a)-crystallographic multiwavelets are a finite set of functions = {ψ i } L i=1 , which generate an orthonormal basis, a Riesz basis or a Parseval frame for L 2 (R d ), under the action of a crystallographic group , and powers of an appropriate expanding affine map a, taking the place of the translations and dilations in classical wavelets respectively. Associated crystallographic multiresolution analysis of multiplicity n (( , a)-MRA) are defined in a natural way. A complete characterization of scaling function vectors which generates Haar type ( , a)-MRA's in terms of ( , a)-multireptiles is given. Examples of ( , a)-MRA crystallographic wavelets of Haar type in dimension 2 and 3 are provided.
Dekking (Adv. Math. 44:78-104, 1982; J. Comb. Theory Ser. A 32:315-320, 1982) provided an important method to compute the boundaries of lattice reptiles as a 'recurrent set' on a free group of a finite alphabet. That is, those tilings are generated by lattice translations of a single tile, and there is an expanding linear map that carries tiles to unions of tiles. The boundary of the tile is identified with a sequence of words in the alphabet obtained from an expanding endomorphism (substitution) on the alphabet. In this paper, Dekking's construction is generalized to address tilings with more than one tile, and to have the elements of the tilings be generated by both translation and rotations. Examples that fall within the scope of our main result include self-replicating multi-tiles, self-replicating tiles for crystallographic tilings and aperiodic tilings.
Let Γ be a crystal group in R d . A function ϕ : R d −→ C is said to be crystal-refinable (or Γ−refinable) if it is a linear combination of finitely many of the rescaled and translated functions ϕ(γ −1 (ax)), where the translations γ are taken on a crystal group Γ, and a is an expansive dilation matrix such thatOne important property of S(ϕ) is, how well it approximates functions in L 2 (R d ). This property is very closely related to the crystal-accuracy of S(ϕ), which is the highest degree p such that all multivariate polynomials q(x) of degree(q) < p are exactly reproduced from elements in S(ϕ). In this paper, we determine the accuracy p from the coefficients dγ . Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function F , which simplifies the classical conditions. Crystal groups and Approximation property and Accuracy and Refinement equation and Composite dilations
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