Narrow Results

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction J_xy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ J_xy with | J_xy |< p^{-1/(p -1)}. (ii) For q∈p ℕ, p≥3 one can choose J_xy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

Refinable functions play an important role in the construction and properties of wavelets. Basically, most of the wavelets are generated from refinable functions. In this paper, a study on the approximation properties of refinable functions on $p$-adic fields is carried out with necessary theoretical background. Various equivalent forms of approximation order and the connection between the approximation order and the Strang–Fix condition are derived. Finally a characterization for the approximation order of a refinable function is given in terms of order of the sum rules associated with the refinement mask.