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 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 ...
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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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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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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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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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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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156 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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329 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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223 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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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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Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
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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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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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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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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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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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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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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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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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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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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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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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134 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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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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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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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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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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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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Equality that should be a consequence of Plancherel formula

I am stuck with this line in my reading of a book: By the Plancherel formula we have: $$\int \frac{\lvert u(x+y)-u(x)\rvert^2}{\lvert y\rvert^{2s+d}}dx = \int \frac{\lvert e^{i(y\xi)} -1\rvert ...
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why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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Involutive fourier transform

The writer here states I am introducing a viewpoint (the involutive convention) which makes the Fourier transform its own inverse (i.e., the Fourier transform so defined is an involution). ...
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Meaning of fractional Fourier transform with imaginary iteration count?

As one may know, the Fourier Transform $$F[f](\nu) = \int_{-\infty}^{\infty} f(t) e^{-2\pi i \nu t} dt$$ can be iterated, and this iteration generalized to fractional iteration count via ...
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Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
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A bijective correspondence induced by Fourier transform

Let $G$ be discrete Abelian group and denote by $\widehat G$ the Pontryagin-Van Kampen dual of $G$. I was reading in a paper due to Justin Peters that Fourier Transform induces a bijection between the ...
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Set theoretic arguments to prove the existence of a certain null set

Let me recall the well-known Carleson's theorem. Theorem (Carleson). Let $f$ be any periodic $L^2[0, 2\pi]$ function. Let $\hat{f}(n)$ be its Fourier coefficients. Then we have $$\lim_{N \to ...
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Fourier matrix - multiplicity of eigenvalues?

This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.) The row and column indices in the $n \times n$ Fourier ...
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Solution of the Dirichlet problem

I'm reading Jones' book Lebesgue integration on Euclidean space. Let $u(x, y)$ be a harmonic function on the half space $\mathbb{R}^n \times (0, \infty)$, with boundary condition $f(x) = u(x, 0)$. On ...
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What is the Fourier transform of a phase modulated signal?

I'm studying some Fourier analysis and have for a while been trying to figure out how to apply the Fourier transform to a phase modulated signal. More rigorously stated, what is $$ ...
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Why are linear functions the natural analogue of exponential functions in a tropical semiring?

I was reading a blog post on the Fourier transform and the Legendre transform as being the same thing over different semirings, in which the author says It's not obvious how to interpret the ...
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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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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} ...
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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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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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55 views

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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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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69 views

Problem using the Fourier transform and convolution to compute an integral

I'm trying to write a subroutine (in Fortran) to compute integrals of the form $$I=\int_{-L}^{L} f(x)g(y-x) \:\mathrm{d}x, $$ using the convolution theorem and fast Fourier transforms. In my routine, ...
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33 views

Existence of Fourier Transform for Implicit function

Given an "explicit" function $f:\mathbb{R}^n\to\mathbb{R}^n$, (e.g $F(x_1,\dots x_n)=\cos(x_n)+x_1^2e^{x_2}$) under some assumptions one can allegedly develop a Fourier transform given by ...
3
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102 views

Pisier's $\epsilon$-net condition

I'm reading a book about Sidon sets and I'm stuck on the following proof. In order to facilitate the comprehension of my problems I will give the full proof and the context. Let $G$ be a compact ...