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Construction of biorthogonal wavelets from pseudo-splines. (English) Zbl 1083.42028

Summary: Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by I. Daubechies, B. Han, A. Ron and Z. Shen [Appl. Comput. Harmon. Anal. 14, No. 1, 1–46 (2003; Zbl 1035.42031)] and I. W. Selesnick [Appl. Comput. Harmon. Anal. 10, No. 2, 163–181 (2001; Zbl 0972.42025)], and their properties were extensively studied by B. Dong and J. Shen [“Pseudo-splines, wavelets and framelets” (2004), preprint]. It was further shown by B. Dong and J. Shen in “Linear independence of pseudo-splines” [Proc. Am. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A15 Spline approximation
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