Difference Equation , B-Spline , Recurrence Relation , Linear Functional
Abstract:
The present paper is to be understood as a revision and continuation of a recent series of publications on recursively defined B-splines, due to C. deBoor and K. Höllig [4] and G. Ciascola [6, 7]. We define B-splines as solutions of a certain difference equation of evolution type, which is equivalent to the well-known B-spline recurrence relation. It turns out that this approach leads to very elementary and direct proofs, in particular without using the Marsden identity, of many important properties of the B-splines. Special emphasis is also laid on the probability theoretic aspects of these functions. Among other results, we prove explicit formulas for the kth moments, the variance and the distribution function of a B-spline.
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