Inhomogeneous refinement equations

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 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 L_p solutions from F ∈ L_p(ℝ), we construct by the cascade algorithm a sequence of functions φ₀ ∈ L_p(ℝ) from a compactly supported initial function ℝ as A necessary and sufficient condition for the sequence {φ_n} to converge in L_p(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. Acknowledgements and Notes. Research supported in part by Research Grants Council and City University of Hong Kong under Grants #9040281, 9030562, 7000741.