TY - JOUR
T1 - Inhomogeneous Refinement Equations
AU - Strang, Gilbert
AU - Zhou, Ding-Xuan
PY - 1998
Y1 - 1998
N2 - Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation φ(t) = σ a(k) φ (2t - k) + F(t), k∈ℤ where a(k) is a finite sequence and F is a compactly supported distribution on ℝ. The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in terms of conditions on the pair (a, F). To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions {φn) from a compactly supported initial function φ0 ∈ Lp(ℝ) as φn(t) = σ a(k) φn-1 (2t - k) + F(t). k∈ℤ A necessary and sufficient condition for the sequence {φn}, to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property of the p-norm joint spectral radius (1 ≤ p ≤ ∈) is presented. Finally, the general theory is applied to some examples and multiple refinable functions.
AB - Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation φ(t) = σ a(k) φ (2t - k) + F(t), k∈ℤ where a(k) is a finite sequence and F is a compactly supported distribution on ℝ. The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in terms of conditions on the pair (a, F). To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions {φn) from a compactly supported initial function φ0 ∈ Lp(ℝ) as φn(t) = σ a(k) φn-1 (2t - k) + F(t). k∈ℤ A necessary and sufficient condition for the sequence {φn}, to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property of the p-norm joint spectral radius (1 ≤ p ≤ ∈) is presented. Finally, the general theory is applied to some examples and multiple refinable functions.
KW - Cascade algorithm
KW - Inhomogeneous refinement equation
KW - Joint spectral radius
KW - Multiple refinable function
KW - Wavelets
UR - http://www.scopus.com/inward/record.url?scp=0032283560&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-0032283560&origin=recordpage
M3 - 21_Publication in refereed journal
VL - 4
SP - 733
EP - 747
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
SN - 1069-5869
IS - 6
ER -