1
vote
0answers
60 views

Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...
2
votes
0answers
57 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform ...
6
votes
0answers
144 views

Trace class norms of special integral operators

Let $\mu$ be a finite compactly supported Borel measure on the real line. On the space $L^2(\mu)$ consider the integral operators $$ (K_a f)(x)=\int k_a(x, y)f(y)d\mu(y) $$ with $$ k_a(x, ...
3
votes
1answer
181 views

Integral operator defined on $two$ distinct dense subspaces

I have decided to edit my post a bit heavily for clarity. I was trying to be fairly general but it's hard to see what I'm asking so I've decided to limit myself to a specific example which will ...
2
votes
3answers
149 views

tranforms that lowers the number of variables of a function

Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
4
votes
0answers
115 views

Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion

Is there a way to solve analytically the Fredholm integral equation of the second kind $$ \int_0^{100} K(s, t) f(s) ds = \lambda f(t) $$ where the kernel has the piecewise 'linear' form \begin{align} ...
3
votes
1answer
105 views

Interpretation of the integral “with respect to a plane wave” in terms of Radon transform

This question might have a formulation in higher dimensions, but for now let's deal with the 2 dimensional Radon transform: $\newcommand{\R}{\mathbb{R}}$ $$ Rf(\varphi,s)=\int_{-\infty}^\infty ...
3
votes
2answers
358 views

Elaborating Mercer's theorem (RKHS) on Cameron-Martin space $k(x,y)=\min(x,y)$

Hi, I'm trying to employ Mercer's theorem on the kernel $k(x,y)=\min(x,y)$. It is known (and easy to verify) that this is a nonnegative-definite kernel over $[0,T]$ for any $T>0$. Fix $T>0$. ...
3
votes
2answers
381 views

One-sided Cauchy principal value

What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely, $ PV \int_a^b f(t) dt = ? $, where the integral is convergent in the upper limit, but ...
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 ...
15
votes
2answers
913 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 ...