Leibniz series for $\pi$ using an integral of the Dirichlet kernel

I'm trying to create a proof of the Leibniz series $\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = \frac{\pi}{4}$ using the Dirichlet Kernel. What I did is start with the kernel $$1+2 \left ( 1+\cos\theta ...
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0answers
19 views

Combining even and odd parts of a Chebyshev series

I imagine this will be an easy problem, perhaps even routine, for some. I am learning to manipulate sums and need insight. I started with a power series $$s(x) = \sum_{n=0}^{\infty} a_n x^n$$ and ...
4
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53 views
+100

Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
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2answers
27 views

fourier series, parsevel's identity

i need to solve the following question using parsevel's identity. $\displaystyle\int_{-\pi}^{\pi} \cos^{4} x\, dx = \frac{3\pi}{4}$. $\displaystyle\int_{-\pi}^{\pi} \cos^{6} x\, dx = ...
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1answer
28 views

Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
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0answers
20 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...
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1answer
50 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
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2answers
32 views

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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0answers
11 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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0answers
20 views

fourier transform extrapolation - formula for one frequency [closed]

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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0answers
13 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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0answers
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
39 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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34 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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16 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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0answers
14 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
32 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) + ...
2
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1answer
34 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
32 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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22 views

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
37 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
24 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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0answers
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
23 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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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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0answers
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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0answers
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
28 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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0answers
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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11 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 ...
3
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1answer
42 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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0answers
29 views

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 ...
2
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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
19 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
30 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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0answers
12 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. ...
2
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1answer
51 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 ...
0
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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 ...