Why are Haar bases in various structures the same?

Abstract

The Haar system [1] is a model example of a wavelet basis in L2([0, 1]). The active study of wavelets began at the end of the twentieth century in connection with the development of multiresolution analysis. Soon interest in wavelet bases defined on various structures arose. In [2] and [3], a theory of multiresolution analysis on the Cantor group was developed, and various wavelet bases substantially different from Haar bases were found. The group of 2-adic numbers is largely similar to the Cantor group (in that they have the same canonical representation of elements and the same metric); the difference is only in the group operation “+,” but this results in dramatically different wavelet theories and multiresolution analyses. In particular, it is shown in [4] that, in contrast to the case of the Cantor group, there does not exist a p-adic multiresolution analysis with orthogonal scaling function other than Haar multiresolution analysis. At the same time, the p-adic Haar basis (first written out in [5]) has exactly the same form as the Haar basis on the Cantor group for p = 2 or on the Vilenkin group for p > 2. Haar bases on the interval [0, 1] with dilation factor pj depending on the level number j were studied in [6]. Haar bases on the rings of integer p-adic numbers and on the projective limits of finite cyclic groups in the oneand many-dimensional cases were constructed in [7]–[10]. The resemblance between the Haar basis on the zero-dimensional group and the Haar basis [6] on [0, 1] was indicated in [8, Remark 2.4]. In the present paper, we give a simple explanation of this resemblance by showing that all the above-mentioned Haar bases can be defined by the same formulas. We describe a sufficiently general scheme for constructing Haar bases which does not even rely on the presence of the group operation “+.” (Although the use of the group operation makes the notation of the Haar functions convenient, it can be completely avoided.) One only needs a “not too bad” topology on a measure space and a suitable sequence of partitions of the space. The construction of the Haar system and the proof of its basis property are exactly the same as for the classical Haar basis. Let μ be a measure on Ω, let Σ be the σ-algebra of measurable sets, and let p = {pj}j=−∞ be a sequence of integers pj > 1. Assume that there exist set collections {Ωjn}, n ∈ Z+, pairwise disjoint for each j ∈ Z and such that μΩ0n = 1 for each n ∈ Z+, Ω = ⋃ nΩjn for each j ∈ Z, and every Ωj−1,n is partitioned into pj subsets Ωj,nk all of which have the same measure, nk = nk(n, j) ∈ Z+. The family of pairs (j, n), numbering the sets Ωjn will be denoted by I, the set {Ωjn}(j,n)∈I will be called an (H,p)-partition, and its elements will be called cells. Let φjn be the characteristic function of the set Ωjn multiplied by a factor normalizing in L2(Ω). Set