On sampling theory and basic Sturm–Liouville systems

AbstractWe investigate the sampling theory associated with basic Sturm–Liouville eigenvalue problems. We derive two sampling theorems for integral transforms whose kernels are basic functions and the integral is of Jackson's type. The kernel in the first theorem is a solution of a basic difference equation and in the second one it is expressed in terms of basic Green's function of the basic Sturm–Liouville systems. Examples involving basic sine and cosine transforms are given

The roots of the third Jackson q-Bessel function

We derive analytic bounds on the roots of the third Jackson Function. This bounds prove a conjecture of M. E. H. Ismail concerning the asymptotic behaviour of the roots

Determinant inequalities for sieved ultraspherical polynomials

Paul Turan first observed that the Legendre polynomials satisfy the inequality Pn2(x)−Pn−1(x)Pn(x)>0, −1<x<1. Inequalities of this type have since been proved for both classical and nonclassical orthogonal polynomials. In this paper, we prove such an inequality for sieved orthogonal polynomials of the second kind