Questions tagged [fourier-transform]

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7 views

How to switch the order of limit and integration here?

I asked it on math.stackexchange but didn’t get an answer. Let $f\in L^1(\mathbb{R}^n)$ and $\phi_\varepsilon(x):=\varepsilon^{-n}e^{-\pi|\varepsilon^{-1}x|^2}$. It is straightforward to see that $\...
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74 views

Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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56 views

(Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
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Fourier transform of a general normal distribution [closed]

How can I calculate the Fourier transform of this equation? Here is a known equation that may be helpful.
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49 views

Conditions for absolute continuity in the Bochner-Schwartz theorem

Suppose that $f$ is a positive-definite Schwartz distribution, that is, $$\langle\phi,f*\phi\rangle\geq0\qquad\text{for every }\phi\in C_0^\infty(\mathbb R^n).$$ By the Bochner-Schwartz theorem, there ...
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178 views

Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$

Can one show that Fourier transform of $$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$ is decreasing in $a$? I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
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32 views

eigenvectors of a graph Laplacian VS Fourier basis

Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
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61 views

Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$

It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
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1answer
112 views

Distribution boundary value of analytic function and wave front sets

Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
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66 views

How to calculate Fourier transformation of eigenstates in CV quantum information [closed]

The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$. And we set there eigenstates as $|s\rangle_q$ and $|s\rangle_p$ with eigenvalue s. In the paper [Phy Rev A. 79, 062318 (2009)],...
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How related are Fourier transforms on finite groups and Fourier transforms on graphs?

Here are two generalizations of the notion of a Fourier transform. I am also aware of the Pontryagin Duality generalization for locally compact abelian groups, though I am personally more concerned ...
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92 views

A close formula for a Fourier transform

I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define $$ I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
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74 views

Laplacian spectrum

I have found (in lecture notes) a method to calculate the spectrum of the operator $$ A:D(A)\subset L^2([0,\pi])\longrightarrow L^2([0,\pi])\text{, such that} $$ $$ Au=\dfrac{\partial^2u}{\partial^2x^...
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230 views

Fourier transform on Minkowski space

Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
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99 views

A Fourier elliptic vector field on a Riemannian manifold

Motivation for this question: Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
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Fourier dimension of radial set

In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
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Cumulant of functions of weakly dependent random variables

Suppose $X_1,\dots,X_4$ are Gaussian random variables with mean and variance $$\mathbf E X_i = 0,\quad \mathbf E X_i^2=1.$$ Furthermore suppose that the random variables have a certain weak ...
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1answer
162 views

Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$

Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...
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1answer
119 views

Wavelet momentum identity

I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31): \...
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222 views

Certain Fourier transforms involving Whittaker function and Bessel functions

I recently meet the following two weird "Fourier transform" questions. (I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ and $\...
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Common eigenvector of the Wigner D matrices for the eigenvalue $\lambda = 1$?

Consider the family of spherical harmonics $Y_{n}^m$ on the sphere $\mathbb{S}^2$, with $n \geq 0$ and $-n \leq m \leq n$. The Wigner matrices are matrices $\mathrm{D}_{\mathrm{R},n} \in \mathbb{R}^{(...
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1answer
80 views

Advantage of fractional Fourier transform over multiscale wavelet

What is the best argument of fractional Fourier transform over multiscale wavelet in data analysis purpose. Optimization of the good time-frequency domain parameter ? "Good" will be, find the %time-%...
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10k views

Why is the Fourier transform so ubiquitous?

Many operations and equivalences in mathematics arise as some sort of Fourier transform. By Fourier transform I mean the following: Let $X$ and $Y$ be two objects of some category with products, and ...
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235 views

The Hubbard-Stratonovich transformation

is there a way to extend the Hubbard-Stratonovich transformation $$e^{\frac{1}{2}Ks^2}=\left(\frac{K}{2\pi}\right)^{1/2}\int_{–\infty}^\infty e^{–Kx^2+Ksx}dx$$ to the case $e^{\frac{1}{2}Ks^p}$ for $...
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1answer
272 views

Large Fourier submatrices with small operator norm

Consider a finite abelian group $G$ (I'm mostly interested in $\mathbb{Z}_2^n$). For two subsets $A$ and $B$ of $G$, one can form a submatrix of the Fourier transform matrix on $G$ by keeping only ...
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1answer
115 views

Fourier transform for $H^2(\mathbb{R}^N)$, $N\geq 5$

How i can prove that if $u\in H^2(\mathbb{R}^N)$ then $u\in \mathcal{F}(L^{p^*}(\mathbb{R}^N))$, where $1/p+1/{p^*}=1,$ $2\leq p<2N/(N-4)$?
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Which set of functions/measures has range $\mathrm{L}^\infty$ under Fourier transformation

I have a question concerning the Fourier transformation. What I know is that $\mathrm{L}^{\infty}=\{\hat{u}:\ u\in Y\}$ for some space $Y$. Now, I want to specify the space $Y$. The question is, is ...
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165 views

Fourier transform derivation from Laurent series

Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$ $$f(z) = \sum^{\infty}_{n=-\infty} c_n(z-a)^n \ \ \ \ (1)$$ where $$c_n = \frac{1} {2\pi i}\int\limits_{\gamma}\frac {f(z)} {...
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Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$

Let $\theta$ be a $C^{\infty}$ (resp. analytic) real-valued function on $\mathbb{T}=[0,2\pi]/\{0,2\pi\}$. When can one find $f \neq 0$, $C^{\infty}$ (resp. analytic) real-valued function on $\...
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69 views

Is harmonic mean of linear functions a Bernstein function?

According to some experiments I've been running, for any $n$ and non-negative $a_1, a_2, \ldots a_n$, the following function: $f(t) = \frac{n}{\sum_{i=1}^n 1/(a_i+t)}$ is a Bernstein function, ...
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74 views

Expressing 1-e^{-z} as a Fourier integral

According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...
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Deriving spectral measure

I am reposting this question from Cross Validated as I have not received any responses. While reading this book, I got stuck on page 266 where the authors found the spectral measure $F(du)$ of the ...
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1answer
53 views

Hilbert transform of a signal to measure skewness and asymmetry of a sinusoidal wave

Thank you for taking the time to read this. I was hoping to get some assistance in understanding how these equations function: $$As=\frac{\langle H(\eta)^3\rangle}{\langle\eta^2\rangle^{3/2}},\qquad ...
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Properties of microlocalization

Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
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1answer
124 views

What subjects of Fourier analysis have had more effect on machine learning? [closed]

What is the salient uses of Fourier analysis in machine learning? What subjects of Fourier analysis have had more effect on machine learning? Please mention the references.
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140 views

Solving an equation of function

How to solve, or at least how to proceed to solve, the following equation for $g(u)$ $$\int_0^{\infty} \{1-\cos(2\pi uh)\} g(u)du = (1+h^{\alpha})^{\beta/\alpha} -1?$$ Here $0<\alpha\leq2$ and $-\...
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1answer
160 views

Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
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Lower Bounds on Oscilatory Integral in Restriction Theory

Consider the oscillatory integral that appears in restriction theory: $$ I(x,y)=\int_{-1}^1 e^{i(x\xi+y\xi^2)}d\xi. $$ Using basic theory on oscillatory integrals, we may bound $$ |I(x,y)|\lesssim |x|^...
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Is there a practical application of natural integral or differintegral?

The following formulas give natural differintegral (that is one with naturally fixed integration constant): $$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$ $$...
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1answer
133 views

Translated version of a Caratheodory article

This excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911. These are: [46] C. Caratheodory. Uber den Variabilitätsbereich der ...
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429 views

Process quicker than Fourier for squares of polynomials

FFT is a quick algorithm for multiplying two polynomials, but given it's a square (i.e. multiplying the polynomial with itself) can we find something better?
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1answer
194 views

Is there (fast) fourier transform for vector convolution?

Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined $$U*V(t)={\sum_{i}} u_iv_{t-i}.$$ Given a list of vectors $u_1,\dots,u_m\in\...
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101 views

Decay of Laplace (or Mellin) transform beyond region of convergence?

Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
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116 views

Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?

All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
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1answer
136 views

Low/high-frequency estimates in $\mathrm{L}^\infty$ for Lipschitz nonlinearities

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. ...
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Approximate method to extract behavior of a Laplace transform in an intermediate region

In the theory of random walks, Tauberian type theorems are often applied to extract the small or large-time behavior from a difficult equation. For example, the Montroll-Weiss formula describing a ...
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1answer
158 views

Origin of the theorem related to the integral transform pair

The development of Fast Fourier transform is attributed to Cooley & Tukey, both have written a lot about it is historical development. Both Cooley and Tukey call it a re-discovery rather. However,...
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1answer
121 views

Image of Fourier transform for finite non-abelian groups

I am working on the Fourier transform over finite non-abelian groups, specifically following Diaconis. He defines it as follows (p.7): Let $P$ be a probability on a finite group $G$. The Fourier ...
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1answer
205 views

Hörmander-Mikhlin theorem on the torus

Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and ...
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3answers
363 views

Limit of an integral vs limit of the integrand

I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral $$ I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...

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