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On Polya Frequency Functions Iv: the Fundamental Spline Functions and Their Limits

The theory of spline functions is developed to a point which allows us to establish the properties of spline interpolation for the case of multiple nodes. The genesis of Polya frequency functions from spline functions is developed. (Author).

The theory of spline functions is developed to a point which allows us to establish the properties of spline interpolation for the case of multiple nodes.

On the Spans of Polynomials and the Spans of a Laguerre-Polya-Schur Sequence of Polynomials

This document proves a conjecture of Meir and Sharma from 1969 determining the least value of the span of a certain derivative P'(x) for> or = 3, if x sub 1 and x sub n are kept fixed. Tools used are the Descartes rule of signs and the inequality> or = H between the arithmetic and harmonic mean. The author also applies his results to the infinite sequences of polynomials introduced by G. Polya and I. Schur in a famous paper from 1914.

This document proves a conjecture of Meir and Sharma from 1969 determining the least value of the span of a certain derivative P'(x) for> or = 3, if x sub 1 and x sub n are kept fixed.