1
vote
0answers
17 views

Sturm-Liouville equation with rational coefficient

I am trying to solve a regular Sturm-Liouville type Ordinary Differential Equation (ODE) with the following form: \begin{equation} \frac{d}{dy} [q(y) \frac{dW(y)}{dy}]+p(y) W(y) = -\lambda W(y) ...
0
votes
0answers
18 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
2
votes
2answers
126 views

Can an integral operator with negative kernel have a positive eigenvalue?

While reading "Integral equations - a reference text" (Zabreiko et al. eds) I came up with this question I cannot answer: Suppose $A:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is a bounded linear integral ...
5
votes
2answers
502 views

integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?

In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain: ...