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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Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
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100 views

Definition of $L^p(\mathbb T)$ with $\mathbb T$ the unit circle

I'm trying to define the $L^p$ spaces in the unit circle(denoted as $\mathbb T$), as Rudin's Real and Complex analysis does in page 88. I've defined a measure in $\mathbb T$ via Riesz's representation ...
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122 views

Accuracy of Real FFTs

I'm not sure if this is the best place to ask, please advice if not. I'm performing the following test, to check the output of real-FFTs, and I'm getting surprisingly high errors. So a real-FFT only ...
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50 views

Moments of Fourier transform

Fix a smooth $\mathcal{C}^{\infty}$ compactly supported function $f$ with the support of $f$ being the unit interval $(-1,1)$ and with $\hat{f} \geq 0$. Is it true that as $k$ goes to infinity $$ ...
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78 views

Function supported on [-1,1] with arbitrary prescribed sub-exponential Fourier decay?

Given $g:[1,\infty) \rightarrow (0,\infty)$ with $g(t) = o(t)$, does there exist $f:\mathbb{R} \rightarrow [0,\infty)$ with support contained in $[-1,1]$ such that $$ \widehat{f}(y) = ...
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156 views

Closed formula Fourier transform of complex exponential

Is there a closed formula for the Fourier transform of the function \begin{equation} f(t) = e^{2\pi i \sqrt{1-t^2}}, \end{equation} where the square root for $|t|>1$ is $i \sqrt{t^2-1}$. This ...
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141 views

Inverse Fourier-cosine transfrom

Suppose we have a function $F(x)$ given by the integral: $$F(x)=\int_{0}^{\infty}f(t)\frac{\cos(t\log x)}{t}dt\;\;\;\;\;(x>1)$$ This looks tantalizingly like a Fourier-cosine transform of ...
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154 views

Fourier transform of projection of spherical cap

I am currently trying to derive an analytical expression for the Fourier transform of the projection of a spherical cap of the unit sphere onto the xy-plane. Setting up the integration in cylindrical ...
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40 views

Fourier transform of a sequence by a matrix

Let $n$ be a positive integer and $H$ the Hilbert space $\ell^2(\mathbb Z^n,\mathbb C^n)$. For $u\in H$, denote by $\mathcal{F}(u)$ the Fourier transform of $u$, defined by $\displaystyle ...
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31 views

An alternative proof without multiplier theorem

Given $f\in \mathcal{S}'$ and $\phi\in C_0^\infty$, with $\operatorname{Supp}\phi = \{1 \leq \xi \leq 2\}$. Define $\phi_j = \phi(2^{-j}\cdot)$. We have if $\phi_k*f\in L^p$, then for all ...
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82 views

The Envelope (wave front) of a span of plane waves.

Imagine we have for every real direction $\mathbf n \in \mathbb R^3$ a plane wave, be it either quantum, eletric, magnetic, acoustic or in my case some sheets of paper I am holding up. Each of these ...
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63 views

What sequence has this Discrete Fourier Transform?

Suppose $$ x[n]= \begin{cases} x_i &, i \in P\\ 0 &, i \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_i \geq 0$. Suppose these equalities hold : $$ ...
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91 views

Output of wavelet transforms

I am working on a time sensitive computer science and fluid dynamics project that requires me to find applications of wavelet analysis. I know that at its core, a wavelet transform simply takes a ...
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48 views

A Fourier Coefficients (Series) problem

Let $\alpha$ be a number such that $\alpha/\pi$ is not rational. Prove that (1) $$\lim_{N\to\infty}\sum_{n=1}^{N} e^{ik(x+n\alpha)}=\frac{1}{2\pi}\int_{-\pi}^{\pi} e^{ikt}\,dt,$$ (2) for any ...
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61 views

The Sinc Function and the Partition of Unity Property

For this question purposes a function, $ f \left( t \right) $ is said to hold the Partition of Unity if: $$ \forall t, \ \sum_{n \in \mathbb{Z}} f \left ( t - n \right ) = 1 $$ Using the Poisson ...
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28 views

Interpolation of linear operators

If $T$ is a bounded linear operator from $L^{p_1}$ into a homogeneous Lipschitz space of order, say $\lambda.$ Further if $T$ is also bounded from $L^{p_2}$ into $L^{q}$ for some $p_1,p_2,$ and ...
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204 views

Convolution property in terms of fft (matlab)

I am working on some signal processing and I have the following data: ...
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56 views

Fourier Analysis and Groups

Consider the Fourier Transform and Inverse Fourier Transform: $$\mathcal{F} f(-s) = \int_{-\infty}^{\infty} e^{-2 \pi i (-s)t} f(t) \ dt = \int_{-\infty}^{\infty} e^{2 \pi ist} f(t) \ dt = ...
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151 views

Is this implication correct? (Proof of Bessel's inequality)

I'm just curious if this implication is correct: $$0 \le\int_a^b \left( f(x)-S_n(x) \right)^2dx \Rightarrow 0 \le \lim_{n\rightarrow \infty}\int_a^b \left( f(x)-S_n(x) \right)^2dx.$$ What needs to ...
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346 views

Help with Hermitian Symmetry and its inverse Fourier transform in MATLAB.

I have tried to impose Hermitian symmetry on the complex number $z$ which is varies with $x$. I need to take its inverse Fourier transform. A hermitian symmetry should give a real valued inverse FT. ...
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208 views

Convert to displacement time signal (from accelerometer) using velocity time signal or displacement frequency signal

Firstly, I try to use ifft to convert displacement frequency into displacement time but the ifft fails to give the original sign.The code as below: ...
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180 views

Fourier transform of convolution in a finite range

Can anyone help me evaluate the Fourier transform of of the following function, $t \in \mathbb{R}$, $\lambda \in \mathbb{C}$, $g:\mathbb{R} \rightarrow \mathbb{R}$, $f(t) = \int_{t_0}^t ...
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52 views

Conditions for matrix operator to preserve complex symmetry on DFT vector?

Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. ...
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57 views

Fourier Transform of $(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$

I would like to calculate the FT of the following function: $$(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$$ Any hint is highly appreciated!
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86 views

2D Fourier transform of exponentials and cosines

I would like to know the 2D FT of the following functions: 1.$$\exp\left(-\frac{(x-y+a)^2}{b^2}\right)$$ ...
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71 views

Inverse fourier transform involving exponentials

I would like to calculate the inverse fourier transform of: $$\hat{f}(k)=\exp(-a k^2+ikv)\cdot \frac{\sinh(m\sqrt{(b+ck^2+ikf)^2-d})}{\sqrt{(b+ck^2+ikf)^2-d}}$$ Any clue? Thanks!
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117 views

Discrete fractional fourier transform

I have written a code for producing matrix of fractional fourier transform with the help of eigen vectors of fourier transfom matrix. Does anyone know the elements of this matrix ( for example a 4 by ...
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93 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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37 views

About this question,I have done something,but I can't go on.

About this question,I have done something,but I can't go on.Firstly,you can examine the validity of my work.If it is right,please go on to compute.If it is a wrong way,please give me your idea.
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107 views

Intervals where the function is similar to the Fourier series

$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$ I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$". ...
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184 views

Fourier Analysis of Prime Counting Function

I was thinking about the following: Denote $\pi(x)$ as the prime counting function such that: $$ \pi(x) = \#\text{ of prime numbers}\leq x $$ It is well known from the prime number theorem that $$ ...
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40 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
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96 views

Inversion of a Fourier Transform

I am told that the functions $f(x), g(x)$ and $h(x)$ satisfy: $\hat{f}(k) = \dfrac{\hat{h}(k)}{A+\hat{g}(k)} $, where $\hat{f}(k)$ is the Fourier Transform of $f(x)$ (likewise for $h(x)$ and $g(x)$ ...
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52 views

What is the relationship between singularities for complex times and high frequency asymptotics?

As said in a paper I am reading on p 2677 in the text directly above FIG3, this should be a standard result about Fourier transforms of analytic functions. In the paper the authors use these methodes ...
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179 views

DFT in complex form and Trigonometric Polynomial Interpolation: why different dimensions of basis vectors set?

I'm trying to figure out, how coefficients of Discrete Fourier Transform in complex form are converted into coefficients of Trigonometric Polynomials Interpolation: Say, I have a function vector with ...
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162 views

Matrix of Discrete fourier transform $F^4$ is identity

I already showed that Discrete fourier transform matrix is unitarian matrix. Now I would like to show that $F^4$ is identity. On wikipedia is written: "This can be seen from the inverse properties ...
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50 views

Discrete time fourier transform of partial sum

I came across the following property of the DTFT: $ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
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79 views

Help with using IFFT to calculate radial distribution function g(r)

I am trying to use ifft function to evaluate the radial distribution function g(r), r is distance (nm), by using the structure factor s(q) , which is function of wave vector q (1/nm) : $$ g(r) = ...
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233 views

Fourier Series on a 2-Torus

Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a ...
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160 views

Fourier Series Expansion of the Partial Differential Equation

Partial differential equation: ...
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44 views

Cancellation of summations

I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider: $$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
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85 views

Fourier transform of $\cos(\sqrt{r^2-t^2})$

What is the Fourier Transform of $\cos(\sqrt{r^2-t^2})$ where r is a constant and t represents time. Please Help.
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61 views

Help in implementing of peak function in Fourier transform

I have a function Peak function I know how to implement it in time range just need to caclulate $r$. first I initial $x$, y with a range and meshgrid them, after it calculate $r$ ...
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Almost Everywhere Convergence of Walsh Series of $L^2$ functions

I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is some notes which presents his argument more ...
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75 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
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83 views

Prove complex Fourier Series in 2D

Prove the complex form of Fourier Series in 2Dimension from periodic function (period $2\pi$) in $x$ and $y$, defined in region $\Omega\subset\mathbb{R^2}$ $$f(x,y)\sim\sum_{-\infty}^{\infty} ...
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34 views

Compute frequency-wavesnumber plot from video

Hey StackExchange Math, I have a question regarding the computation of a wavenumber-frequency graph from a video. In the paper "Simulating Ocean Water" by Jerry Tessendorf (link), it is stated that a ...
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36 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
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70 views

Questions about Fourier transform.

I am reading the notes Lecture notes on representation theory. I have some difficulty in proving a) in Exercise 1.2. We need to prove that $$ \mathcal{F}^2(f)(x) = qf^{-}. $$ I compute as follows: ...
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$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...