Lp-Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.

L p –Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form $ \varphi (x) = {\sum\limits_{\alpha \in \mathbb{Z}^{s} } {a(\alpha )\varphi {\left( {Mx - \alpha } \right)},x \in \mathbb{R}^{s} } }, $ where the vector of functions ? = (?1, . . . ,?r) ^T is in (L _p (? ^s )) ^r , 0 < p ≤ ∞, a(α), α ∈ ? ^s ¸ is a finitely supported sequence of r× r matrices called the refinement mask, and M is an s × s integer matrix such that lim_{n→ ∞} M ^–n = 0. In this article, we characterize the existence of an L _p –solution of the refinement equation for 0 < p ≤ ∞. Our characterizations are based on the p–norm joint spectral radius.