A Family of nonseparable Smooth compactly Supported Wavelets

We construct smooth nonseparable compactly supported refinable functions that generate multiresolution analyses on L2(R), d > 1. Using these refinable functions we construct smooth nonseparable compactly supported orthonormal wavelet systems. These systems are nonseparable, in the sense that none of its constituent functions can be expressed as the product of two functions defined on lower dimensions. Both the refinable functions and the wavelets can be made as smooth as desired. Estimates for the supports of these scaling functions and wavelets, are given. 1. Definitions and preliminary results The purpose of this paper is to construct smooth nonseparable wavelet systems of compact support. The idea underlying our construction is to start with a suitable orthonormal wavelet system obtained by tensor products, and then make an appropriate change of variables. Other articles studying nonseparable frame or orthogonal wavelets include Ayache [1, 2, 3, 4], Belogay and Wang [5], Karoui [9], Kovac̆ević and Vetterli [10], Lai [11, 12], Li [13], and Yang and Xue [16, 17]. In this section we summarize the notation, definitions and well–known preliminary results that will be used. Section 2 is devoted to the construction of compactly supported nonseparable smooth scaling functions that generate an MRA in L2(R), d > 1. In Section 3 we construct smooth nonseparable compactly supported orthonormal wavelet systems. The sets of strictly positive integers, integers, and real numbers will be denoted by N, Z and R respectively. Given c ∈ R and x = (x1, x2, . . . , xd) ∈ R, we define cx := (cx1, cx2, . . . , cxd) and c(Z) := {ck : k ∈ Z}, and 0d will denote the zero vector in R; if there is no risk of ambiguity we may omit the index d. A function φ ∈ L2(R), d > 1 is said to be separable if there exist a ∈ N, 1 ≤ a < d, and two functions φ ∈ L2(R) and θ ∈ L2(Rd−a) such that φ can be expressed as (1) φ(x1, · · · , xa, xa+1, · · · , xd) = φ(x1, · · · , xa)θ(xa+1, · · · , xd) a.e. on R. If φ ∈ L2(R) is not a separable function, then it is called a nonseparable function. A set of functions Ψ = {ψ1, . . . , ψN} ⊂ L2(R) is called an orthonormal wavelet system if {2ψ`(2x + k); j ∈ Z,k ∈ Z, 1 ≤ ` ≤ N}