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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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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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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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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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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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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 [on hold]

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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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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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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2answers
66 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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119 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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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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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
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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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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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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
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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 ...
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1answer
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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
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$L^2$ 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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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 \...
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Positive function for which the sequence $\int_{\mathbb{R}}f(t)\frac{\sin^2((2n+1)t)}{\sin^2(t)}dt$ is bounded (on $n\in\mathbb{N}$).

Is there an example of a "big" function $f:\mathbb{R}\to\mathbb{R}_+$ (say, not exponentially decaying at infinity) for which the sequence $$\int_{\mathbb{R}}f(t)\frac{\sin^2((2n+1)t)}{\sin^2(t)}dt$$ ...
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1answer
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What is the Fourier cosine transform in complex notation and what is the conjugate of the Fourier cosine transform?

Suppose the Fourier cosine transform is given by: \begin{align} F_c(k)=\mathcal{F}(f(x))&=\sqrt{\frac{2}{\pi}}\int_0^{\infty} f(x) \cos(kx) \end{align} or any other form I'm not particular ...
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What is the meaning of $T$ in the power of a periodic signal

I know that power of a periodic signal is defined as follows: $$P =\lim_{T\rightarrow\infty} \frac{1}{2T} \int_{-T}^{T}{{|x(t)|}^2dt}$$ or $$P =\lim_{T\rightarrow\infty} \frac{1}{T} \int_{-\frac{T}{2}...
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How to compute newtonian gravitation from an infinite array of attractors?

In a flat toric universe (up connects down, right connects left and front connects back), every points repeats at $size_x$, $size_y$ and $size_z$ intervals. In such case the Newtonian gravitational ...
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Prerequisites for reading “Automorphic Forms and Representations”

I am hoping to read (at least the first chapter of) "Automorphic forms and Representations" by D. Bump. As per the introductions, prerequisites are "basic knowledge of algebraic number theory, the ...
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Prove $\int_{-\infty}^{\infty} |F(k)|^2 \ dk=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} |f(x)|^2 \ dx$ (Parseval's theorem)

I would like to know how to prove Parseval's theorem, $$\int_{-\infty}^{\infty} |F(k)|^2 \ dk=\int_{-\infty}^{\infty} |f(x)|^2 \ dx.$$ The definition of the Fourier transformed used is $$F(k)=\mathcal{...
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1answer
22 views

discontinuity of $f(x) = \sum_{n\geq 1} \sin((n+ \alpha /n)x)/n$

It is well known that the function defined by $$g(x) = \sum_{n\geq 1 } \sin(nx )/n $$ is a piece-wise linear function $x$, and has jumps at $x = 2 m \pi $. Its picture is Now let us consider a ...
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Density of a class of function in $L^2(\mathbb{R}, e^x\,dx)$

Consider the class of function defined by $$\mathcal{G}=\operatorname{Span}\left\{e^{-\frac{(x+a)^2}{2}}-e^{-x}e^{-\frac{(x+a)^2}{2}}\mid a\in\mathbb{R}\right\}.$$ Is $\mathcal{G}$ dense in $L^2(\...
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20 views

Fourier Transform of Rectangular Impulse Function possible in specific Form?

Rectangular Impulse Function The above Rectangular Impulse Function is given. It's height is $A$ and it's width is $T$. The question is the following: If the Fourier Transform of the above function ...
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21 views

Haar Matrix is orthogonal

How can I prove that Haar Matrix is orthogonal ($H_{2^n}$ for all $n$)? $H_1,H_2,H_4,H_8, \dots, H_{2^n}, \dots$
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1answer
19 views

Fourier-Transformation of $\exp(-|t|)$ [closed]

Using the Fourier-inversion-theorem I have to show that $$\frac{\pi}{2}\exp(-|t|)=\int_0^\infty\frac{\cos(\omega t)}{1+\omega^2}\mathrm d\omega$$ Can anyone give me a hint on how to show it? ...
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0answers
8 views

Bessel coefficients decay

Is it possible to bound the decay of the coefficients of the bessel function $J_n(j^n_lr)$, where $j_l^n$ is the n-th zero of $J_n$? I want to know if there is a bound in terms of $n$ and $l$
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24 views

Fourier transform in spherical space

Recall that in 3 dimensions, the Fourier transforms are defined as following: $$ \tilde{f}(\textbf{k})= \frac{1}{(2\pi)^{3/2}} \int_{0}^{\infty}f(\textbf{x}) e^{-i\textbf{k}\cdot\textbf{x}} d^3x$$ $$...
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1answer
41 views

A lemma in the proof of Fourier's Main Theorem.

Assume $f,g:[-\pi,\pi]\to\mathbb{R}$ are continuous functions. We define the $L^2$ norm of $f$, and the scalar product between $f$ and $g$ as $$\|f\|_{L^2}=\sqrt{\int_{-\pi}^{\pi}|f(x)|^2\,dx}$$ and $$...
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1answer
67 views

How to extract Fourier coefficients from a Fourier transform

Suppose that I have a time-limited pulse $x(t)$ and that I know the pulse's Fourier transform $X(\omega)$. Suppose further that I use the pulse to construct the repeating waveform $\sum_{k=-\infty}^{\...