TY - JOUR
T1 - Vector subdivision schemes and multiple wavelets
AU - Jia, Rong-Qing
AU - Riemenschneider, S. D.
AU - Zhou, Ding-Xuan
PY - 1998/10
Y1 - 1998/10
N2 - 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.
AB - 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.
KW - Joint spectral radii
KW - Multiple refinable functions
KW - Multiple wavelets
KW - Refinement equations
KW - Transition operators
KW - Vector subdivision schemes
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-0032384592&origin=recordpage
M3 - 21_Publication in refereed journal
VL - 67
SP - 1533
EP - 1563
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 224
ER -