Notes on spline functions I. The limits of the interpolating periodic spline functions as their degree tends to infinity

Abstract

In the reference [3, 126] the author conjectured the following result: Let S n ( x) be the periodic spline function of degree 2 n−1, period 2 π, that interpolates a prescribed set of k periodic data. 1. If k=2 h+1 and T( x) is the trigonometric polynomial of order h interpolating the data, then (1) S n ( x) → T( x) as n → ∞. 2. If k=2 h and T( x) is among all interpolating trigonometric polynomials of order h the one such the terms of highest frequency have least amplitude, then again (1) holds. Here these results are established.