Limits of generalized periodic D-splines

Abstract

Results of Schoenberg and others on limits of periodic splines as their order,m, goes to infinity are extended to sequences of D^m-splines determined by the powers of an unbounded non-negative self-adjoint operator D on a Hilbert space,H, and an evaluation mapL fromH toR ⁿ. All such limits lie in the lowest frequencyn-dimensional invariant subspace for D,T* _n . When each term in the sequence is the D^m-spline whose image underL matches a fixed vector,y, (anL-interpolant), then the limit is theL-interpolant toy fromT* _n . When the terms aresmoothing splines derived fromy then the limit exists when the smoothing parameter goes to 0 ast ^−m. Ift is not an eigenvalue,α _i, of D, the limit is theL-least squares best fit toy fromT* _l ,l=card {j: α _j<t}.