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

AbstractIn the reference [3, 126] the author conjectured the following result: Let Sn(x) be the periodic spline function of degree 2n−1, period 2π, that interpolates a prescribed set of k periodic data. 1. If k=2h+1 and T(x) is the trigonometric polynomial of order h interpolating the data, then (1) Sn(x) → T(x) as n → ∞. 2. If k=2h 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