Abstract
In this paper, we construct a new family of refinable functions from generalized Bernstein polynomials, which include pseudo-splines of Type II. A comprehensive analysis of the refinable functions is carried out. We then prove the convergence of cascade algorithms associated with the new masks and construct Riesz wavelets whose dilation and translation form a Riesz basis for $L_{2}(\mathbb{R})$ . Stability of the subdivision schemes, regularity and approximation orders are obtained. We also illustrate the symmetry of the corresponding refinable functions.
Highlights
During the last decades, the study of refinable functions has attracted considerable attention
It is natural to consider a new extension of pseudo-splines, which has compact support and exponential decay, in particular, masks of new refinable functions derived from generalized Bernstein polynomials [ ] by substitution and summation
Convergence of cascade algorithms is implemented, which guarantees the existence of refinable functions
Summary
The study of refinable functions has attracted considerable attention. It is natural to consider a new extension of pseudo-splines, which has compact support and exponential decay, in particular, masks of new refinable functions derived from generalized Bernstein polynomials [ ] by substitution and summation. A compactly supported function φ ∈ L (R) is refinable if it satisfies the refinement equation φ = τ (k)φ( · –k), k∈Z where τ , called the refinement mask of φ, is a finitely supported sequence. By the iteration of equation ( ), the corresponding refinable function φ can be written in terms of its Fourier transform as. We will provide three lemmas about the relations of the quantities ρτ (a, ∞) associated with masks and a condition of the convergence of cascade algorithms which are necessary for the following theorem. By Lemma , the cascade algorithm associated with the mask τ m,l,α(ω) converges in the space L ,∞(R)
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