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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Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
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1k views

Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
9
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179 views

Half Solved: A problem on the heat operator not being elliptic with a weakened version of elliptic regularity

I should first mention this: in my studies of Sobolev spaces I have completed all the questions of chapter 9 from Folland's real analysis with the help of this site and this is my last one, which is ...
9
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111 views

Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$ f(x)=\sum_{n\geq ...
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156 views

What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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75 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
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125 views

Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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618 views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in ...
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115 views

Tridiagonal matrix w/trigonometric eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
7
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113 views

How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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179 views

Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
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218 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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161 views

number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
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340 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
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414 views

Fourier transform of integral related to zeta function

In this MO question here, I asked about the Fourier transform of the zeta function. The second answer lists the following as a representation for $\zeta(s)$, with $E(x)$ as the floor function: ...
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114 views

If the Fourier transform of a measure is zero then the measure is zero

If $\mu$ is a complex finite Borel measure on a separable real Hilbert space $H$ be such that $$\hat \mu (x) = \int \limits _H \Bbb e ^{\Bbb i \langle x, y \rangle} \Bbb d \mu _{(y)} = 0, \ \forall ...
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262 views

How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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155 views

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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85 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
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129 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
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52 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
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56 views

For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$.

As the title states: For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$ where $C_c^\infty(\mathbb{R})$ is the ...
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Necessary to assume $f\in C^\infty$ in this Fourier transform problem?

Consider the following problem. Is the hypothesis that $f\in C^\infty$ necessary, or could we weaken it and assume just that $f$ is continuous? Let $\hat f$ denote the Fourier transform of the ...
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98 views

Intuition behind the proof of the Inverse Fourier Transform?

I am interested in the proof of the Inverse Fourier Transform for absolutely integrable real valued functions. The proof I have read asks you to consider an auxiliary function $g_{a}(x)$ defined as ...
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104 views

How to find Green's function using Fourier-Bessel expansion

The Green's function satisfies the non homogeneous Bessel equation can be written as $xg''+g'+\left(k^2x-\frac{m^2}{x}\right)g=-\delta(x-\xi)$ where $m\geq0$ and an integer. The boundary conditions ...
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114 views

Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
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148 views

Inverse Fourier transform using Residues for a ratio of hyperbolic functions.

I'm new and glad to be here. I have a problem relating to an inverse Fourier transform. I have $$g(w)= \frac{\sinh{w(a-b)}}{w \cosh{wa}}$$ and want to find $$G(t)$$. I cannot find this in tables so I ...
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302 views

Prove the equation: $\frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \ldots $

Prove the following equation: \begin{equation} \frac{2}{\pi} \int_0^\infty \frac{\cos kr - ak \sin kr}{k^2a^2 +1} \left (\int_0^\infty \cos kr' \left [u(r')-au'(r') \right] dr' \right ) dk =u(r) . ...
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123 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
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453 views

How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?

I am trying to numerically solve the Possion PDE in cylindrical coordinate system. $$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
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523 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...
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102 views

multipliers on $H^{1}$

I'm begining to study the hardy space $H^{p}(\mathbb{R}^{n})$. First recall that a $L^{\infty}$ function is called a $H^{1}$ multiplier if the associated operator ...
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248 views

Fourier dimension of a measure restricted to an open set

Suppose that the measure $\mu$ on $\mathbb{R}^n$ has Fourier dimension $\beta$, which is to say that \begin{equation*} \beta= \sup\left\{\gamma \leq n : |\hat{\mu}(x)| \leq ...
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270 views

Fourier dimension of sets of positive Lebesgue measure

Let $K$ be a compact set in $\mathbb{R}$ with positive Lebesgue measure. My question is whether there exists a probability measure $\mu$ supported on $K$ such that $\hat{\mu}(\xi)$, the ...
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686 views

Fourier transform of vector-valued functions (e.g. differential forms)

Consider $L^2(\mathbb R^n, \mathbb R^m)$. There should be a Fourier transform for these functions, like in the case $L^2( \mathbb R^n, \mathbb R )$. I wonder how these can be defined. The application ...
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76 views

Hilbert space in Papa Rudin

In Rudin's Real and Complex Analysis, there is a problem in Chapter 4 on a Hilbert space $X = \text{span} \{e^{ist} \, \mid \, s \in \mathbb{R}\}$ with the inner product $$(f,g) = \lim_{T \to \infty} ...
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Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
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65 views

Rigorous proof of this limit

I have shown that the function $$f(x):=\int_{[-\pi,\pi]^n} \frac{e^{-i\langle k,x \rangle}}{1-\frac{1}{n} \sum_{i=1}^n \cos(k_i)}dk$$ exists everywhere for $n \ge 3$. Now, I want to show that ...
4
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85 views

Continuous Littlewood-Paley Inequality

I am trying to prove Q13 from Terence Tao's Fourier Analysis notes (number 4): For every $t>0$, let $\psi_{t}:\mathbb{R}^{d}\rightarrow\mathbb{C}$ be a function obeying the estimates ...
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86 views

Asymptotics of Null Solution to Heat Equation

In the book J. Rauch, Partial Differential Equations, the author claims that for $\alpha\in(1/2,1)$, the function $u$ defined by ...
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63 views

Computing the Fourier transform of the distribution $\|x\|^{-\alpha}$.

Question: Suppose we are given the tempered distribution $\|x\|^{-\alpha}$. We want to compute the Fourier transform $\mathcal{F}[\|x\|^{-\alpha}](\xi)$. What techniques are available for ...
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145 views

A lower bound for an arithmetic function

Let $N \in \mathbb{N}$ such that $\phi(N) \sim N$, where $\phi$ is the Euler's totient function. Let $A \subset [N] := \{1, 2, \ldots, N\}$. For $n \in \mathbb{N}$ define the function $$ C_A(n) = \#\{ ...
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59 views

Linear independence of primitive Dirichlet characters and convolution

This is not an exercise but merely a question I have. Fix $N \in \mathbb{N}$ and suppose there exist some values $a_k \in \mathbb{C}$, for $k \in \mathbb{Z}_N$, such that $$ \sum_{k \in \mathbb{Z}_N} ...
4
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112 views

Asymptotic form of an integral to an power law decaying function

$$ f(x)=\frac{1}{2}+\frac{1-x^2}{4x}\ln\left|\frac{1+x}{1-x}\right| $$ This function is not analytic at $x=1$. The plot is shown: The integral is: $$ I=\int_0^\infty g(x) \sin(2b rx) dx $$ where ...
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74 views

Fourier Transform of Gaussian-like function

I need to find the Fourier transform of the following Gaussian-like function: $$f(x)=\frac{1}{\sqrt{2\pi\sigma^2(x)}}\,e^{-\frac{x^2}{2\sigma^2(x)}}$$ where $\sigma(x) = \sigma_0 ...
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45 views

How are shaft motions defined in cyclic-symmetry?

When investigating the modal properties of cyclic structures (composed by a repetition of N identical sectors) such as bladed-disk assemblies, modes are often sorted by nodal diameters (or spatial ...
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48 views

Newton series and Fourier transform, is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
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80 views

Fourier transform of $\frac{1}{r}$ (Coulomb potential)

When calculating the Fourier transform of a function of the form $f(\vec{r}) = \frac{1}{4 \pi \left|\vec{r}\right|}$, one encounters the problem that the resulting integral does not converge, i.e. ...
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66 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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160 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...