Xufa Fang

A refinement equation, which can be expressed as a functional equation of the form φ ( x ) = ∑ j ∈ Z a ( j ) φ ( m x − j ) , x ∈ R , where φ is the unknown function defined on R, { a ( j )| j ∈Z} is a finitely supported sequence (called the refinement mask), and m> 1 is an integer dilation factor. Solutions to such equations, known as refinable functions, play crucial roles in spline theory, interpolation schemes for curve construction, wavelet analysis with compactly supported bases, fractal generation, and probability theory. In this paper, we show the minimum cardinality of the support of the nonnegative mask for a univariate cascade algorithm, which ensures the convergence of the corresponding algorithm.

In this paper, we characterize the convergence of the multivariate cascade algorithm with an [ ] dilation matrix [ ] and a nonnegative finite mask, which generalizes the result obtained recently. Moreover, we present two examples, which demonstrate the power and applicability of our main result.