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 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 L_p solutions from F ∈ L_p(ℝ), we construct by the cascade algorithm a sequence of functions {φ_n) from a compactly supported initial function φ₀ ∈ L_p(ℝ) as φ_n(t) = σ a(k) φ_n-1 (2t - k) + F(t). k∈ℤ 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.