Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two–scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m + 1 and defect r, first considered by P.R. Lipow and I.J. Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degree m and defect r. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution. 1991 Mathematics Subject Classification. Primary 41A15, 42A38, 65D07.