Splines and the logarithmic function

Donald J. Newman and I. J. Schoenberg

Vol. 61 (1975), No. 1, 241–258

DOI: 10.2140/pjm.1975.61.241

Abstract

The paper studies a spline function S_n(x)(0 < x < ∞) of degree n, with knots at the points of the geometric progression x_k = q^k (q is fixed > 1, k = 0,±1,±2,⋯ ), which in shown to be uniquely defined by the following two properties: 1^∘. S_n(x) interpolates the function f(x) = log x∕log q at all the knots q^k, 2^∘. S_n(x) satisfies the functional equation S_n(qx) = S_n(x) + 1 for x > 0. S_n(x) is explicitly determined and shown to share with f(x) some of its global properties. The main point is the detailed study of the somewhat surprising behavior of S_n(x) as n →∞.

Mathematical Subject Classification 2000

Primary: 41A15

Milestones

Received: 22 May 1975

Published: 1 November 1975