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 [P. Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940) 180–186], Kahane [J.-P. Kahane, Sur la distribution de certaines séries aléatoires, in: Colloque de Théorie des Nombres, Univ. Bordeaux, Bordeaux, 1969, Mém. Soc. Math. France 25 (1971) 119–122 (in French)] and Solomyak [B. Solomyak, On the random series ∑ ± λ n (an Erdös problem), Ann. of Math. (2) 142 (1995) 611–625] 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 [A.M. Garsia, Arithmetic properties of Bernoulli convolutions, Trans. Amer. Math. Soc. 102 (1962) 409–432].