Multivariate box spline wavelets in higher-dimensional Sobolev spaces

We construct wavelets and derive a density condition of MRA in a higher-dimensional Sobolev space. We give necessary and sufficient conditions for orthonormality of wavelets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{s}(\mathbb {R}^{d})$\end{document}Hs(Rd). We construct nonseparable orthonormal wavelets in a higher-dimensional Sobolev space by using multivariate box spline.


Introduction
Box splines are refinable functions, and we can easily choose various directions to have a box spline function with a desired order of smoothness. Naturally, they have been used to construct various wavelet functions. Mathematically box splines offer an elegant toolbox for constructing a class of multidimensional elements with flexible shape and support. In multivariate setting, box splines are often considered as a generalization of B-splines [1]. Theoretically, the computational complexity of a box spline is lower than that of an equivalent B-spline, since its support is more compact and its total polynomial degree is lower. To investigate this potential in practice, several attempts were made. Recurrence relation [1,2] is the most commonly used technique for evaluating box splines at an arbitrary position. There are many papers on multivariate spline wavelet theory, in particular, on orthogonal spline wavelets [3], compactly spline prewavelets [4][5][6], bivariate and trivariate compactly supported biorthogonal box spline wavelets [7,8], and multivariate compactly supported tight wavelet frames [9].
Wavelets in a Sobolev space and their properties were instigated by Bastin et al. [10,11], Dayong and Dengfeng [12], and Pathak [13]. Regular compactly supported wavelets and compactly supported wavelets of integer order in a Sobolev space by B-spline are given in [10,11]. Further, bivariate box splines in a Sobolev space were introduced in [14].
Inspired by the works mentioned, in this paper, we study nonseparable wavelets in a higher-dimensional Sobolev space by using a multivariate box spline. To the best of our knowledge, no previous studies of multivariate box spline wavelets exist in higherdimensional Sobolev spaces. This paper is organized as follows. In Sect. 2, we hereby present construction of wavelets and density conditions of MRA in a higher-dimensional Sobolev space. Also, we give necessary and sufficient conditions for the orthonormality of wavelets in H s (R d ). In Sect. 3, we construct nonseparable wavelets in a higher-dimensional Sobolev space by using a multivariate box spline.

Sobolev space H s (R d )
For any real number s, the Sobolev space H s (R d ) consists of tempered distributions in S (R d ) such that where · denotes the Euclidean norm in R d , and the corresponding inner product is where x, ξ is the inner product of two vectors x and ξ in R d .

Multiresolution analysis
To adapt classical theory of MRA over H s (R d ), we first derive an orthonormality and density condition. The main problem is that H s -norm is not dilation invariant. We also don't achieve orhtonormality at each level of dilation by a single scaling function. This lead us to a more general construction of MRA, where the scaling function depends on the level of dilation. Throughout this paper, the superscript j of a function ϕ (j) represents level j.

Proposition 2.1
If s is a real number, ϕ (j) ∈ H s (R d ), and j is an integer, then the distributions ϕ almost everywhere. It follows that we have the bound converges almost everywhere, belongs to L 1 loc (R d ), and is 2πZ d -periodic, that is, Proof Let us prove the first part with h ∈ C ∞ 0 (R d ). By the definition of P j we get Moreover, since h and ϕ (j) belong to H s (R d ), Hence, using the Parseval formula in , and the fact that h belongs to the Schwartz space S(R d ) (i.e., |ĥ(ξ )| ≤ C(1 + ξ 2 ) -α for any α > 0), we obtain that the sum of the other ones is bounded by which is valid for every ε > 0 and any seminorm. For any χ ∈ C ∞ 0 (R d ), we have By the same way, we can obtain a similar lower bound. To prove that the left-hand side converges to 0 as j converges to +∞, we first take ε sufficiently small. Then we choose χ approximating h and finally j large.
For the second part, we have to prove that, for every h ∈ C ∞ 0 (R d ), P j h converges to zero in H s (R d ) as j → -∞. We use the last expression of P j h s obtained previously. We first estimate the sum over q without the integral. By the Cauchy-Schwarz inequality and Proposition 2.1 we have We know that The last expression converges to zero as j converges to -∞.
Proof We know from Proposition 2.1 that if the system is orthonormal, then With the help of (2) and Theorem 2.4, we may define ϕ (j) bŷ

Theorem 2.5 The distributions ψ
and they are orthogonal to V j if (ξ + γ q π)| = 1, γ q ∈ E d , and k = l. Now we prove second part of the theorem: Now we define unitary matrix with the help of our theorems, j,k,p by (4) for p = 1, 2, . . . , 2 d -1 and j ∈ Z. Then W j = W j,1 ⊕ W j,2 ⊕ · · · ⊕ W j,2 d -1 with W j,p = span{2 jd/2 ψ (j) is an orthonormal basis for H s (R d ).

Multivariate box spline
Now we give an example of multivariate box splines in a Sobolev space. Using them, we construct a wavelet in H s (R d ).
Let D be the direction matrix of order d × d+1 i=1 m i , m i ∈ N 0 , ∀i, whose column vectors consist of (m 1 , m 2 , . . . , m d+1 ) copies of the following d + 1 column vectors: 1e -i k j ,ξ i k j , ξ m j , k j ∈ D, m j ∈ N 0 , ∀j.