Cardinal Hermite spline interpolation: convergence as the degree tends to infinity

Abstract

Let 52m,r denote the class of cardinal Hermite splines of degree 2m - I having knots of multiplicityr at the integers. For f(x) E C'-'(R), the cardinal Hermite spline interpolant tof(x) is the unique element of S2m,, which interpolates f(x) and its first r - 1 derivatives at the integers. For y = (y?, .... ,yr-1) an r-tuple of doubly-infinite sequences, the cardinal Hermite spline interpolant toy is the unique S(x) E 52,,. satisfying SO(v) y, ,s =0, 1 . .., r- 1, and v an integer. The following results are proved: If f(x) is a function of exponential type less than rsr, then the derivatives of the cardinal Hermite spline interpolants to f(x) converge uniformly to the respective derivatives of f(x) as m -oo. For functions from more general, but related, classes, weaker results hold. If y is an r-tuple of lP sequences, then the cardinal Hermite spline interpolants toy converge to Wr(y), a certain generalization of the Whittaker cardinal series which lies in the Sobolev space WP- (R). This convergence is in the Sobolev norm. The class of all such Wr(y) is characterized. For small values of r, the explicit forms of Wr(y) are described.