Cardinal interpolation by discrete splines (Q2736130)

Functions are defined on the integers. Natural numbers \(p\), \(n\) and \(\nu\), with \(n= 2\nu+1\), are fixed. Functions \(B_1,\dots, B_p\) on \(\mathbb{Z}\), called discrete \(B\)-splines are defined by: \(B_1\) is the characteristic function of \(\{j\in\mathbb{Z}: -\nu\leq j\leq \nu\}\) and \(B_r= B_1* B_{r-1}\) for \(r= 2,\dots, p\). Then \(B_p\) is a piecewise polynomial function of degree \(p-1\) but is not the restriction to \(\mathbb{Z}\) of a continuous \(B\)-spline. By a discrete spline is meant a linear combination of translates by integers of some discrete \(B\)-spline. \({\mathcal S}_p\) is the set of discrete splines of the form \(S= \sum_{l\in\mathbb{Z}} c(l) B_p(\cdot-ln)\) where \((c(l):l\in\mathbb{Z})\) is the sequence of Fourier coefficients of some \(2\pi\)-periodic distribution. Properties of discrete \(B\)-splines are established using the \(Z\)-transform. Euler-Frobenius trigonometric polynomials and exponential splines \(E(x,\cdot)= \sum_{l\in\mathbb{Z}} e^{-ilx}B_p(\cdot-ln)\) are used in showing that, if \((z(k): k\in\mathbb{Z})\) is a sequence of not more than polynomial growth then there exists a unique \(S\in{\mathcal S}_p\) such that \(S(kn)= z(k)\) for \(k\in\mathbb{Z}\). The final section discusses `father' spline wavelets. The work is motivated by reference to digital signal processing and is related to the first author's work on continuous splines [Integral representation of slowly growing equidistant splines and spline wavelets. Technical Report 5-96. School of Mathematics, Tel-Aviv University, 1996].