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The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure μ ρ , Hu and Lau [Spectral property of the B ernoulli convolutions, Adv. Math.219(2) (2008) 554–567] showed that L 2 (μ ρ) contains infinite orthogonal exponentials if and only if ρ becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure μ ρ except the contraction ratio ρ being some algebraic numbers called odd-trinomial number. When ρ is an odd-trinomial number, we provide an exponential and polynomial estimations of the upper bound of the spectral number related to the algebraic degree of ρ. Some examples on odd-trinomial number via generalized Fibonacci numbers are provided such that the spectral number of them can be determined. Our study involves techniques from polynomial theory, especially the decomposition theory on trinomial. [ABSTRACT FROM AUTHOR]

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 (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]. [Copyright &y& Elsevier]