Convergence of cascade algorithms by frequency approach

Starting with an initial function φ0, the cascade algorithm generates a sequence {Q_aⁿφ₀} _n=1^n∞ by cascade operator Q_a defined by Qaf = ∑_α∈ℤ^d a(α) f(M · - α). A function φ is refinable if it satisfies Qaφ = φ. The refinable functions play an important role in wavelet analysis and computer graphics. The cascade algorithm is the main approach to approximate the refinable functions and to study their properties. This note establishes a sufficient condition, in terms of Fourier transforms of the initial function φ₀ and the refinable function φ, for the convergence of cascade algorithm. Our results apply to the case where neither the initial function is compactly supported nor the refinement mask is finitely supported. As a byproduct, we prove that any compactly supported refinable function has a positive Sobolev regularity exponent provided it is in L₂.