Acta Polytechnica Hungarica Vol. 12, No. 2, 2015 Sturm–Liouville problem and I –Bessel sampling Dragana Jankov Maˇ sirevi´ c Department of Mathematics, University of Osijek, Trg Lj. Gaja 6, 31000 Osijek, Croatia e-mail: djankov@mathos.hr Abstract: The main aim of this article is to establish summation formulae in form of the sam- pling expansion series building the kernel function by the samples of the modified Bessel func- tion of the first kind I ν , and to obtain a sharp truncation error upper bound occurring in the derived sampling series approximation. Summation formulae for functions I ν +1 /I ν , 1/I ν , I 2 ν and the generalized hypergeometric function 2 F 3 are derived as a by–product of these results. The main derivation tools are the Sturm–Liouville boundary value problem and various prop- erties of Bessel and modified Bessel functions. Keywords: Bessel function of the first kind J ν , modified Bessel function of the first kind I ν , sampling series expansions, Sturm–Liouville boundary value problems, generalized hyperge- ometric function 2 F 3 , Fox-Wright generalized hypergeometric function p Ψ ∗ q , sampling series truncation error upper bound. 1 Introduction and motivation The historical background of sampling theorems, various applications in many branches of science and engineering, especially in signal analysis and reconstruction and/or its up-to-date results in different areas of mathematics like approximation theory and interpolation are well–covered among others by Jerri’s ”IEEE 1977 paper” [13], by survey articles of Khurgin–Yakovlev [14] and Unser [24], by the monographs of Higgins [9], an edited monograph by Higgins and Stens [10], the book by Seip [22] and numerous references therein. Thus, by skipping an outline of the facts from the aforementioned references we can focus on our main goal – establishing the I –Bessel sampling expansion result via the appropriate Strum–Liouville boundary value problem and the related sampling expansion series truncation upper bound, which yields the precise convergence rate in this kind of approximation procedures. Here and in what follows B–Bessel sampling is called a sampling expansion pro- cedure for some input function f , when the underlying sampling kernel function is built up in terms of samples of B being a Bessel or modified Bessel function, and the sampling nodes correspond to the zeros b k of B used in the expansion formula. – 231 –