5
votes
1answer
260 views

Is this inverted integral transform valid?

I have the following transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ with the following conditions: $f(x)$ and $F(y)$ must ...
2
votes
0answers
187 views

Solving a nonlinear integral equation

Consider the integral equation $$f=g^2+H[g]^2$$ where $f\colon\mathbf R\to \mathbf R$ is an even and integrable function, $g$ is the function to be solved for, and $H[g]$ is the Hilbert transform of ...
5
votes
1answer
363 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
3
votes
2answers
944 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 ...