Construction of biorthogonal wavelets from pseudo-splines

  • Authors:

  • Bin Dong;Zuowei Shen

  • Affiliations:

  • Department of Mathematics, University of California, Los Angeles, CA and Department of Mathematics, National University of Singapore, Singapore;Department of Mathematics, National University of Singapore, Singapore

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2006

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Abstract

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 Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1-46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon, Anal. 10(2) (2001) 163-181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. 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.