On a minimal property of cubic periodic Lagrangian splines

Abstract

We prove the following theorem concerning cubic periodic spline interpolation: If l j denote the Lagrangian splines in cubic periodic spline interpolation with period N on the grid Z , then the sum of the squares of the l j , j = 0,…, N − 1, is bounded by one. An analogous result for the space P n of algebraic polynomials of degree n and for the interval ¦−1, 1¦ was given by Fejér.