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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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Can analytic continuation be applied to real functions? e.g. $f(x) = a\sin x + b\cos x$ in $[0,17]$

I've a Fourier series up to the 3rd harmonic. It looks like this: f(x) = 7.833333335 - 0.327444444*cos(0.349*x) - 0.882182222*cos(0.698*x) + 0.0000355555*cos(1.047*x) - 5.150566667*sin(0.349*x) - 2....
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Fourier cosine series of even periodic extension

I've been posed the following question: screenshot of the question I have tried using the formula for the Fourier series of an even function but don't think this is the correct approach. I have tried ...
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Fourier Series Representation for piecewise function

I've been posed the following question: $$ f(x)= \begin{cases} 1-x^2, & 0 \leqslant|x|<1,\\ 0, & 1\leqslant|x|<2\\ \end{cases} $$ I'm trying to find the Fourier series representation ...
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function in $\mathcal{L}^2$ space periodically.

Any idea or hint to prove this theorem ? The book says the hint is by drawing a picture, but I don't really get it. If $f\in\mathcal{L}^2([-\pi,\pi])$, extended periodically in $\mathbb{R}$, then $$\...
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1answer
35 views

On the liminf and limsup of the fourier coefficient of non-increasing function

Let $f:(0,\infty)\to\mathbb{R}_+$ be a non-negative non-increasing function such that $\int_0^1 xf(x)\,dx<\infty$. Define $A(b)=\int_0^\infty e^{-x}\sin(bx)f(x)\,dx$. Is it possible that $\limsup_{...
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How to comprehense the relation between convolution and each Fourier coefficient of f?

I am learning Fourier Analysis by Elias.M.Stein.I don’t understand the motivation of convolution.From my point of view,the convolutions corresponds to the “weighted averages” if we write it in ...
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Were the classic “high school” trigonometric formulae known and derived before Fourier transforms?

We all probably learn the famous trigonometric formulae: $$\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)\\\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha) \sin(\...
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Non-negative trigonometric polynomials to exponential form

I've been working on an exercise and have gotten stuck: Suppose that $T(x)=\sum_{n=0}^N a_n\cos(2\pi nx)+b_n\sin(2\pi nx)$ is non-negative on $[0,1]$. Show that there exist $c_0,...,c_N\in \mathbb{C}$...
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Compare R Programming with Sagemath for Fourier analysis? [on hold]

For Fourier mathematical simulation harmonics which one we will use R or Sagemath.
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Show that $\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$

Show that$$(1)\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$$ where $f_{ab}(x)$ and $\hat{f}_{ab}(\mu)$ are: More clarifications: Let the ...
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Is my proof about the behavior of Convolutions as $||x|| \rightarrow \infty$ correct?

I've written a proof to the following statement and would appreciate if somebody could look over it and give some feedback. My apologies if this is the wrong place for this. Statement Let $f \in L^...
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How do i get fourier series of signals given below [on hold]

how can i calculate and plot $Ae^{-at}$ amplitude and phase spectrum? how can i determine the trigonometric fourier series of the signal given below ? Your help is appreciated !! Thanks
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Extending a diffeomorphism of $S^1$ to a diffeomorphism of $D^2$ with fourier series.

I'm reading this and there are some details that are missing. I'm asking for those. First let $f:S^1\to S^1$ a diffeomorphism. By Dini criterion we can write $f(e^{i\theta})=\sum_n \hat{f}(n)e^{in\...
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Modification of Shannon wavelets for compactly supported maximally “square” wavelets?

The Shannon wavelets are maximally square in the sense that they are ideal band-pass filters in the Fourier domain. Their amplitudes are literally box functions on frequency line and time shifts as we ...
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Show that $\int_{-\infty}^{\infty}f(x)\overline {g(x)}dx = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\mu)\overline{\hat{g}(\mu)}d\mu.$

Given: Show that if f(x) is defined as: The Fourier transform $\hat(\mu)$ of a function $f(x)$ specified on $\mathbb R$ is often defined by the formula: $$\hat{f}(\mu) = \int_{-\infty}^{\infty}e^{i\...
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Compactness when mapping into a higher $L^p$ space and then back

Question: Let $q>p \ge 1$ and let $T:L^p[0,1] \to L^q[0,1]$ be a bounded linear operator. Let $i: L^q[0,1] \to L^p[0,1]$ be the inclusion map (which is bounded). Is the composition $i\circ T$ ...
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Understanding the relation between the dominated convergence theorem and uniform convergence

Problem: Let $f \in L ^ { 1 } , | \widehat { f } | \in L ^ { 1 }$,$$ u ( x , t ) = \int _ { - \infty } ^ { + \infty } \widehat { f } ( \xi ) e ^ { 2 \pi i \xi x - 4 \pi ^ { 2 } a ^ { 2 } \xi ^ { 2 } t ...
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What is $\sup_\limits{E\subset[-\pi,\pi)}\left|\int_Ee^{it}\,dt\right|$?

I couldn't think of a descriptive title other than the problem statement itself. What is $$\sup_\limits{E\subset[-\pi,\pi)}\left|\int_Ee^{it}\,dt\right|?$$ (The question seems possibly relevant to ...
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+150

Show that $\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\hat{f}(\mu)|^2d\mu$

Given: Let $a_1 \lt b_1 \le a_2 \lt b_2 \le ... \le a_{n-1} \lt b_{n-1} \le a_n \lt b_n$ and let $$f(x) = \sum_{j=1}^nc_jf_{a_jb_j}(x).$$ Show that, $$(*)\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}...
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Fourier series Parseval equality with partial sums

Let $f \in \mathcal{R}_\left[-\pi,\pi\right]$ be function with period $2\pi$. We denote n-th partial Fourier series sum of function $f$ with $S_n(x)$. Prove that: $$ \int_{-\pi}^{\pi}\left(f(x)-S_n(...
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meaning of double headed arrow

I am studying properties of discrete time Fourier transform and I encountered a notation shown highlighted in attached photo What is meant by this notation? Is it meaning equality? screenshot of ...
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1answer
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What is the Fourier transform of $\frac{x}{(x^2+y^2)^{n/2}}$?

We have the following Fourier transforms: $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{1/2}}\right] = 1/\sqrt{k_x^2+k_y^2} $$ $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{3/2}}\right] = -\sqrt{k_x^2+k_y^2} $$ $$ {\...
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1answer
35 views

Fourier Stieltjes series of a positive measure

I am reading Katznelson's book on harmonic analysis, and I reached the following statement regarding a characterization of the Fourier-Stieltjes series of positive measures: Taken from page 38: (7.5) ...
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how to reconstruct a function from its convolution with a good kernel?

I have a question about approximations to identity. Thm A continuous function on $S^1$ with $\hat{f}(n)=0$ for all $n$ is identically zero. The Kernel they use is $p_k(\theta)= [\epsilon + \cos \...
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1answer
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How to understand the convergence of Fourier Series in $L^p$

My professor told me that Suppose that $f \in L^p(-\pi, \pi)$ (i.e. $f$ is 2$\pi$-periodic and $\|f\|_{L^p} < \infty$). If $1<p<\infty$, then the Fourier series of $f$ converges to $f$ in $...
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Projection estimator in hilbert space [closed]

Let $ f \in \mathbb{L}^2[0,1] $ $ Y_i = f(i/n) + \epsilon_i $ with $ \epsilon$ independent and centered With basis $ \phi_1 (x) = 1 , \: \: \phi_{2k} (x) = \sqrt{2}cos(2\pi kx) \: \: \phi_{2k+1} (...
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1answer
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Basis for subset of $\mathcal{L}^2(\mathbb{R})$ with the help of Fourier transform

Let $W \subset \mathcal{L}^2(\mathbb{R})$ be a linear subspace. I want to show that for a certain $\phi \in W$, $\{ \phi_m : m \in \mathbb{Z}\}$ is a basis for $W$. Here $\phi_m$ describes a ...
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Does wavelet transform have any corresponding homomorphism space?

It is known that the Fourier transform $\mathcal{F}: L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an algebra homomorphism from $L^1$ into a sub-algebra $\mathcal(L^1(\mathbb{R}))$ of the space $C_0$ (...
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Simulate a sample function with known power spectrum

There is a known power spectrum $$C_l=\frac1{2l+1}\sum_{m=-l}^l |a_{lm}|^2$$ for some $l=0,1,\dots,l_{max}$ where $$ a_{lm}=\int_{-\pi/2}^{\pi/2} \mathrm \int_0^{2\pi}f(\theta,\phi)Y^*_{lm}(\theta,\...
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Modified Airy functions

The question is quite formal. I recall the definition of Airy function $$Ai(\tau^{2/3}\zeta)=\frac{\tau^{1/3}}{2\pi}\int e^{i(\sigma^3/3+\sigma\zeta)}d\sigma,\quad Ai'(\tau^{2/3}\zeta)=\frac{i\tau^{1/...
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Analytical expression for PSD

Is it possible to obtain an analytical expression for the PSD? The PSD is defined as follows, $S(\omega) = lim_{T \rightarrow \infty} \frac{1}{T} |Y(\omega)|^2 $ Assuming there is $Y(\omega)$ ...
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1answer
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Given a time-dependent function, complete it and find $c_0$

Complete graphically and analytically the function $f(t)$ so that the coefficients of the exponential Fourier series are pure imaginary: $$f(t)=\begin{cases}2t+1&\text{if $0\leq t\leq2$},\\\...
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1answer
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Finding the sum of another Fourier series using Parseval's identity

So...I previously found the series for $ f(t)= \begin{cases} 0&\text{if}\, -\pi\leq t\lt -\pi/2\\ \cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\ 0&\text{if}\, \pi/2\lt t\leq \pi\\ \...
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Prove $\sum_{0<|n|\leq N}\frac{e^{inx}}{n}$ is uniformly bounded in $N$ and $x\in[-\pi,\pi]$.

Prove $\sum_{0<|n|\leq N}\frac{e^{inx}}{n}$ is uniformly bounded in $N$ and $x\in[-\pi,\pi]$ by using the fact that \begin{eqnarray*} &&\frac{1}{2i}\sum_{0<|n|\leq N}\frac{e^{inx}}{n}=...
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Fourier Analysis an Introduction Chap4 12

12: A change of variables in (8) $$(8)\ u(x,t)=\sum_{n=-\infty}^{\infty}a_{n}e^{-4\pi^{2}n^{2}t}e^{2\pi inx}=(f*H_{t})(x)$$ $$H_{t}(x)=\sum_{n=-\infty}^{\infty}e^{-4\pi^{2}n^{2}t}e^{2\pi inx}$$ ...
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Having trouble with undefined terms in a Fourier series

So...I am trying to find the Fourier series for the following function: $ f(t)= \begin{cases} 0&\text{if}\, -\pi\leq t\lt -\pi/2\\ \cos(t)&\text{if}\, -\pi/2\leq t\leq \pi/2\\ 0&\...
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67 views

Showing $\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}d\omega=\pi$

Given the function $f$ with $f(t)=1$ for $|t|<1$ and $f(t)=0$ otherwise, I have to calculate its Fourier-transform, the convolution of $f$ with itself and from that I have to show that $$\int_{-\...
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1answer
22 views

Prove the following inequality involving a sum.

Suppose that $m,n, q\in \mathbb{N}$ such that $$\lambda_{n,m}=\frac{m+1/2}{(m+1/2)^2−n^2} \text{ and } \sigma_{q,m} = \sum_{k=0}^q \lambda_{k,m}.$$ Furthemore we also know that, $$\frac{\lambda_{0,m}}...
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Showing a scheme is convergent through Von Neumann stability analysis

Show that the box scheme $$\frac{1}{2k}\Big[(U_j^{n+1}+U_{j+1}^{n+1})-(U_j^n + U_{j+1}^n)\Big]+\frac{a}{2h}\Big[(U_{j+1}^{n+1}-U_{j}^{n+1})+(U_{j+1}^n - U_{j}^n)\Big]$$ is convergent for the ...
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1answer
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Calculating $\frac{1}{\sqrt{2\pi}}\sum_{r=-N}^{N-1}e^{-i2arq}\int_a^{2a}Ae^{iqu}du$

I have a problem calculating $$\frac{1}{\sqrt{2\pi}}\sum_{r=-N}^{N-1}e^{-i2arq}\int_a^{2a}Ae^{iqu}du$$ Calculating the integral gives me $$\frac{A}{iq\sqrt{2\pi}}\left(e^{-iqa}-e^{-i2qa}\right)\...
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1answer
120 views

Inverse Fourier Transform of a Constant

The Fourier transform and its inverse can be defined as $$\mathcal{F}(f(x))=F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-ikx} \ dx \ \ \text{and} \ \ \mathcal{F}^{-1}(F(k))=\frac{1}{\sqrt{...
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1answer
141 views

Asymptotic behaviour of Laplace transform

If the functions $x(t)$ and its derivatives $x'(t), x''(t), \ldots, x^n(t)$ are continuous* and $x(0^+) = x'(0^+) = x''(0^+) \ldots = x^{n-1}(0^+)=0$ ($0^+$ denotes the right side limit when the ...
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2answers
69 views

Calculate the given sum

Problem: $$\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$$ This is a problem from a course of Fourier Series and the only hint was to use partial fraction decomposition. I'm sure that this is meant to be ...
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1answer
47 views

Convolution of functions $f,g\in L^1([0,1])$

Problem: convolution ($f*g) $ of functions $f,g\in L^1([0,1])$, where: $$f(x) = \frac{3}{5-4\cos{4\pi x}},$$ $$g(x) = \frac{2\cos{2\pi x}}{5-4\cos{4\pi x}},$$ and $$(f * g)(x) = \int_{0}^{1}f(x-y)g(...
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0answers
18 views

Inverse Fourier Transform of a Spiral?

I am trying to minimize a free energy by plugging in this variation spin structure (a vector field): $$\mathbf{S}(\mathbf{r}) = {1 \over \sqrt{2}} \left( \mathbf{S_k} e^{i\mathbf{k \cdot r}} + \mathbf{...
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43 views

DFT is not a sampling of FT?

From wikipedia: The discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time ...
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1answer
89 views

Proving $ \sum_{n \in \mathbb{Z} } \left[\frac{\sin (n \alpha + \theta) }{ n \alpha + \theta} \right]^2 = \frac{\pi}{\alpha} $

Proving $$ \sum_{n \in \mathbb{Z} } \left[\frac{\sin (n \alpha + \theta) }{ n \alpha + \theta} \right]^2 = \frac{\pi}{\alpha} \,\, \forall \alpha , \theta \in \mathbb{R} $$ My attempt It is ...
2
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1answer
30 views

Showing that an orthonormal set becomes a basis for the Hilbert space

This is an exercise from Folland Real Analysis Chapter 8 that I am stuck at. I am actually stuck at (b). I succeeded in showing that $H_a$ is a Hilbert space and the given set is an orthonormal set of ...
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1answer
29 views

$L^2$ functions with compactly supported Fourier transforms form a Hilbert space

Given a fixed compact subset of $\mathbb{R}$, I want to show that square integrable functions on the real line whose fourier transforms are supported in the given compact set form a Hilbert space in ...
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0answers
23 views

Fourier series for a periodic vector field

In usual Fourier analysis, we have a periodic function like $f: \mathbb{R}^3 \rightarrow \mathbb{C}$. The function can be decomposed into a countable basis (i.e. $\{\mathrm{e}^{i\mathbf{k_n}\cdot \...