Vector subdivision schemes and multiple wavelets
TL;DR: The 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, which leads to a class of continuous orthogonal double wavelets with symmetry.
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Abstract: We consider solutions of a system of refinement equations written in the form formula math where the vector of functions Φ = (Φ 1 ,...,Φ r ) T is in (L p (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 Q a defined on (L p (R)) r by Q a f:= Σ α ∈ 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 (Q a n f) n=1,2... in the L p -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 L 2 -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 explicitly 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.
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References
A study of orthonormal multi-wavelets
Charles K. Chui,Jian-ao Lian +1 more
TL;DR: In this paper, a general scheme for constructing symmetric and/or antisymmetric compactly supported orthonormal multi-scaling functions and multi-wavelets is introduced, where the main emphasis is on maximum order of polynomial-reproduction by the scaling functions, or equivalently maximum number of vanishing moments for the corresponding wavelets.
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Subdivision schemes inL p spaces
TL;DR: The Lp-convergence of a subdivision scheme is characterized in terms of thep-norm joint spectral radius of two matrices associated with the corresponding mask, which describes various properties of the limit function, such as stability, linear independence, and smoothness.
Wavelet analysis of refinement equations
TL;DR: In this article, the spectral radius of a linear operator acting on a wavelet basis is determined by using the wavelet's spectral radius as a measure of the Besov regularity of a compactly supported refinement equation.
164
Refinable function vectors
TL;DR: Regularity of function vectors gives smoothness orders in the time domain and decay rates at infinity in the frequency domain and regularity criteria are established in terms of the vanishing moment order of the matrix mask.
162