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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Confused by certain interpretation of expected value…

I read the following in Stein / Shakarchi's Fourier Analysis book, where they discussed the notion of expectation of a probility density. "Consider the simpler (idealized) situation where we are ...
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To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz

I have asked this question on mathoverflow also. (my question, I wasn't sure if its ok ask at another similar forum, on stack exchange, but I hope it would reach more people). It is well known how to ...
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26 views

Positiveness of partial sums of type $ \psi * D_N $

In his paper about Extremal Functions for the Fourier Transform (see, for example, here? https://projecteuclid.org/download/pdf_1/euclid.bams/1183552525), Jeffrey Vaaler, while trying to build ...
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About the causality of the signal whose frequency spectrum is not continuous as follows

Consider the signal in frequency domain: $$ \alpha(\omega) = \begin{cases} 1, & |\omega|<\omega_c \\ 0, & |\omega|\ge\omega_c \end{cases} $$ $$ =A(-j\omega)A(j\omega) $$ $$ =|A(j\omega)|^2 ...
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Show there exist a constant $c\in \Bbb{C}$ such that $\int_{0}^{1}|{f-c}|^2<{1\over 36}$

Let $f:\Bbb{R}\to \Bbb{C}$ be a $1$-periodic function, $f\in C^1$ and $\int_{0}^{1}|f'|^2\le 1$. a. Show $\sum_{k\ne 0}|{\hat{f}(n)}|^2\le {1\over 4\pi^2}$ (I did it already, and that question is ...
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The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is positive. We can observe that $|f|^2$ has some nice property : ...
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Fourier transform of periodic function

Is it possible to Fourier transform a periodic function f(x) = f(x+L) with period L, numerically, only over the range x = 0 to L and use periodic boundary conditions to enforce the periodicity of the ...
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Deriving expression for an integral that arose in Fourier analysis.

Note : This question arose when i am trying to solve this question. I am making this question self contained, and not to depend on the MO question, but one can look at MO question for understanding ...
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45 views

Does the Fourier series converge at $x=0$?

Let $f(x)$, a $2\pi$ periodic funciton such that $f(0) = 1$ and for every $0\ne x\in[-\pi,\pi]$: $f(x) = 1 + \sin \frac{\pi^2}{x}$. Is the Fourier series of $f(x)$ converges at $x=0$? If so, what ...
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21 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
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Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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36 views

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
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17 views

Applying the inversion formula to Hardy's theorem

So, in the proof of Hardy's theorem, who says that $\zeta$ has infinite zeros on the critical line, we have eventually that ...
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49 views

Question about the Fourier Inversion Formula

We have $$\hat{f}(\xi)=\mathcal{F}f(\xi):= \int_{-\infty}^{\infty}f(x)e^{-2\pi i\xi x}dx,$$ with $f\in L^{1}$, and the Fourier inversion formula says that ...
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35 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and ...
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18 views

Fast Way to Compute DFT with index summation subject to a constraint

I really appreciate if anyone can help me with this problem. Problem: Let $W_n=e^{\frac{2\pi i}{N}}$ which is the $N$th root of unity. The backward Discrete Fourier Transform of a complex vector ...
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25 views

2D Discrete Fourier Transform on an Image - Example with numbers (rgb)

I am trying to write my own function that takes an image, an pixel by pixel it calculates that pixel value that will produce a 2D Fourier Transform image. I have no idea about signal processing, my ...
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41 views

Orthonormal Basis of $L^2$

Theorem: ' ' The Orthonormal family $e_n(x)=e^{2\pi i n x},\ n\in\mathbb{N}$ is a basis for $\mathcal{L}^2([0,1])$.`` In this case, $\{e_n(x)\}_{n\in\mathbb{N}}$ being a basis would mean that any ...
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37 views

What is the Fourier transform of Riemann Zeta function?

All: Is there an explicit form of Fourier Transform of Riemann Zeta function ? Also, is there an discrete Fourier Transform (DFT) of Riemann Zeta function ? I remembered I had seen something like ...
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46 views

Fourier transform of exponent

I need to count the Fourier transform of the following function but it does not seem so obvious for me. $f(x)=(e^{-ab})-1$ for $x\ge0$ and $f(x)=0$ for $x<0$ where: $a=1$ and $b=-1$ I don't ...
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34 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
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136 views

Do Electrical Engineering Researchers Usually Know Higher Math? e.g. Measure and Distribution Theory, Functional Analysis

I am an EE undergraduate student, and I recently took two courses in signals and systems. We were exposed to things like the Dirac delta (without mention of distributions), Fourier series (without ...
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62 views

Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$ [on hold]

Let $f\in C^{\infty}(Ω)$ for some open set $Ω \subset R^n$ that contains $0$. Show that if $f$ is differentiable as to function $x\mapsto ||x||$ with $x\in R$,then $f'(0)=0$. I found this problem in a ...
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29 views

Fourier transform of a $H(x)$ product distribution

So I am given this simple example, where $T \in \mathcal{S}(\mathbb{R})$: \begin{equation} T=(\mu +\lambda x+\beta x^2)H(x) \end{equation} where $H(x)\in \mathcal{S}(\mathbb{R})$ (also notated as the ...
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46 views

Orthonormal Basis of a function

An Orthonormal Family $\{e_k\}_{k\in\mathbb{N}}$ is a basis if and only if $$f=\sum^\infty_{n=1}\hat{f}(n)e_n \ \ \ \text{in} \ \mathcal{L}^2(\mathbb{R})$$ where $f\in\mathcal{L}^2(\mathbb{R})$ ...
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1answer
30 views

Find the Fourier coefficients of $g(x)$

Let $f:\mathbb{R}\to\mathbb{C}$, $2\pi$ periodic function and $f\in C^1$, such that the n-th Fourier coefficient is: $\hat{f}(n) = 3^{-n^2}$. Find the Fourier coefficients of $g(x) = \pi ...
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20 views

A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$ a_1 * f_1 + a_2 * f_2 = 0, $$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
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1answer
58 views

Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
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42 views

How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
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32 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
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Fractional derivative of $e^{-x^2/2}$ using Fourier transform and Taylor series

I am not familiar with fractional calculus, so I want to know what I am doing wrong. The convention I use $$\int^\infty_{-\infty}e^{-\frac{x^2}{2}}e^{-i k x}dx=\sqrt{2 \pi}e^{-\frac{k^2}{2}}$$ I am ...
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Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
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On the Fourier transform of $f(x)=\ln(x^2+a^2)$

I would like to derive the Fourier transform of $f(x)=\ln(x^2+a^2)$, where $a\in \mathbb{R}^+$ by making use of the properties: \begin{equation} \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ ...
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1answer
18 views

Property of complex borel measures with absolutely convergent Fourier series (Wiener algebra)

Let $\mu$ be a complex Borel measure on the circle $\mathbb{R}/\mathbb{Z}$ with $$ \sum_{n \in \mathbb{Z}} \lvert\hat{\mu}(n)\rvert < \infty. $$ How does it follow that $d\mu(x) = f(x) dx$ for $f$ ...
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Computing the laplacian Green function by Fourier transform and analytic continuation

I know that the Green function for the laplacian operator in $d$ space dimensions $$\Delta \equiv \sum_{i=1}^d \frac{\partial^2}{\partial x^2_i}$$ is given by $$ \Delta^{-1}(x-x')=\begin{cases} ...
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Find the fourier series of the function

Find the fourier series of the function $g(x) = \sum\limits_{n=1}^\infty \frac{sin(nx)}{6^n sin(x)}$ for $x \not= k\pi$, and $g(k\pi) = \lim_{x\to k\pi} g(x)$, $(k \in \mathbb{Z})$
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Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
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Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$ \int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a) $$ where $$ F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert $$ and $$ G(x) = \frac{\sin x-x\cos x}{x^4} $$ EDIT: ...
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Inverse fourier transform - Where did the Heaviside function come from?

I asked this question on another forum but no answers so I'm copy/pasting it here in hopes that someone can help out
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49 views

Fourier transform of $f(x)=1/x$

I would like to compute the Fourier transform for the function: \begin{equation} f(x)=\begin{cases} 1/x&, x\in [a,b] \\ 0,& x \notin [a,b]\end{cases} \end{equation} but I cannot do the ...
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FFT Hyperbolic Distribution R

This is my first posting so forgive me if it is not 100% in line with this forum's best practices. I am completing an analysis using ICA as the decomposition technique. I am keeping 4 of the 10 ...
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26 views

On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
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Fourier transform with $\sin(t^2)$

This exercise gave me nightmares this night. I have $$ x(t)=\sin(t^2)e^{-2|t-2|} $$ to Fourier transform. First I though about solving the integral. (should I divide the signal in $2$, first for ...
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58 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...
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58 views

On the Fourier transform of $f(x)=e^{-x^2+2x}$

So, I have the $f(x)=e^{-x^2+2x}$ and to take the FT of it, I complete the square: \begin{equation} f(x)=e^{-x^2+2x \pm1}=e^{-(x-1)^2}e \end{equation} Then, by knowing that the FT of $g(x)=e^{-x^2}$ ...
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Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
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58 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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1answer
6 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
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14 views

How to introduce a new equivalent using two separate boxes in variables?

I am reading a paper in mathematics about Fourier Transform. It consider two boxes A and B with centers of $ x_0(A) $ and $p_0(B)$ respectively. It introduce a new function $R^{AB}(x,p)$ as follows: ...
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1answer
34 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...