Questions tagged [fourier-analysis]

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Fourier analysis - limit calculation

The following is a question from an exam i had a while ago, didn't manage to find any direction to solve it. I think it's supposed to be solved using the Riemann-Lebesgue lemma, where $\lim_{n\to \...
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How Fourier series using Fourier integral is equivalent to expanding signal to its exponential form, when finding Fourier series coefficients?

I'm studying an introductory course to Signal and systems.And we were taught to find Fourier series coefficients using both integration and by expanding the signal (i.e., $~\sin/\cos~$ terms) to its ...
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Can't follow the proof of Fourier coefficient of distribution iff $\sum_\eta(1+|\eta|^2)^{-k}|S_\eta|^2<\infty$

This is the proof that $S_i$ is fourier coefficients of a distribution iff $\sum_{i\in Z}(1+i^2)^{-k}S_i<\infty$. Let $S_i$ be fourier coefficients of a distribution $S$. Then $\exists C,\forall ...
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Why do we put $w=s$ and $x=t$ while solving Fourier transform questions

$F(s)=$$\int f(t) \sin wt\, dt$ but we put $t=x$ and $w=s$ any reason behind this
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Why must the length of a sequence under the discrete Fourier transform be equal to the input sequence length?

Let's consider a continuous signal, $f(t)$, which has been sampled $N$ times, with spacing $T$ between samples. We denote the $N$ samples $f[0], f[1], ..., f[N-1]$. The Fourier transform of the ...
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Does the use of $\delta(\omega-\omega_1)$ for retrieving amplitude of a single Fourier-component of a periodic function violate Fubuni's requirements?

I've got a non-negative function $f(x)=a+b\cos(\omega_1 x)$ for $a,b,\omega_1 \in \mathbb R$ and I need to represent $b^2$ as a square of the Fourier-transform of a whatever function. As a physicist I ...
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Help with an example from Rudin's RCA. $\sum_{n=0}^N |b_n|^2 r^{2n}=\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(z_0+re^{i\theta})|^2d\theta$? [duplicate]

I am looking at an example from Rudin's RCA, however, there is an equation I cannot figure out. Below, why do we get $\sum_{n=0}^N |b_n|^2 r^{2n}=\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(z_0+re^{i\theta})|^...
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Help with finding the function that satisfies conditions of a Fourier Sine Integral Coefficient

I understand that the following function is the Fourier Sine Coefficient for the function. However, I am having trouble finding the function that can satisfy such conditions. How do I go about finding ...
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Convergence of Fourier transform in $L^p$

Consider a sequence $f_n \to f$ convergent in $L^2(\mathbb{R}^N) \cap L^1(\mathbb{R})$. Prove that the respective Fourier transforms $$ \hat{f_n} \to \hat{f} $$ in $L^p(\mathbb{R}^N) \ \forall p > ...
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Preserving continuity of periodic functions under fractional-integration-type transformations

Assume that $f$ is a continuous and periodic function over $\mathbb{T}=[0,1)$, and denote by $f_n$, $n \in \mathbb{Z}$, its Fourier series. Let $(a_n)_{n\in \mathbb{Z}}$ be a sequence such that $n^a ...
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Discrete Fourier transform of $(1,1,1,1)$

I am asked to determine the Fourier transform of $(1,1,1,1)$. In the solution I found this: I don't get how is he transitioning from the $\omega$'s to $-i, i, -1, 1$ etc... How to break it down, so ...
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Good kernels that exist in the real world.

A Good kernel (bounded approximate identity) in $L^1(\mathbb{T})$ is a sequence of functions $\{g_n\}$ satisfying: (i) $\frac{1}{2\pi}\int_0^{2\pi}g_n(t)dt=1~~$ (ii) $||g_n||_1=O(1)~~$ (iii) $~~\...
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Discrete Fourrier transform

I'm learning the DFT (Discrete Fourrier Transform), and came across a mathematical sign, that I didn't understand. They say (Inverse of DFT Matrix) F^-1 * c = v And then, there is an equality that ...
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Peter-Weyl Theorem on the Sphere

The Peter-Weyl theorem says that the matrix coefficients of the unitary irreps of a compact topological group $G$ form an orthonormal basis for $L^2(G)$. Similarly, spherical harmonics provide an ...
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Fourier Transform An FM synthesized wave.

Greetings StackExchange, (I hope this question comes under Mathematics and not Physics) I've been attempting a Fourier Transform on an FM synthesized wave (as below). After a long time (about 8 days) ...
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A piecewise function as the output signal of an LTI system

In an LTI system, consider the following: The input signal: $$ x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases} $$ And the unit impulse ...
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how to find the discrete time fourier series of sin function

I have this discrete signal sin($0.1 \cdot \pi n$) and I want to find the DTFS of it. first I have found its fundamental period N which is 20 but what am struggling with is how am I going to find 29 ...
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Fourier transforms and the angular frequency

In the context of Fourier transforms, is the following true? $$ \mathcal{F} \left( F(t) \right) =2\pi \, f(-\omega ) $$ If it is not, what small changes can I make to the equation to make it true?...
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Can someone help me with Fourier? [closed]

can someone give me a hand with the section c) and d)? I can not do it ... I think the point is that I do not understand how to integrate the function and then draw what it asks for. Problem
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Rewriting infinity as a limit to infinity (in terms of Fourier series)

Put informally: When writing down the complex Fourier series of a function, is it proper to write $$\displaystyle\sum_{n=-\infty}^\infty \tag*{(1)}$$ or $$\displaystyle\lim_{k\to\infty}\displaystyle\...
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On an exercise of Katznelson's book

My question is concerned with exercise one from section 1. It suggests two different functions on the interval $[-\pi,\pi]$: $$f(t)=\sqrt{2\pi}\chi_{(-\frac12,\frac12)}~~~,~~~\Delta(t)=(1-|t|)\chi_{(-...
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Unit impulse response of a discrete-time LTI system

The problem: Consider a discrete-time LTI system. If the output signal is: $$ y[n]=5 \left( \frac{1}{5} \right) ^n u[n] -2^{-n} u[n] $$ , then the input signal will be: $$ x[n]=\left( \frac{...
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Conceptual Doubt regarding principle of superposition for Linear PDE(s)

This is in regards to a problem I have been trying to solve. I have already posted a question regarding it previously, but the part of the problem I describe here leans more towards conceptual ...
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Reference request for decay of Fourier transform

I am interested in conditions on $f: \mathbb{R}^d \rightarrow \mathbb{R}$ that lead to decay of the Fourier transform of $f$. A standard one is $f\in C^k$ and $\partial^\alpha f \in L^1$ for $\vert \...
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Linear Transformation from one function to another.

Let's say we have two functions $f:[-1,1]\rightarrow \mathbb{R}$ and $g:[-1,1]\rightarrow \mathbb{R}$. Suppose furthermore that one can write down $f,g$ in terms of a linear sum of basis functions ...
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Derivative of the Fejer Kernel?

I'm looking to calculate the derivative of the Fejer kernel: $$ F_n(x) = \frac1{n+1}\left(\frac{\sin\left(\frac{n+1}{2}x\right)}{\sin\left(\frac x2\right)}\right)^2 $$ However, I'm running into some ...
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Bessel decomposition from Fourier Transform

I am looking to decompose a signal in terms of Bessel functions. I'm aware of Hankel transforms; however, for computational reasons, I have to use Fourier transforms. Essentially, my question is: ...
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$\int_{-\infty}^\infty \exp(iqy')dy' \int_{-\infty}^\infty |k|\exp(ik(y-y'))dk$ in 2 way give different results

I have integration whose result change depending on the way of calculation. I want to compute the integration below $$I=\int_{-\infty}^\infty \exp(iqy')dy' \int_{-\infty}^\infty |k|\exp(ik(y-y'))dk$$ ...
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Fourier transform of function defined on finite interval

Let $f(t)$ be a function defined on the finite interval $[t_1,\, t_2]$. Is the Fourier transform of such a function uniquely defined? In the sense that there exists only one function $\hat{f}(\omega)$ ...
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Energy spectral density in an LTI system

Consider the LTI system: $$x\left( t \right) \to LTI\, system \to y\left( t \right) $$ The graph of the $x(t) $ signal is the following: Remark: The graph when $t>0$ is a quarter of a circle. Q1:...
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Fourier transform of matrix over polynomial field?

I know we can do (Discrete) Fourier Transform of vectors of polynomial coefficients. This is useful for example when multiplying polynomials, since convolution turns into multiplication in Fourier ...
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Can we calculate the integral $\int_{-a}^a \frac{\sin(\pi x)}{\pi x}e^{2\pi i xt} \, dx$?

It is well known, that $\frac{\sin(\pi x)}{\pi x}$ (the sinc function) is the Fourier transform of the characteristic function on $[-1/2,1/2]$. Is there a way to calculate the integral $$\int_{-a}^a \...
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Essential supremum of Fourier transform.

Suppose $K\in L^2(\mathbb R)$ is symmetric s.t. $$ \text{ess sup}_{t\in \mathbb R/\{0\}} \frac{|1-\hat K(t)|}{|t|^{\beta}}\leq A, $$ where $\hat K$ is the Fourier transform. Theorem 1.5 in "...
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Periodic coefficients of Fourier series

If we have a continuous function $f$ on $[-\pi,\pi]$ and its complex Fourier coefficients are periodic, i.e. $$c(n) = c(n+k)$$ for some $k\in \mathbb{N}$, can we prove that $f$ is identically the ...
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Integral of a given complex function

Consider the following integral with $α,β,γ$ constant coefficients: \begin{equation}\int_{-\infty}^{\infty}\frac{e^{iαx}}{βx^2+iγx}dx\end{equation} We can say that this is an inverse Fourier ...
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Bandlimited reconstruction of sampled periodic functions.

This has to do with the Nyquist-Shannon sampling and reconstruction theorem and the so-called Whittaker–Shannon interpolation formula. I had previously asked an ancillary question about this here but ...
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Frequency spectrum and unit impulse response

In the context of signal processing, consider the following system: $$ x\left( t \right) \to System\, A \to z\left( t \right) \to System \, B \to y\left( t \right) .$$ Let the global system be ...
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On the orthogonal bases in $L^2[0,1]$

Let $\{\phi_n\}$ be an orthonormal base in $L^2[0,1]$ such that for every continuous functions $f$ on $[0,1]$ one may find a sequence of complex numbers $\{\lambda_n\}$ with $f(x)=\sum \lambda_n\...
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(Motivation behind) Orthogonality of functions

I'm interested in understanding the usual inner product on functions spaces more deeply than I already do. That is, the inner product $\int f(t) \;g^*(t) \;dt$, where $f$ and $g$ are complex valued ...
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Lower Bound on Oscillatory Integral

Let $p,y\in\mathbb{R}^d\setminus\{0\},\beta>0$ be given and fixed and define for all $\alpha>0$, $$I(\alpha) := \int_{x\in\mathbb{R}^d}\exp(\mathrm{i}\alpha p\cdot x-\alpha\beta \|x-y\|^2)f(x)\...
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Fourier transformation and differential equation 2nd order

Applying Fourier Transfomation in the following differential: $$ m\ddot{x}+c\dot{x}+kx =f(t)\tag1 $$ we arrive at the general form for the particular solution: $$x(t)=f(t)* \int_{-\infty}^\infty{\...
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Can we bound the value $\int_{[-A,A]^c} | \hat f(k)| dk$ where $A>0$ and $\hat f$ is the Fourier transform?

Assume $f \in L^1$ and $\hat f \in L^1$. Then it is easy to see that for $A>0$ the value $$D := \int_{[-A,A]^c} | \hat f(k)| dk$$ tends to zero for big $A$ ($[-A,A]^c$ is the complement of the ...
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Shifting therome and 2d fouire transform on appeture functions

I am currently looking over some slides for my course and found something that I cant quite get my head around. As you can see from my slide below it gives a results for when you apply the shifting ...
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While Fourier series solve heat equation on a finite interval, can Fourier transform solve heat equation on infinite line?

Consider the heat equation with initial value $f$ $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\ \ \ u(x,0)=f(x).$$ We look at two cases, one with $x\in[-\pi,\pi]$ (heat equation ...
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Inner product of Hermite polynomial

How to prove the Fourier Hermite series $$\int H_n(x) f(x) \phi(x) dx=\int f^{(n)}(x) \phi(x) dx $$ where $\phi(x)=\frac{1}{\sqrt{2\pi}} e^{-x^2/2}$, $f^{(n)}(x)$ is the $n$th derivative of the ...
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How to explain Fourier series in linear algebra context

I recently read about how the Fourier series is essentially a projection of a function $f$ onto the basis $\{1, \sin(x),\cos(x)\}$. Since a vector $w$ in $\mathbb{R}^2$ can be represented as $$w = \...
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Miswritten exponential sum definition. What did my professor likely mean to write?

In my analytic number theory course, in a lesson about exponential sums, my professor defined a particular exponential sum. I believe that I copied it down correctly, because I scrutinized it for a ...
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The Fourier series $\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log(n)}$

Prove that the series $\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log(n)}$ represented a function $f(x)$ that is not Lebesgue integrable. My anwser: Suppose that $f(x)=\sum_{n=2}^{\infty} \frac{\sin(nx)}{\...
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Sequence of polynomials approximating sines and cosines

Suppose that you create a sequence of polynomials by first stating that $$ P_1(x) = \pi x $$ and then describing successive polynomials by: $P_{n+1}(x)$ is the indefinite integral of $2\pi P_n(x)$, ...
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Proof of Parseval´s Theorem

Could someone point me to a proof of Parseval's Theorem? That is, $$\sum_{n=-\infty}^{\infty} a_n\bar b_n = \int_{-\pi}^{\pi} A(x)\bar B(x) dx$$ where $A(x)=\sum_{n=-\infty}^{\infty} a_ne^ {inx}$ ...