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Equivalence of A-Approximate Continuity for Self-Adjoint

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Equivalence of A-Approximate Continuity for Self-Adjoint Equivalence of A-Approximate Continuity for Self-Adjoint Expansive Linear Maps a,1 b, ,2 Sz. Gy. R´ev´esz , A. San Antol´ın ∗ aA. R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary bDepartamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Abstract Let A : Rd Rd, d 1, be an expansive linear map. The notion of A-approximate −→ ≥ continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of A-approximate continuity at a point x – or, equivalently, the definition of the family of sets having x as point of A-density – depend on the expansive linear map A. The aim of the present paper is to characterize those self-adjoint expansive linear maps A , A : Rd Rd for which 1 2 → the respective concepts of Aµ-approximate continuity (µ = 1, 2) coincide. These we apply to analyze the equivalence among dilation matrices for a construction of systems of MRA. In particular, we give a full description for the equivalence class of the dyadic dilation matrix among all self-adjoint expansive maps. If the so-called “four exponentials conjecture” of algebraic number theory holds true, then a similar full description follows even for general self-adjoint expansive linear maps, too. arXiv:math/0703349v2 [math.CA] 7 Sep 2007 Key words: A-approximate continuity, multiresolution analysis, point of A-density, self-adjoint expansive linear map. 1 Supported in part in the framework of the Hungarian-Spanish Scientific and Technological Governmental Cooperation, Project # E-38/04. 1 The author was supported in part by the Hungarian National Foundation for Scientific Research, Project # T-049301. 1 This work was accomplished during the author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927. 2 Supported in part in the framework of the Spanish-Hungarian Scientific and Technological Governmental Cooperation, Project # HH 2004-0002. Preprint submitted to Elsevier 25 July 2021 1 Introduction A multiresolution analysis (MRA) is a general method introduced by Mallat [15] and Meyer [16] for constructing wavelets. On Rd (d 1) equipped with the Euclidean norm , an MRA means a sequence of subspaces≥ V , j Z k·k j ∈ of the Hilbert space L2(Rd) that satisfies the following conditions: (i) j Z, V V ; ∀ ∈ j ⊂ j+1 (ii) j Z, f(x) Vj f(2x) Vj+1; ∀ ∈ 2∈ d ⇔ ∈ (iii) W = j∈Z Vj = L (R ); (iv) there exists a scaling function φ V , such that φ(x k) k Zd is an S ∈ 0 { − } ∈ orthonormal basis for V0. We could consider MRA in a general context, where instead of the dyadic dilation one considers a fixed linear map A : Rd Rd such that A is an expansive map, i.e. all (complex) eigenvalues have absolute→ value greater than 1, and A(Zd) Zd, (1) ⊂ i.e., the corresponding matrix of A with respect to the canonical basis has every entries belonging to Z. Given such a linear map A one defines an A MRA as 2 d − a sequence of subspaces Vj, j Z of the Hilbert space L (R ) (see [14], [9], [20], [23]) that satisfies the conditions∈ (i), (iii), (iv) and (ii ) j Z, f(x) V f(Ax) V . 1 ∀ ∈ ∈ j ⇔ ∈ j+1 A characterization of scaling functions in a multiresolution analysis in a general context was given in [2], where the notion of A-approximate continuity is introduced as a generalization of the notion of approximate continuity. In this work G d denotes the d-dimensional Lebesgue measure of the set G Rd, and B :=| | x Rd : x < r stands for the ball of radius r with⊂ r { ∈ k k } the center in the origin. Also, we write F + x0 = y + x0 : y F for any d d { ∈ } F R , x0 R . ⊂ ∈ Definition 1 Let an expansive linear map A : Rd Rd be given. It is said that x Rd is a point of A density for a measurable−→ set E Rd, E > 0 0 ∈ − ⊂ | |d ∗ Corresponding author. Telf: 0034 91 497 5253, fax: 0034 91 497 4889 Email addresses: [email protected] (Sz. Gy. R´ev´esz), [email protected] (A. San Antol´ın). 1 Present address: Institut Henri Poincar´e, 11 rue Pierre et Marie Curie, 75005 Paris, France 2 Partially supported by # MTM 2004-00678 2 if for all r> 0, E (A−jB + x ) lim | r 0 |d =1. (2) j−→∞ A−jB + x |T r 0|d d d d Given an expansive linear map A : R R , and given x0 R , we define −→ ∈ the family of A-dense sets at x0 as d (x0)= E R measurable set : x0 is a point of A density for E . EA { ⊂ − } Furthermore, we will write when x0 is the origin. EA Definition 2 Let A : Rd Rd be an expansive linear map and let f : Rn → n −→ C be a measurable function. It is said that x0 R is a point of A-approximate continuity of the function f if there exists a measurable∈ set E Rn, E > 0, ⊂ | |n such that x0 is a point of A-density for the set E and lim f(x)= f(x0). (3) x→x0,x∈E The relation between the behavior of the Fourier transform φ of the scaling function φ in the neighborhood of the origin and the condition (iii) is described in the following theorem of [2]. b 2 d Theorem A. Let Vj be a sequence of closed subspaces in L (R ) satisfying the conditions (i), (ii1) and (iv). Then the following conditions are equivalent: 2 d (A) W = j∈Z Vj = L (R ); ∗ (B) SettingS φ(0) = 1, the origin is a point of A -approximate continuity of the function| φ |. b | | As it was observedb in [2, Remark 5, p. 1016], that the definition of points of A-approximate continuity depends of the expansive linear map A. The aim of the present paper is to study the following problem: Problem 1: Characterize those expansive linear maps A , A : Rd Rd for 1 2 → which the concept of A1-approximate continuity coincides with the concept of A2-approximate continuity. Remark 1. ¿From the definition of point of A-approximate continuity of a d d measurable function on R , it is easy to see that given x0 R and given a d ∈ measurable set E R , the point x0 is a point of A-approximate continuity ⊂ 3 for the function 1 if x E x0 f(x)= χE∪{x0}(x)= ∈ { } 0 if x / E S ∈ if and only if E (x0). Therefore, it suffices to study the notion of A-density, ∈E and once (x0) = (x0), also the notions of A -approximate continuity EA1 EA2 1 and A2-approximate continuity coincide. Moreover, clearly, E if and only if E + x0 (x0). ∈EA ∈EA Thus, we can simplify Problem 1 in the following way. Problem 1’: Describe under what conditions on two expansive linear maps A , A : Rd Rd, we have that = . 1 2 −→ EA1 EA2 In Corollary 20 we solve the problem for expansive self-adjoint linear maps on Rd, without the extra condition (1). In the last section we discuss the additional, essentially number theoretical restrictions, brought into play by condition (1). Characterization of expansive matrices satisfying (1) have been studied by several authors. In [13] a complete classification for expanding 2 2-matrices satisfying (1) and det M = 2 is given. Their result is the following.× | | Call two integer matrices A and M integrally similar if there exists an integer unimodular matrix C such that C−1AC = M. Now, denote 0 2 0 2 A1 = A2 = 1 0 1 0 − 1 1 0 2 A3 = A4 = . 1 1 1 1 − − Lemma B. Let M be an expanding 2 2-matrix satisfying (1). If det M = 2 × − then M is integrally similar to A1. If det M =2 then M is integrally similar to A , A , A . 2 ± 3 ± 4 On the other side, a complete characterization for expanding 2 2-matrices satisfying (1) and × M l = nI for some l, n N (4) ∈ 4 is given in [6]. Their answer is given in the following theorem where they do not write the trivial case that M is a diagonal matrix. Theorem C. Given l, n N, an expanding 2 2-matrix which satisfy (1) ∈ × and (4) exists if and only if there exist two numbers λ1 = λ2 whose sum and l l 6 product are integral and satisfy λ1 = λ2 = n. Furthermore, then λ1 and λ2 are the eigenvalues of the matrix M in (4). The following corollary is a classification of the expanding matrix M satisfying (1), det M < 0 and (4). Corollary D. Let M be an expanding 2 2-matrix satisfying (1) with det M < 0. Then M satisfies condition (4) if and× only if trace M =0 and det M = n. Especially − M 2 = nI. Moreover, they give a classification of the expanding matrix M satisfying (1), det M > 0 and (4). They write the following theorem with the restriction to the case that the eigenvalues are complex numbers because in other way, by (4), M is a diagonal matrix. Here, l will denote the minimal index for which (4) holds, i.e., they neglect the trivial cases generated by powers of (4). Theorem E. Let M be an expanding 2 2-matrix satisfying (1) with det M > × 0 and eigenvalues λi / R, i = 1, 2. Then condition (4) can only hold for l =3, 4, 6, 8 and 12.
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