Continuity of nonseparable quincunx wavelets

Abstract

We characterize the continuous compactly supported solutions to the bidimensional refinement equation where the dilation matrix corresponds to a multiplication by √2 followed by a rotation of π/4. The exact Hölder exponent is found in terms of the spectral radius of an operator acting on a subspace of ℓ1(ℤ2). The corresponding wavelet basis is generated by a single function ψ, and the existence of such an orthonormal basis for L2(ℝ2), where ψ is continuous and compactly supported, follows from estimates of the above spectral radius. © 1994 Academic Press, Inc.