Refinable functions with non-integer dilations

Abstract

Refinable functions and distributions with integer dilations have been studied extensively since the pioneer work of Daubechies on wavelets. However, very little is known about refinable functions and distributions with non-integer dilations, particularly concerning its regularity. In this paper we study the decay of the Fourier transform of refinable functions and distributions. We prove that uniform decay can be achieved for any dilation. This leads to the existence of refinable functions that can be made arbitrarily smooth for any given dilation factor. We exploit the connection between algebraic properties of dilation factors and the regularity of refinable functions and distributions. Our work can be viewed as a continuation of the work of Erdös [6], Kahane [11] and Solomyak [19] on Bernoulli convolutions. We also construct explicitly a class of refinable functions whose dilation factors are certain algebraic numbers, and whose Fourier transforms have uniform decay. This extends a classical result of Garsia [9].

Stable Filtering Schemes with Rational Dilations

“…The frame bounds of V (Ψ) are determined by the eigenvalues of G Ψ (ξ) over σ Ψ , which, by (5), are seen to be identical to the eigenvalues of G Φ;p (ξ) over σ Φ;p .…”

“…This idea was hinted at in Daubechies' treatment of the a = 3 2 case, where it was pointed out that the rational filtering schemes studied by Kovačević and Vetterli in [9] could not arise from standard MRA constructions. More recently, refinability with rational dilations has been studied by Dai, Feng, and Wang with particular attention paid to the regularity of the refinable functions and distributions [5].…”

Summation and intersection of refinable shift invariant spaces

“…We refer the readers to Daubechies [11], Cavaretta et al [5], and the references therein. Refinable distributions also play important roles in fractal geometry, self-affine tiles and Bernoulli convolutions, see for instance, Falconer [12], Lagarias and Wang [20], and Dai et al [6][7][8]. In this paper, we focus mainly on the shift invariant spaces V (f ) and V 0 (f ) generated by a compactly supported refinable distribution f , where…”

Refinement equations and spline functions

“…In [2], Cavaretta et al extended this result to higher dimensions by a matrix method. When λ is non-integer, 'the regularity question becomes more complicated and perhaps more interesting from the viewpoint of pure analysis' [4]. Moreover, the refinable functions with non-integer dilations play an important role in the construction of wavelets with non-integer dilations [1].…”

“…In general, one characterizes the regularity of f (x) by considering the decay of f (w). In [4], Dai et al considered the uniform decay of the Fourier transform of the refinable functions with noninteger λ. In particular, they give an elegant answer to the following question:…”