Vol. 61, No. 1, 1975

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Splines and the logarithmic function

Donald J. Newman and I. J. Schoenberg

Vol. 61 (1975), No. 1, 241–258
Abstract

The paper studies a spline function Sn(x)(0 < x < ) of degree n, with knots at the points of the geometric progression xk = qk (q is fixed > 1, k = 0,±1,±2, ), which in shown to be uniquely defined by the following two properties: 1. Sn(x) interpolates the function f(x) = log x∕log q at all the knots qk, 2. Sn(x) satisfies the functional equation Sn(qx) = Sn(x) + 1 for x > 0. Sn(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 Sn(x) as n →∞.

Mathematical Subject Classification 2000
Primary: 41A15
Milestones
Received: 22 May 1975
Published: 1 November 1975
Authors
Donald J. Newman
I. J. Schoenberg