Vector subdivision schemes and multiple wavelets

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

  • Rong-Qing Jia
  • S. D. Riemenschneider
  • Ding-Xuan Zhou

Related Research Unit(s)

Detail(s)

Original languageEnglish
Pages (from-to)1533-1563
Journal / PublicationMathematics of Computation
Volume67
Issue number224
Publication statusPublished - Oct 1998

Abstract

We consider solutions of a system of refinement equations written in the form φ = Σα∈ℤ a(α)φ(2·-α), where the vector of functions φ = (φ1, . . . , φr)T is in (Lp(ℝ))r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(ℝ))r by Qaf := Σα∈ℤ a(α)f(2·-α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Qnaf)n=1,2.... in the Lp-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L2-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme exp icitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry.

Research Area(s)

  • Joint spectral radii, Multiple refinable functions, Multiple wavelets, Refinement equations, Transition operators, Vector subdivision schemes

Citation Format(s)

Vector subdivision schemes and multiple wavelets. / Jia, Rong-Qing; Riemenschneider, S. D.; Zhou, Ding-Xuan.
In: Mathematics of Computation, Vol. 67, No. 224, 10.1998, p. 1533-1563.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review