@article{1e043c4332814c32b070a9d39667a45c, title = "Inhomogeneous Refinement Equations", abstract = "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.", keywords = "Cascade algorithm, Inhomogeneous refinement equation, Joint spectral radius, Multiple refinable function, Wavelets", author = "Gilbert Strang and Ding-Xuan Zhou", year = "1998", language = "English", volume = "4", pages = "733--747", journal = "Journal of Fourier Analysis and Applications", issn = "1069-5869", publisher = "Springer Basel AG", number = "6", }