Sampling theorems for Jackson–Nörlund transforms associated with Hahn-difference operators

Abstract

For 0<q<1 and ω>0, the q,ω-Hahn difference operator approximates the classical derivative as q→1− and ω→0+. A q,ω-Hahn–Sturm–Liouville theory is established in the regular setting. The present paper introduces a couple of sampling theorems of Lagrange-type interpolation for q,ω-integral transforms, whose kernels are either solutions or Green's function of the q,ω-Hahn–Sturm–Liouville problem. The new theorems are illustrated in numerical examples, indicating that they approximate the classical results as q→1− and ω→0+.