Cardinal Hermite interpolation using positive definite functions
Cardinal Hermite interpolation problem (CHIP) was first studied by \textit{P. R. Lipow} and \textit{I. J. Schoenberg} [Linear Algebra Appl. 6, 273-304 (1973; Zbl 0246.41015)]. This paper is a continuation of this study. The main result is the following: Denote by \(S_{2m-1,r}\) the set of all functions that are polynomials of degree \(2m-1\) in each of the intervals \([j, j+1]\) \((j\in\mathbb{Z})\). Then for any sequence of data \(d_ 1= (d_{1,j}),\dots, d_ r= (d_{r,j})\), \(j\in \mathbb{Z}\), of polynomial growth there is a unique \(s_{2m-1} (x)\in S_{2m-1}\) such that \(s_{2m-1}^{(k)} (j)= d_{k,j}\) \((k=0,1,\dots, r-1)\) for all integers \(j\). This paper is a continuation of the study of CHIP. It contains necessary and sufficient conditions for solvability of CHIP for \(\ell^{(p)}\)-data \((p=1,\infty)\) and further study of the ``solvability'' condition. Several examples are presented here and the relation of results of the paper to other papers in this field is explained.
scientific article; zbMATH DE number 640745
scientific article