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2
votes
1answer
159 views

Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution

I am trying to calculate the Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution. I am ...
8
votes
2answers
1k views

When is an integral transfrom trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
3
votes
2answers
1k views

Best Numerical Method for Evaluating a Hilbert transform

I have to evaluate a Hilbert transform for some $\mathcal{L}^p(\mathbb{R},\mathbb{C})$-function ($1\leq p<\infty$). I know there are a number of algorithms out there to do it, but I don't have a ...
1
vote
1answer
633 views

Writing an integral equation as a partial differential equation

Hello, Can anyone help me see how one can get from the following integral $$\lambda\int_{\beta=0}^{y}\int_{\alpha=x}^{Q-\beta}e^{-\mu(\alpha-x)}f(\alpha,\beta)d\alpha ...
0
votes
1answer
128 views

Floquet tranform of the derivative of a function $f(r)$

The derivative of the Floquet transform equals the Floquet tranform of the derivative. But can the Floquet tranform of the derivative of a function $f(r)$ can be expressed in terms of the Floquet ...
1
vote
0answers
141 views

Integral Transform Wanted

Hello! Imagine I have a function $f(r_1, r_2) = \int_{-\infty}^\infty g(|r_1 - r_3|) g(|r_1 - r_4|) g(|r_2 - r_3|) g(|r_2 - r_4|) g(|r_3 - r_4|) \; d r_3 d r_4$ Is there a way I could get rid of ...
8
votes
2answers
379 views

The relationship between Crofton formula and Radon transform.

The famous Crofton formula says that the length of a curve can be calculated by integral of the `line crossing' over the space of all oriented lines. My question is, is there a way to treat this ...
3
votes
2answers
465 views

Fonction spéciale

Est-ce qu'il y a une fonction spéciale sous la forme suivante: $$\int e^{-st}e^{-(\alpha/(1+t))}t^{\beta-1}(1+t)^{-(\beta-1)-\gamma}dt$$ ou $$\int ...
15
votes
2answers
962 views

Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform.

The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator $$ \mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy. $$ It satisfies ...
2
votes
1answer
388 views

Integral kernel of form $e^{-<x,y>^2}$

Let $K(x,y): \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by $K(x,y) = e^{-< x,y>^2}$ where $<\cdot,\cdot>$ denote the canonical inner product. Define integral operator ...
8
votes
3answers
1k views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $f(x) ...
10
votes
3answers
585 views

MacWilliams Identity for Asymptotic Weight Spectrum of a Code

Introduction Let $C$ be a code of block length $n$ having $A_i^C$ words of Hamming weight $i$, for $i\in [n]$, where $[n]:=\{0,\ldots,n\}$. Then, the sequence $\{ A_i^C \}_{i \in [n]} $ is called the ...
0
votes
1answer
2k views

From an integral equation to a differential equation

Hello, I am wondering whether it is possible to convert the following integral equation to a partial differential equation. where $J_0(t,x)$ is some given nonnegative function and $\nu>0$ is ...
0
votes
1answer
182 views

Completeness of an “infinite mixture of gaussians” representation

Is there a complete "infinite mixture of gaussians representation" for densities? What I mean is, is there, for any reasonably big class of densities $\phi(x)$ I can come up with a function $c(\mu, ...
3
votes
0answers
294 views

inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...