Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions

I asked the question below on Math Stack Exchange,, but [despite it having 6 upvotes and two favourites] received no answer even after a 300-point bounty. Therefore, I am re-posting it here. Let f:R→R be a bounded L1 function that is piecewise-smooth (with the boundaries of the pieces having no accumulation points), but not … Read more

Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?

As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, while if the derivative of a bandlimited function has L1 Fourier transform (and hence is bounded) then so do all higher derivatives. Isaac Pesenson et … Read more

Fourier coeffients of Cantor measure

For 0<θ<12, denote by μθ the uniform Cantor measure with dissection ratio θ. It is not hard to show that the Fourier–Stieltjes transform of μθ is ˆμθ(ξ)=∞∏k=1cos(πθkξ) (up to scaling and constant multiple). It is well known that ˆμθ(ξ)→0 as ξ→∞ if and only if θ is a Pisot number. Now we are concentrated on … Read more

Given θ\theta, find ff such that ∫Teiθcos(h⋅f)=0,\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0, for all h∈Nh \in \mathbb{N}

Let θ be a C∞ (resp. analytic) real-valued function on T=[0,2π]/{0,2π}. When can one find f≠0, C∞ (resp. analytic) real-valued function on T such that ∫[0,2π]eiθ(s)cos(h⋅f(s))ds=0, for all h∈N∪{0}? Setting n=0, one obvious necessary condition is ∫[0,2π]eiθ(s)ds=0. A sufficient condition I found, which is far from being explicit, is the existence of a reparametrization v:T→T that switches … Read more

Does the Fourier transform preserve the separation property?

The space of Schwartz functions on the plane is denoted by S. The usual multiplication and the convolution multiplication on S are denoted by m1 and m2, respectively. The Fourier transform F on S give a bijective correspondence between the m1-subalgebras of S and m2– subalgebras of S. We say that a subset A of … Read more

Distribution boundary value of analytic function and wave front sets

Assume f(z) is analytic in the tube domain Rn⊕iC, where C⊂Rn is a convex cone. Under the assumption |f(x+iy)|≤1/|y|k, we know by a Theorem of Martineau (see also Hormander, volume 1, Theorem 3.1.15) that the limit limy→0,y∈Cf(x+iy) exists as a tempered distribution f(x) on Rn, uniformly in proper cones y∈C′⊂C. The convergence is in the … Read more

Integral solving request

Dear all, please help me solve the following integral. I need to solve this integral for one of my problems. $$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho r)L*Sinc(\frac{Lk_z}{2})e^{jk_zz}}{(\frac{\omega_0}{c})^2-(\rho^2+k_z^2)}\rho\operatorname{d}k_z\operatorname{d}\rho$$ $J_0(x)$ is first kind bessel function and $Sinc(x)=sin(x)/x$. Is this integral have an analytical solution? I tried to solve it with Mathematica and Matlab but they even can not solve the … Read more

Fourier transform in terms of special function?

I have a Fourier integral \int\limits_{-\infty}^{\infty}\mathrm{d}t\,\frac{1}{t^2}\exp\left({\mathrm{i}\frac{t^3}{3}+\mathrm{i}Yt+\frac{\mathrm{i}\lambda^2}{4t}}\right), where Y and \lambda are arbitrary real parameters. Is it possible to express this integral in terms of some special functions, say hypergeometric functions or confluent hypergeometric functions? Answer I(Y,\lambda)=\int\limits_{-\infty}^{\infty}\mathrm{d}t\,\frac{1}{t^2}\exp\left(\frac{\mathrm{i}t^3}{3}+\mathrm{i}Yt+\frac{\mathrm{i}\lambda^2}{4t}\right) =2\int\limits_{0}^{\infty}\mathrm{d}u\,\cos\left(\frac{1}{3u^3}+\frac{Y}{u}+\frac{u\lambda^2}{4}\right) I had initially hoped for a simple answer, but was mistaken; since I started an answer, let me … Read more

2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question is: (1) As $\mu$ and $\nu$ are distinct, hence $W_2(\mu,\nu)\neq 0$. Is it possible that there is a sequence of distributions $\mu_i, i\geq 1$ on $\mathbb{R}^n$ such … Read more

If $f$ is non-prime, can we say $|f|$ is also a non-prime; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < \infty \}).$$ My question is: Let $f \in L^{2}(\mathbb T) \ast L^{2}(\mathbb T)$ that is, $f=g \ast h$ for some $g, h \in L^{2}(\mathbb T)$. Can we expect $|f| \in … Read more