On the convergence of periodic splines of arbitrary degree

Abstract

H. ter Morsche [12] presented a unified theory of interpolation by periodic splines of degree m on a uniform mesh with mesh size h = 1 n . He obtained error bounds of the form ∥ D rƒ − D rφ∥ ∞ ⩽Kh m−r∥D mƒ∥ ∞ (0⩽r⩽m ) for ƒ ∈ C m ( R ) such that D jf(0) = D iƒ(1) (0⩽j⩽m), i.e., ƒ∈C 0 m . This extends results of Quade and Collatz [13] and Subbotin [14]. We will establish error estimates of the form ∥ D rƒ − D rφ∥ ∞ ⩽Kh m+1−r∥D m+1ƒ∥ ∞ (0⩽r⩽m ) for ƒ∈C 0 m+1 . This generalizes special results of Dubeau and Savoie [5, 6] to arbitrary degree m.