On fundamental and interpolating Spline functions[MatSciRep: 82]

Abstract

This report is the outcome of a series of lectures given at Matscience, Since the lntroduction of SpLine functions in 1946 by L.J.Schoenberg, many new and significant results have been unearthed, of which not a few can be traced, to Schoenberg himself. Consequently, it was felt that a systematic study of the contributions of I.J.Schoenberg to the theory of splines would be beneficial to the students of mathematics. This report contains results from two of his earliest papers on 1) On Polya Frequency Functions III: The Positivity of Translation Determinants with an Application to the Interpolation Problem by Spline Curves and 2) On Polya Frequency functions IV: The Fundamental Spline Functions and their Limits. The later papers are discussed in the subsequent reports.