Wavelets associated with Nonuniform Multiresolution Analysis on positive Half-Line

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,3 has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L(R+). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V0 of L(R+) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x λ)}λ∈Λ+ where Λ+ = {0, r/N} + Z+, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime and Z+ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen’s condition for the nonuniform multiresolution analysis on the positive half-line.