In this paper we study univariate two-scale refinement equations φ(x) = ∑κ∈ℤcκφ (2x - κ), where the coefficients cκ∈ ℂ satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: ∑κ(-1)κκncκ= 0, n = 0, . . . , L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.

A complete solution characterizing smooth refinable functions

Protasov, Vladimir
2000-01-01

Abstract

In this paper we study univariate two-scale refinement equations φ(x) = ∑κ∈ℤcκφ (2x - κ), where the coefficients cκ∈ ℂ satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: ∑κ(-1)κκncκ= 0, n = 0, . . . , L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11697/123667
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 7
  • ???jsp.display-item.citation.isi??? ND
social impact