Orthonormality of cardinal Chebyshev B-spline bases in weighted Sobolev spaces
There is a celebrated result of \textit{U. Reif} [SIAM J. Math. Anal. 28, No.~5, 1258-1263 (1997; Zbl 0884.41009)] which shows that the classical polynomial (univariate) B-splines satisfy an orthonormality property with respect to a weighted Sobolev-inner product. This makes them even more useful for least-squares approximation which is facilitated by the provision of orthogonal bases. In this article, the authors generalise the theorem to Chebyshev B-splines which emerge from certain differential operators, closely related to the L-splines by \textit{C. A. Micchelli} [Stud. Spline Funct. Approx. Theory, 203-250 (1976; Zbl 0343.41008)]. The differential operators are expressed as products of shifts of derivative operators, and both cases when the shifts are either real or complex are studied. The B-splines themselves are differences of Green functions of the said differential operators.
scientific article; zbMATH DE number 1885727