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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54 views

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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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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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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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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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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Orthonormal basis for $\mathcal{L}^2([0,1])$

$\textbf{Theorem:}$ The orthonormal family $\{e_n(x):\, n\in\mathbb{N}\}$, where $e_n(x)=e^{2\pi inx}$, is a basis for $\mathcal{L}^2([0,1])$. In this case, $\{e_n(x):\, n\in\mathbb{N}\}$ being a ...
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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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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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The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
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Can someone help me with Fourier? [on hold]

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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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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1answer
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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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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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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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oscillatory integrals in several variables with nonstationary pahse

Consider the oscillatory integral $$I(\lambda):=\int_{\mathbb{R}^{n}} \mathrm{e}^{\lambda\dot{\imath}\phi{(x)}} \psi{(x)}dx $$ where the phase $\phi$ and the amplitude $\psi$ are smooth real-valued ...
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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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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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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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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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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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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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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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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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1answer
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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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What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
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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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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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Fourier Transform of $~f(ix)~$ [closed]

What is the Fourier Transform of $~f(ix)~$? Where I've looked: In my formulary, on Wolfram Alpha, on this site. (it might be duplicate anyways, since searchers don't like things like " $~f(ix)~$")
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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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2answers
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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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1answer
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How to find fundamental frequency of two signals?

I am facing difficulty with finding fundamental frequency of signals I mean by fundamental frequency $=\frac{1}{\text{Time period}}$ Correct me if I am wrong consider two continuous time signals ...
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2answers
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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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1answer
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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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1answer
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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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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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1answer
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Order of evaluation for summing over integers

In Fourier analysis, how should I interpret sums of the form $$ \sum_{n \in \mathbb{Z}} a_n = \sum_{n=-\infty}^\infty a_n?$$ Is it $$\lim_{N \to \infty} \sum_{|n| < N} a_n$$ or $$\sum_{n=1}^\infty ...
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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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1answer
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$f$ is continuous on the circle, and its Fourier series is absolutely convergent. Then its Fourier series converges uniformly to $f$.

This Corollary's proof (2.3 in Stein's book), is not sufficiently detailed for me to fully understand. I can see why the Fourier series uniformly converges to some continuous $g$ but how can I ...
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How to calculate the Fourier Transform of a constant?

The definition of the FT in engineering is: $$\int_{-\infty}^{\infty}f(x)e^{-j2\pi ft}dt$$ I'm having trouble calculating the FT of a constant, such as $\frac{1}{2}$: $$\int_{-\infty}^{\infty}\frac{...
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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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1answer
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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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1answer
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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 $A(x)=\sum_{n=-\infty}^{\infty} a_ne^ {inx}$ ...