Characterization of compactly supported refinable splines. (English) Zbl 0828.41006

Lawton, Wayne; Lee, S. L.; Shen, Zuowei

Adv. Comput. Math. 3, No. 1-2, 137-145 (1995).

Summary: We prove that a compactly supported spline function \(\varphi\) of degree \(k\) satisfies the scaling equation \(\varphi (x)= \sum_{n=0}^N c(n) \varphi (mx- n)\) for some integer \(m\geq 2\), if and only if \(\varphi (x)= \sum_n p(n) B_k (x-n)\) where \(p(n)\) are the coefficients of a polynomial \(P(z)\) such that the roots of \(P(z) (z-1)^{k+1}\) are mapped into themselves by the mapping \(z\to z^m\), and \(B_k\) is the uniform \(B\)-spline of degree \(k\). Furthermore, the shifts of \(\varphi\) form a Riesz basis if and only if \(P\) is a monomial.

MSC:

41A15	Spline approximation
41A30	Approximation by other special function classes
65D07	Numerical computation using splines

Keywords:

refinable spline; \(B\)-spline; Riesz basis

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