A spectral set is a subset 0 of R with Lebesgue measure 0<+(0)< such that there exists a set 4 of exponential functions which form an orthogonal basis of L(0). The spectral set conjecture of B.… (More)

We consider functions φ ∈ L(R) such that {| det(D)| 12φ(Dx−λ) : D ∈ D, λ ∈ T } forms an orthogonal basis for L(R), where D ⊂Md(R) and T ⊂ R. Such a function φ is called a wavelet with respect to the… (More)

In this paper we show that a class of sets known as the Rauzy fractals, which are constructed via substitution dynamical systems, give rise to self-affine multi-tiles and self-affine tilings. This… (More)

A compact domain in R is a spectral set if there is some subset of R such that fexp(2 ih ; xi) : 2 g when restricted to gives an orthogonal basis of L( ). The set is called a spectrum for . We give a… (More)

For a self similar or self a ne tile in R we study the following questions What is the boundary What is the convex hull We show that the boundary is a graph directed self a ne fractal and in the self… (More)

A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a… (More)

In this paper we show that a class of sets known as the Rauzy fractals, which are constructed via substitution dynamical systems, give rise to self-affine multi-tiles and self-affine tilings. This… (More)

Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent… (More)

Deyu Qian 1,* ID , Nong Zhang 1,*, Dongjiang Pan 1 ID , Zhengzheng Xie 1, Hideki Shimada 2, Yang Wang 1,3, Chenghao Zhang 1 and Nianchao Zhang 4 1 Key Laboratory of Deep Coal Resource Mining,… (More)

A Borel measure μ in R is called a spectral measure if there exists a set Λ ⊂ R such that the set of exponentials {exp(2πiλ · x) : λ ∈ Λ} forms an orthogonal basis for L(μ). In this paper we prove… (More)