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Multivariate box spline wavelets in higher-dimensional Sobolev spaces

Journal of Inequalities and Applications20182018:246

https://doi.org/10.1186/s13660-018-1839-z

  • Received: 3 April 2018
  • Accepted: 6 September 2018
  • Published:

Abstract

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 \(H^{s}(\mathbb {R}^{d})\). We construct nonseparable orthonormal wavelets in a higher-dimensional Sobolev space by using multivariate box spline.

Keywords

  • Wavelets
  • Box Splines
  • Multiresolution analysis
  • Sobolev space

1 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 [46], 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 higher-dimensional 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 \(H^{s}(\mathbb {R}^{d})\). In Sect. 3, we construct nonseparable wavelets in a higher-dimensional Sobolev space by using a multivariate box spline.

1.1 Sobolev space \(H^{s}(\mathbb{R}^{d})\)

For any real number s, the Sobolev space \(H^{s}(\mathbb {R}^{d})\) consists of tempered distributions in \(S'(\mathbb {R}^{d})\) such that
$$\Vert f \Vert _{s}^{2}:=\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{f}(\xi) \bigr\vert ^{2}\,d\xi, $$
where \(\|\cdot\|\) denotes the Euclidean norm in \(\mathbb {R}^{d}\), and the corresponding inner product is
$$\langle f, g \rangle_{s}:=\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\hat{f}(\xi)\overline{\hat{g}(\xi)}\,d\xi. $$
The Fourier transform of \(f\in L^{1}(\mathbb {R}^{d})\) is defined as
$$\hat{f}(\xi):= \int_{\mathbb {R}^{d}}e^{-i \langle x,\xi \rangle}f(x)\,dx, $$
where \(\langle x,\xi \rangle\) is the inner product of two vectors x and ξ in \(\mathbb {R}^{d}\).

2 Multiresolution analysis

To adapt classical theory of MRA over \(H^{s}(\mathbb {R}^{d})\), 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 \(\varphi^{(j)}\) represents level j.

Proposition 2.1

If s is a real number, \(\varphi^{(j)}\in H^{s}(\mathbb {R}^{d}) \), and j is an integer, then the distributions \(\varphi _{j,k}^{(j)}(x)=2^{{jd}/2}\varphi^{(j)}(2^{j}x-k), k\in\mathbb {Z}^{d}\), are orthonormal in \(H^{s}(\mathbb {R}^{d}) \) iff
$$\begin{aligned} \sum_{k\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2k\pi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{\varphi}^{(j)}(\xi+2k\pi) \bigr\vert ^{2}=1 \end{aligned}$$
(1)
almost everywhere. It follows that we have the bound
$$\bigl\vert \hat{\varphi}^{(j)} \bigl(2^{-j}\xi \bigr) \bigr\vert \leq \bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{-s/2}. $$

Proof

Since \(\varphi_{j,k}^{(j)}(t)\in H^{s}(\mathbb {R}^{d})\), the series
$$M(\xi)=\sum_{r\in\mathbb {Z}^{d}}{ \bigl\vert \hat{\varphi}^{(j)}(\xi+2\pi r) \bigr\vert }^{2}{ \bigl(1+2^{2j}{ \bigl\Vert (\xi+2\pi r) \bigr\Vert }^{2} \bigr)}^{s} $$
converges almost everywhere, belongs to \(\mathbb {L}_{\mathrm{ loc}}^{1}(\mathbb {R}^{d})\), and is \(2\pi\mathbb {Z}^{d}\)-periodic, that is, \(M(\xi )\in L^{1}(\mathbb {T}^{d})\), where \(\mathbb {T}^{d}=[0,2\pi]^{d}\) is the d-dimensional torus. Moreover, for every \(l\in\mathbb {Z}^{d}\), we have
$$\begin{aligned} &\int_{\mathbb {T}^{d}}M(\xi)e^{-i\langle\xi,(k-l)\rangle}\,d\xi \\ &\quad=\sum_{r\in\mathbb {Z}^{d}} \int_{\mathbb {T}^{d}}{ \bigl\vert \hat{\varphi}^{(j)}(\xi +2\pi r) \bigr\vert }^{2}{\bigl(1+2^{2j}{ \bigl\Vert (\xi+2\pi r) \bigr\Vert }^{2}\bigr)}^{s}e^{-i\langle\xi ,(k-l)\rangle}\,d\xi \\ &\quad= \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\varphi}^{(j)}(\nu) \bigr\vert }^{2}{\bigl(1+2^{2j}{ \bigl\Vert (\nu) \bigr\Vert }^{2}\bigr)}^{s}e^{-i\langle\nu,(k-l)\rangle}\,d\nu \\ &\quad=2^{-jd} \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\varphi}^{(j)} \bigl(2^{-j}u\bigr) \bigr\vert }^{2}{\bigl(1+{ \Vert u \Vert }^{2}\bigr)}^{s}e^{-i 2^{-j}\langle u,(k-l)\rangle}\,du \\ &\quad= \int_{\mathbb {R}^{d}}{\bigl(1+{ \Vert u \Vert }^{2} \bigr)}^{s}e^{-i 2^{-j}\langle u,k\rangle }2^{-jd/2}\hat{\varphi}^{(j)} \bigl(2^{-j}u\bigr) \overline{e^{-i 2^{-j}\langle u,l\rangle}2^{-jd/2}\hat{\varphi}^{(j)}\bigl(2^{-j}u\bigr)}\,du \\ &\quad= \int_{\mathbb {R}^{d}}{\bigl(1+{ \Vert u \Vert }^{2} \bigr)}^{s}\mathcal{F} \bigl[ \varphi _{j,k}^{(j)}(t) \bigr](u)\mathcal{F} \bigl[ \varphi_{j,l}^{(j)}(t) \bigr](u)\,du \\ &\quad={(2\pi)^{d}}\bigl\langle \varphi_{j,k}^{(j)}(t), \varphi _{j,l}^{(j)}(t)\bigr\rangle _{s}. \end{aligned}$$
Since \(\{1/{(2\pi)^{d}}e^{-i\langle\xi,(k-l)\rangle}:k,l\in \mathbb {Z}^{d}\}\) is an orthonormal basis for \(L^{2}(\mathbb {T}^{d})\), we have
$$\begin{aligned} \frac{1}{{(2\pi)}^{d}} \int_{\mathbb {T}^{d}}M(\xi)e^{-i\langle\xi ,(k-l)\rangle}\,d\xi=\bigl\langle \varphi_{j,k}^{(j)}(t), \varphi _{j,l}^{(j)}(t) \bigr\rangle _{s}=\delta_{k,l} \end{aligned}$$
if \(M(\xi)=1\).
From (1) we get
$$\bigl(1+2^{2j} \Vert \xi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{\varphi}^{(j)}(\xi) \bigr\vert ^{2}\leq1, $$
which implies
$$\bigl\vert \hat{\varphi}^{(j)}(\xi) \bigr\vert \leq \bigl(1+2^{2j} \Vert \xi \Vert ^{2} \bigr)^{-s/2}. $$
 □

Proposition 2.2

Let \(\varphi^{(j)}, j\in\mathbb {Z}\), be a sequence of elements of \(H^{s}(\mathbb {R}^{d})\) such that, for every j, the distributions \(\varphi _{j,k}^{(j)}(x)=2^{{jd}/2}\varphi^{(j)}(2^{j}x-k), k\in\mathbb {Z}^{d}\), are orthonormal in \(H^{s}(\mathbb {R}^{d})\). If \(P_{j}\) is the orthogonal projection from \(H^{s}(\mathbb {R}^{d})\) onto \(V_{j}:=\overline{\operatorname{span}}\{ \varphi_{j,k}^{(j)}: k\in\mathbb {Z}^{d}\}\), then, for every \(h\in H^{s}(\mathbb {R}^{d})\), we have
$$\lim_{j\rightarrow{+\infty}} \biggl( \Vert P_{j}h \Vert _{s}^{2}-\frac{1}{(2\pi)^{d}} \int _{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr) \bigr\vert ^{2}\,d\xi \biggr)=0. $$
Moreover, if there are \(A,\alpha>0\) such that
$$\int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert \bigr)^{\alpha} \bigl\vert \hat{\varphi}^{(j)}(\xi) \bigr\vert ^{2}\,d\xi \leq A $$
for every \(j\leq0\), then \(\bigcap_{j={-\infty}}^{j=\infty}V_{j}=\{0\}^{d}\).

Proof

Let us prove the first part with \(h\in C_{0}^{\infty}(\mathbb {R}^{d})\). By the definition of \(P_{j}\) we get
$$\Vert P_{j}h \Vert _{s}^{2}=\sum _{k\in{\mathbb {Z}^{d}}} \bigl\vert {\bigl\langle h, \varphi _{j,k}^{(j)}\bigr\rangle }_{s} \bigr\vert ^{2}=\frac{2^{-jd}}{(2\pi)^{2d}}\sum_{k\in\mathbb {Z}^{d}} \biggl\vert \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\hat{h}(\xi)\overline{\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr)}e^{i2^{-j}\langle k,\xi\rangle}\,d\xi \biggr\vert ^{2}. $$
Moreover, since h and \(\varphi^{(j)}\) belong to \(H^{s}(\mathbb {R}^{d})\),
$$\begin{aligned} & \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\hat{h}(\xi)\overline{\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr)}e^{i2^{-j}\langle k,\xi\rangle}\,d\xi \\ &\quad = \int_{]0,{2^{j}}2\pi[^{d}}e^{i2^{-j}\langle k,\xi\rangle}\sum_{p\in \mathbb {Z}^{d}} \bigl(1+ \bigl\Vert \xi+2^{j}2\pi p \bigr\Vert ^{2} \bigr)^{s}\hat{h}\bigl(\xi+2^{j}2\pi p\bigr)\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi+2\pi p\bigr)}\,d\xi. \end{aligned}$$
Hence, using the Parseval formula in \(L^{2}({ ]0,2^{j}2\pi [}^{d})\), we get
$$\begin{aligned} & \Vert P_{j}h \Vert _{s}^{2} \\ & \quad=\frac{1}{(2\pi)^{d}} \int_{]0,{2^{j}}2\pi[^{d}} \biggl\vert \sum_{p\in\mathbb {Z}^{d}} \bigl(1+ \bigl\Vert \xi+2^{j}2\pi p \bigr\Vert ^{2} \bigr)^{s}\hat{h}\bigl(\xi+2^{j}2\pi p\bigr)\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi+2\pi p\bigr)} \biggr\vert ^{2}\,d\xi \\ &\quad =\frac{1}{(2\pi)^{d}}\sum_{q\in\mathbb {Z}^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\bigl(1+ \bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s}\hat{h}(\xi)\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr)} \\ &\qquad{} \times\overline{\hat{h}\bigl(\xi+2^{j}2\pi q\bigr)}\hat{\varphi}^{(j)}\bigl(2^{-j}\xi +2\pi q\bigr) \,d\xi \\ & \quad=\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi ) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr) \bigr\vert ^{2} \\ &\qquad{} +\frac{1}{(2\pi)^{d}}\sum_{q\in\mathbb {Z}^{d}\backslash\{0\}^{d}} \int _{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\bigl(1+ \bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s}\hat{h}(\xi )\overline{\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr)} \\ & \qquad{}\times\overline{\hat{h}\bigl(\xi+2^{j}2\pi q\bigr)}\hat{\varphi}^{(j)}\bigl(2^{-j}\xi +2\pi q\bigr) \,d\xi. \end{aligned}$$
The term associated with \(q=\{0\}^{d}, \{0\}^{d}=(0,0,\ldots,0)\in\mathbb {Z}^{d} \) is used as an approximation for \(\|P_{j}h\|_{s}^{2}\). Using Proposition 2.1, the inequality \(|\varphi^{(j)}(2^{-j}\xi)|\leq(1+\|\xi\|^{2})^{-s/2}\), and the fact that ĥ belongs to the Schwartz space \(S(\mathbb {R}^{d})\) (i.e., \(|\hat{h}(\xi)|\leq C(1+\|\xi\|^{2})^{-\alpha}\) for any \(\alpha>0\)), we obtain that the sum of the other ones is bounded by
$$\begin{aligned} &\sum_{q\in\mathbb {Z}^{d}\backslash\{0\}^{d}} \int_{\mathbb {R}^{d} }\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s/2}\bigl(1+ \bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s/2} \bigl\vert \hat{h}(\xi)\overline{\hat{h} \bigl(\xi +2^{j}2\pi q\bigr)} \bigr\vert \,d\xi \\ &\quad\leq C\sum_{q\in\mathbb {Z}^{d}\backslash\{0\}^{d}}\frac{1}{(1+ \Vert 2^{j}2\pi q \Vert ^{2})^{2}} \int_{\mathbb {R}^{d} }\frac{1}{(1+ \Vert \xi \Vert ^{2})^{2}}\,d\xi \\ &\quad\leq C\sum_{q\in\mathbb {Z}^{d}\backslash\{0\}^{d}}\frac{1}{( \Vert 2^{j}2\pi q \Vert ^{2})^{2}} \int_{\mathbb {R}^{d} }\frac{1}{(1+ \Vert \xi \Vert ^{2})^{2}}\,d\xi \\ &\quad\leq C2^{-4(j+1)} \biggl(\sum_{q\in\mathbb {Z}^{d}\backslash\{0\}^{d}} \frac {1}{ \vert q \vert ^{2}} \biggr) \int_{\mathbb {R}^{d} }\frac{1}{(1+ \Vert \xi \Vert ^{2})^{2}}\,d\xi, \end{aligned}$$
where \(|q|= (\sum_{r=1}^{d}|q_{r}|^{2} )^{1/2}, q=(q_{1},q_{2},\ldots ,q_{d})\in\mathbb {Z}^{d}\). This expression converges to 0 as \(j\rightarrow{+\infty}\).
Now let \(h\in H^{s}(\mathbb {R}^{d})\). Recall the inequality
$$\Vert f+g \Vert ^{2}\leq(1+\varepsilon) \Vert f \Vert ^{2}+ \biggl(1+\frac{1}{\varepsilon} \biggr) \Vert g \Vert ^{2}, $$
which is valid for every \(\varepsilon>0\) and any seminorm. For any \(\chi \in C_{0}^{\infty}(\mathbb {R}^{d})\), we have
$$\begin{aligned} &\Vert P_{j}h \Vert _{s}^{2}-\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr) \bigr\vert ^{2}\,d\xi \\ &\quad\leq(1+\varepsilon) \Vert P_{j}\chi \Vert _{s}^{2}+ \biggl(1+\frac{1}{\varepsilon} \biggr) \bigl\Vert P_{j}(h-\chi) \bigr\Vert ^{2} \\ &\qquad{} -\frac{1}{(2\pi)^{d}(1+\varepsilon)} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{\chi}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr) \bigr\vert ^{2}\,d\xi \\ &\qquad{} +\frac{1}{(2\pi)^{d}(\varepsilon)} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi)-\hat{\chi}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi \bigr) \bigr\vert ^{2}\,d\xi \\ &\quad\leq(1+\varepsilon) \biggl( \Vert P_{j}\chi \Vert _{s}^{2}-\frac{1}{(2\pi)^{d}} \int _{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{\chi}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j}\xi\bigr) \bigr\vert ^{2}\,d\xi \biggr) \\ &\qquad{} + \biggl(1+\frac{2}{\varepsilon} \biggr) \Vert h-\chi \Vert _{s}^{2}+ \biggl(1+\varepsilon-\frac{1}{(1+\varepsilon)} \biggr) \Vert \chi \Vert _{s}^{2}. \end{aligned}$$
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 \(h\in C_{0}^{\infty}(\mathbb {R}^{d})\), \(P_{j}h\) converges to zero in \(H^{s}(\mathbb {R}^{d})\) as \(j\rightarrow{-\infty}\). 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
$$\begin{aligned} & \sum_{q\in\mathbb {Z}^{d}}\bigl(1+\bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s} \bigl\vert \hat{h}\bigl(\xi+2^{j}2\pi q\bigr)\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi+2\pi q\bigr)} \bigr\vert \\ &\quad\leq \biggl(\sum_{q\in\mathbb {Z}^{d}}\bigl(1+ \bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s} \bigl\vert \hat{h}\bigl(\xi +2^{j}2\pi q\bigr) \bigr\vert ^{2} \biggr)^{1/2}. \end{aligned}$$
We know that
$$\begin{aligned} \sum_{q\in\mathbb {Z}^{d}}\bigl(1+ \bigl\Vert \xi+2^{j}2\pi q \bigr\Vert ^{2}\bigr)^{s} \bigl\vert \hat{h}\bigl(\xi+2^{j}2\pi q\bigr) \bigr\vert ^{2} \bigl(2^{j+1}\pi\bigr)^{d}\rightarrow \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{h}(\xi) \bigr\vert ^{2}\,d\xi \end{aligned}$$
if \(j\leq-1\). It follows that
$$\begin{aligned} \Vert P_{j}h \Vert _{s}^{2}\leq{}& \frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{h}(\xi)\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr) \bigr\vert 2^{-jd}C \Vert h \Vert _{s}\,d\xi \\ \leq&{}\frac{2^{-jd}C \Vert h \Vert _{s}}{(2\pi)^{d}} \biggl( \int_{\mathbb {R}^{d}}\bigl(1+2^{-j} \Vert \xi \Vert \bigr)^{\alpha} \bigl\vert \hat{\varphi}^{(j)}\bigl(2^{-j} \xi\bigr) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} \\ &{}\times \biggl( \int_{\mathbb {R}^{d}}\bigl(1+2^{-j} \Vert \xi \Vert \bigr)^{-\alpha}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2} \\ \leq{}&\frac{C\sqrt{A} \Vert h \Vert _{s}}{(2\pi)^{d}} \biggl( \int_{\mathbb {R}^{d}}\bigl(1+2^{-j} \Vert \xi \Vert \bigr)^{-\alpha}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{2s} \bigl\vert \hat{h}(\xi) \bigr\vert ^{2}\,d\xi \biggr)^{1/2}. \end{aligned}$$
The last expression converges to zero as j converges to −∞. □

Now we construct wavelets in \(H^{s}(\mathbb {R}^{d})\) with the help of previous propositions.

By definition, \(V_{j}\) is the set of all \(f\in H^{s}(\mathbb {R}^{d})\) such that
$$\hat{f}(\xi)=m \bigl(2^{-j}\xi \bigr)\hat{\varphi}^{(j)} \bigl(2^{-j}\xi \bigr), $$
where \(m\in L_{\mathrm{loc}}^{2} (\mathbb {R}^{d})\) is \(2\pi\mathbb {Z}^{d}\)-periodic. This follows immediately from the fact that the Fourier transform of \(2^{{jd}/2}\varphi ^{(j)}(2^{j}x-k)\) is
$$2^{{-jd}/2}e^{-i2^{-j}\langle k,\xi\rangle}\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr). $$
We have \(V_{j}\subset V_{j+1}\) for every \(j\in\mathbb {Z}^{d}\) iff there are \(2\pi\mathbb {Z}^{d}\)-periodic functions \(m_{0}^{(j)}\in L_{\mathrm{loc}}^{2} (\mathbb {R}^{d})\) such that the following scale relation holds:
$$\begin{aligned} \hat{\varphi}^{(j)}(2\xi)=m_{0}^{(j+1)} ( \xi )\hat{\varphi}^{(j+1)} (\xi ); \end{aligned}$$
(2)
moreover, \(\varphi^{(j)}\) and \(\varphi^{(j+1)}\) satisfy the hypothesis of Proposition 2.1. Now, using our theorems and propositions, we develop the definition of MRA in \(H^{s}(\mathbb {R}^{d})\).

Definition 2.3

Let s be a real number. The MRA of \(H^{s}(\mathbb {R}^{d})\) is a sequence \(V_{j}, j\in\mathbb {Z}\), of closed linear subspaces of \(H^{s}(\mathbb {R}^{d})\) such that
  1. (a)

    \(V_{j}\subset V_{j+1}\),

     
  2. (b)

    \(\bigcup_{j=-\infty}^{j=\infty}V_{j}= H^{s}(\mathbb {R}^{d})\),

     
  3. (c)

    \(\bigcap_{j=-\infty}^{j=\infty}V_{j}= \{0\}^{d}\), and

     
  4. (d)

    for every j, there is a function \(\varphi^{(j)}\) such that the distributions \(2^{{jd}/2}\varphi^{(j)}(2^{j}x-k)\), \(k\in\mathbb {Z}^{d}\), form an orthonormal basis for \(V_{j}\).

     

Before giving a necessary condition for the orthonormality, we define \(E_{d}:=\{0,1\}^{d}\) as the unit cube in the d-dimensional Euclidean space.

Theorem 2.4

If \(\varphi^{(j)}\) and \(\varphi^{(j+1)}\) satisfy the hypothesis of Proposition 2.1, then
$$\sum_{q=0}^{2^{d}-1} \bigl\vert m_{0}^{(j+1)}(\xi+\gamma_{q}\pi) \bigr\vert =1,\quad \gamma_{q}\in E_{d}, q=1,2,\ldots, 2^{d}-1. $$

Proof

We know from Proposition 2.1 that if the system is orthonormal, then
$$\begin{aligned} \delta_{k,l}={}&\bigl\langle \varphi_{j,k}^{(j)}, \varphi_{j,l}^{(j)} \bigr\rangle _{s} \\ ={}&\frac{2^{-jd}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr) \bigr\vert }^{2}e^{-i2^{-j} \langle\xi,(k-l) \rangle}{ \bigl(1+{ \Vert \xi \Vert }^{2}\bigr)}^{s}\,d\xi \\ ={}&\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\varphi}^{(j)}(u) \bigr\vert }^{2}e^{-i \langle u,(k-l) \rangle}{\bigl(1+2^{2j}{ \Vert u \Vert }^{2}\bigr)}^{s}\,du \\ ={}&\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert m_{0}^{(j+1)}(u/2) \bigr\vert }^{2}{ \bigl\vert \hat{\varphi}^{(j+1)}(u/2) \bigr\vert }^{2}e^{-i \langle u,(k-l) \rangle}{\bigl(1+2^{2j}{ \Vert u \Vert }^{2}\bigr)}^{s}\,du \\ ={}&\frac{2^{d}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert m_{0}^{(j+1)}(\nu) \bigr\vert }^{2}{ \bigl\vert \hat{\varphi}^{(j+1)}(\nu) \bigr\vert }^{2}e^{-i 2 \langle\nu,(k-l) \rangle}{\bigl(1+2^{2(j+1)}{ \Vert \nu \Vert }^{2}\bigr)}^{s}d\nu \\ ={}&\frac{1}{(\pi)^{d}} \int_{\mathbb {T}^{d}}{ \bigl\vert m_{0}^{(j+1)}(\nu) \bigr\vert }^{2}\sum_{r\in\mathbb {Z}^{d}}{ \bigl\vert \hat{\varphi}^{(j+1)}(\nu+2\pi r) \bigr\vert }^{2} \\ &{} \times{\bigl(1+2^{2(j+1)}{ \Vert {\nu+2\pi r} \Vert }^{2} \bigr)}^{s}e^{-i 2 \langle\nu,(k-l) \rangle}\,d\nu \\ ={}&\frac{1}{(\pi)^{d}} \int_{\mathbb {T}^{d}}{ \bigl\vert m_{0}^{(j+1)}(\nu) \bigr\vert }^{2}e^{-i 2 \langle\nu,(k-l) \rangle}\,d\nu \\ ={}&\frac{1}{(\pi)^{d}} \int_{[0,\pi)^{d}}\sum_{q=0}^{2^{d}-1} \bigl\vert m_{0}^{(j+1)}(\nu +\gamma_{q}\pi) \bigr\vert ^{2}e^{-i 2 \langle\nu,(k-l) \rangle}\,d\nu, \end{aligned}$$
which implies that
$$\sum_{q=0}^{2^{d}-1} \bigl\vert m_{0}^{(j+1)}(\xi+\gamma_{q}\pi) \bigr\vert ^{2}=1,\quad \gamma_{q}\in E_{d}, $$
if \(k=l\). □
With the help of (2) and Theorem 2.4, we may define \(\varphi^{(j)}\) by
$$\begin{aligned} \hat{\varphi}^{(j)}(\xi)&= m_{0}^{(j+1)}(\xi/2)\hat{\varphi}^{(j+1)}(\xi /2) \\ &=\prod_{t=1}^{J} m_{0}^{(j+t)} \bigl(\xi/2^{t}\bigr)\hat{\varphi}^{(j+J)}\bigl(\xi/2^{J} \bigr) \\ &=\cdots=\frac{1}{(1+ \Vert \xi \Vert ^{2})^{s/2}}\prod_{t=1}^{+\infty} m_{0}^{(j+t)}\bigl(\xi/2^{t}\bigr) \end{aligned}$$
(3)
for \(j\in\mathbb {Z}\). For \(V_{j}\), let \(W_{j}\) be the orthogonal complement of \(V_{j}\) in \(V_{j+1}\). We have
$$\begin{aligned} \psi^{(j)}_{j,k,p}:=2^{jd/2} \psi^{(j)}_{p}\bigl(2^{j}x-k\bigr) \in V_{j+1} \end{aligned}$$
(4)
if there are \(2\pi\mathbb {Z}^{d}\)-periodic functions \(m^{(j)}_{1},m^{(j)}_{2},\ldots,m^{(j)}_{2^{d}-1}\in L^{2}_{\mathrm{loc}}(\mathbb {R}^{d})\) such that
$$\hat{\psi}^{(j)}_{p}\bigl(2^{-j}\xi \bigr)=m_{p}^{(j+1)}\bigl(2^{-j-1}\xi\bigr)\hat{\varphi}^{(j+1)}\bigl(2^{-j-1}\xi\bigr),\quad p=1,2,\ldots, 2^{d}-1. $$

Theorem 2.5

The distributions \(\psi_{j,k,p}^{(j)}(x)=2^{jd/2}\psi_{p}^{(j)}(2^{j}x-k)\) are orthonormal if
$$\sum_{q=0}^{2^{d}-1} \bigl\vert m_{p}^{(j+1 )}(\xi+\gamma_{q}\pi) \bigr\vert =1,\quad \gamma_{q}\in E_{d}, \forall p=1,2,\ldots, 2^{d}-1, $$
and they are orthogonal to \(V_{j}\) if
$$ \sum_{q=0}^{2^{d}-1}m_{p}^{(j+1)}( \xi+\gamma_{q}\pi)\overline{m_{0}^{(j+1)}(\xi + \gamma_{q}\pi)}=0,\quad \gamma_{q}\in E_{d}, \forall p=1,2,\ldots, 2^{d}-1. $$
(5)

Proof

$$\begin{aligned} &\bigl\langle \psi_{j,k,p}^{(j)},\psi_{j,l,p}^{(j)} \bigr\rangle _{s} \\ &\quad=\frac{2^{-jd}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\psi}_{p}^{(j)} \bigl(2^{-j}\xi\bigr) \bigr\vert }^{2}e^{-i2^{-j} \langle\xi,(k-l) \rangle }{ \bigl(1+{ \Vert \xi \Vert }^{2}\bigr)}^{s}\,d\xi \\ &\quad=\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert \hat{\psi}_{p}^{(j)}(u) \bigr\vert }^{2}e^{-i \langle u,(k-l) \rangle}{\bigl(1+2^{2j}{ \Vert u \Vert }^{2}\bigr)}^{s}\,du \\ &\quad=\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert m_{p}^{(j+1)}(u/2) \bigr\vert }^{2}{ \bigl\vert \hat{\varphi}^{(j+1)}(u/2) \bigr\vert }^{2}e^{-i \langle u,(k-l) \rangle }{\bigl(1+2^{2j}{ \Vert u \Vert }^{2}\bigr)}^{s}\,du \\ &\quad=\frac{2^{d}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{ \bigl\vert m_{p}^{(j+1)}(\nu) \bigr\vert }^{2}{ \bigl\vert \hat{\varphi}^{(j+1)}(\nu) \bigr\vert }^{2}e^{-i2 \langle\nu,(k-l) \rangle}{\bigl(1+2^{2(j+1)}{ \Vert \nu \Vert }^{2}\bigr)}^{s}d\nu \\ &\quad=\frac{1}{(\pi)^{d}} \int_{\mathbb {T}^{d}}{ \bigl\vert m_{p}^{(j+1)}(\nu) \bigr\vert }^{2}\sum_{r\in \mathbb {Z}^{d}}{ \bigl\vert \hat{\varphi}^{(j+1)}(\nu+2\pi r) \bigr\vert }^{2} \\ &\qquad{} \times{\bigl(1+2^{2(j+1)}{ \Vert {\nu+2\pi r} \Vert }^{2} \bigr)}^{s}e^{-i 2 \langle\nu ,(k-l) \rangle}\,d\nu \\ &\quad=\frac{1}{(\pi)^{d}} \int_{\mathbb {T}^{d}}{ \bigl\vert m_{p}^{(j+1)}(\nu) \bigr\vert }^{2}e^{-i 2 \langle\nu,(k-l) \rangle}\,d\nu \\ &\quad=\frac{1}{(\pi)^{d}} \int_{[0,\pi)^{d}}\sum_{q=1}^{2^{d}-1} { \bigl\vert m_{p}^{(j+1)}(\nu+\gamma_{q}\pi) \bigr\vert }^{2}e^{-i 2 \langle\nu,(k-l) \rangle}\,d\nu \\ &\quad=\frac{1}{(\pi)^{d}} \int_{[0,\pi)^{d}}e^{-i2 \langle\nu,(k-l) \rangle}\,d\nu. \end{aligned}$$
Therefore
$$\begin{aligned} \bigl\langle \psi_{j,k,p}^{(j)},&\psi_{j,l,p}^{(j)} \bigr\rangle _{s}=1 \end{aligned}$$
if \(\sum_{q=0}^{2^{d}-1}|m_{p}^{(j+1 )}(\xi+\gamma_{q}\pi)|=1, \gamma_{q}\in E_{d}\), and \(k=l\).
Now we prove second part of the theorem:
$$\begin{aligned} 0={}&\bigl\langle \psi_{j,k,p}^{(j)},\varphi_{j,l,p}^{(j)} \bigr\rangle _{s} \\ ={}&\frac{2^{-jd}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{\bigl(1+{ \Vert \xi \Vert }^{2} \bigr)}^{s}{\hat{\psi}_{p}^{(j)}\bigl(2^{-j} \xi\bigr)}\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi \bigr)}e^{-i2^{-j} \langle\xi,(k-l) \rangle}\,d\xi \\ ={}&\frac{1}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{\bigl(1+2^{2j}{ \Vert \xi \Vert }^{2}\bigr)}^{s}{ m_{p}^{(j+1)}(\xi/2)} \overline{m_{0}^{(j+1)}(\xi/2)} { \bigl\vert \hat{\varphi}^{(j+1)}(\xi/2) \bigr\vert ^{2}}e^{-i \langle\xi,(k-l) \rangle }\,d\xi \\ ={}&\frac{2^{d}}{(2\pi)^{d}} \int_{\mathbb {R}^{d}}{\bigl(1+2^{2(j+1)}{ \Vert \xi \Vert }^{2}\bigr)}^{s}{ m_{p}^{(j+1)}(\xi)} \overline{m_{0}^{(j+1)}(\xi)} { \bigl\vert \hat{\varphi}^{(j+1)}(\xi) \bigr\vert ^{2}}e^{-i2 \langle\xi,(k-l) \rangle}\,d\xi \\ ={}&\frac{1}{(\pi)^{d}} \int_{\mathbb {T}^{d}}m_{p}^{(j+1)}(\xi) \overline{m_{0}^{(j+1)}(\xi)} \\ &{} \times\sum_{r\in\mathbb {Z}^{d}}{\bigl(1+2^{2(j+1)}{ \Vert \xi+2r\pi \Vert }^{2}\bigr)}^{s}{ \bigl\vert \hat{\varphi}^{(j+1)}(\xi+2r\pi) \bigr\vert ^{2}}e^{-i2 \langle\xi,(k-l) \rangle }\,d \xi \\ ={}&\frac{1}{(\pi)^{d}} \int_{[0,\pi)^{d}}\sum_{q=1}^{2^{d}-1} m_{p}^{(j+1)}(\xi+\gamma_{q}\pi) \overline{m_{0}^{(j+1)}(\xi+\gamma_{q}\pi )}e^{-i 2 \langle\xi,(k-l) \rangle}\,d\xi, \end{aligned}$$
which implies
$$\sum_{q=0}^{2^{d}-1}m_{p}^{(j+1)}( \xi+\gamma_{q}\pi)\overline{m_{0}^{(j+1)}(\xi + \gamma_{q}\pi)}=0,\quad \gamma_{q}\in E_{d}, \forall p=1,2,\ldots, 2^{d}-1. $$
 □
Now we define unitary matrix with the help of our theorems,
$$ \begin{bmatrix} m_{0}^{(j)}(\xi+\gamma_{0}\pi) & m_{0}^{(j)}(\xi+\gamma_{1}\pi) & \cdots& m_{0}^{(j)}(\xi+\gamma_{2^{d}-1}\pi) \\ m_{1}^{(j)}(\xi+\gamma_{0}\pi) & m_{1}^{(j)}(\xi+\gamma_{1}\pi) & \cdots& m_{1}^{(j)}(\xi+\gamma_{2^{d}-1}\pi) \\ \ddots& \ddots& \ddots& \ddots&\\ m_{2^{d}-1}^{(j)}(\xi+\gamma_{0}\pi) & m_{2^{d}-1}^{(j)}(\xi+\gamma_{1}\pi) & \cdots& m_{2^{d}-1}^{(j)}(\xi+\gamma_{2^{d}-1}\pi) \end{bmatrix} . $$
(6)

Theorem 2.6

Suppose that the scaling function \(\varphi^{(j)}, j\in \mathbb {Z}\), generate an MRA \(\{V_{j}\}\) of \(H^{s}(\mathbb {R}^{d})\) and \(\varphi _{j,k}^{(j)}, k\in\mathbb {Z}^{d}\), form an orthonormal basis for \(V_{j}, j\in\mathbb {Z}\). Suppose that, for each \(j\in\mathbb {Z}, m^{(j)}_{p}\) for \(p=1,2,\ldots,2^{d}-1\) are such that matrix (6) is unitary. Define \(\psi_{j,k,p}^{(j)}\) by (4) for \(p=1,2,\ldots,2^{d}-1\) and \(j\in\mathbb {Z}\). Then \(W_{j}=W_{j,1}\oplus W_{j,2}\oplus\cdots\oplus W_{j,2^{d}-1}\) with \(W_{j,p}=\overline{\operatorname{span}}\{2^{jd/2}\psi ^{(j)}_{p}(2^{j}x-k): k\in\mathbb {Z}\}, p=1,2,\ldots,2^{d}-1\), is perpendicular to \({V_{j}}\) in \(V_{j+1}\), and \(V_{j+1}=V_{j}\oplus W_{j}\). Therefore
$$2^{jd/2}\psi^{(j)}_{p}\bigl(2^{j}x-k \bigr), \quad k\in\mathbb {Z}, p=1,2,\ldots,2^{d}-1, $$
is an orthonormal basis for \(H^{s}(\mathbb {R}^{d})\).

Proof

First,we show that \(\psi_{j,k,p}^{(j)}\bot V_{j}\), for all \(k\in\mathbb {Z}^{d}\) and \(p=1,2,\dots, 2^{d}-1\). Indeed,
$$\begin{aligned} & (2\pi)^{d}\bigl\langle \psi_{j,k,p}^{(j)}(x), \varphi_{j,k}^{(j)}(x)\bigr\rangle _{s} \\ &\quad=(2\pi)^{d}\bigl\langle 2^{jd/2}\psi_{p}^{(j)} \bigl(2^{j}x-k_{1}\bigr),2^{jd/2}\varphi _{j,k}^{(j)}\bigl(2^{j}x-k_{2}\bigr)\bigr\rangle _{s} \\ &\quad= \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\mathcal {F}\bigl(2^{jd/2}\psi _{p}^{(j)} \bigl(2^{j}\xi-k_{1}\bigr)\bigr)\overline{\mathcal {F} \bigl(2^{jd/2}\varphi _{j,k}^{(j)}\bigl(2^{j} \xi-k_{2}\bigr)\bigr)}\,d\xi \\ &\quad=2^{-jd} \int_{\mathbb {R}^{d}}\bigl(1+ \Vert \xi \Vert ^{2} \bigr)^{s}\hat{\psi}_{p}^{(j)}\bigl(2^{-j} \xi \bigr)\overline{\hat{\varphi}^{(j)}\bigl(2^{-j}\xi \bigr)}e^{2^{-j}\langle\xi ,(k_{2}-k_{1})\rangle}\,d\xi \\ &\quad= \int_{\mathbb {R}^{d}}\bigl(1+2^{2j} \Vert \xi \Vert ^{2}\bigr)^{s}m_{p}^{(j+1)}(\xi/2)\hat{\varphi}^{(j+1)}(\xi/2) \\ &\qquad{} \times\overline{m_{0}^{(j+1)}(\xi/2)\hat{\varphi}^{(j+1)}(\xi /2)}e^{2^{-j}\langle\xi,(k_{2}-k_{1})\rangle}\,d\xi \\ &\quad= \int_{\mathbb {T}^{d}}\sum_{l\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2\pi l \Vert ^{2} \bigr)^{s}m_{p}^{(j+1)}(\xi/2+\pi l)\hat{\varphi}^{(j+1)}(\xi/2+\pi l) \\ & \qquad{}\times\overline{m_{0}^{(j+1)}(\xi/2+\pi l)\hat{\varphi}^{(j+1)}(\xi/2+\pi l)}e^{2^{-j}\langle\xi,(k_{2}-k_{1})\rangle}\,d\xi \\ &\quad = \int_{\mathbb {T}^{d}} \Biggl[\sum_{q=0}^{2^{d}-1}m_{p}^{(j+1)}( \xi/2+\gamma _{q}\pi)\overline{m_{0}^{(j+1)}(\xi/2+ \gamma_{q}\pi)} \Biggr]e^{2^{-j}\langle\xi,(k_{2}-k_{1})\rangle}\,d\xi \end{aligned}$$
by Proposition 2.1. This expression is equal to zero because matrix (6) is unitary. Similarly, we can show that \(W_{j,p_{1}}\bot W_{j,p_{2}}\) for all \(p_{1},p_{2}\in\{1,2,\dots,2^{d}-1\} \).
We know show that \(V_{j+1}=V_{j}\oplus W_{j,1}\oplus W_{j,2}\oplus\cdots \oplus W_{j,2^{d}-1}\) for any \(f\in V_{j+1}\). We write
$$\hat{f}(\xi)=B\bigl(2^{-j-1}\xi\bigr)\hat{\varphi}^{(j+1)} \bigl(2^{-j-1}\xi\bigr). $$
We will demonstrate that there exist \(2\pi\mathbb {Z}^{d}\)-periodic functions \(G(2^{-j}\xi)\) and \(H_{p}(2^{-j}\xi)\) such that
$$\hat{f}(\xi)=G\bigl(2^{-j}\xi\bigr)\hat{\varphi}^{(j)} \bigl(2^{-j}\xi\bigr)+\sum_{p=1}^{2^{d}-1}H_{p} \bigl(2^{-j}\xi\bigr)\hat{\psi}_{p}^{(j)} \bigl(2^{-j}\xi\bigr). $$
Now, we have
$$\begin{aligned} B(\xi/2)\hat{\varphi}^{(j+1)}(\xi/2)&=G(\xi)\hat{\varphi}^{(j)}( \xi)+\sum_{p=1}^{2^{d}-1}H_{p}(\xi) \hat{\psi}_{p}^{(j)}(\xi) \\ &=G(\xi)m_{0}^{(j+1)}(\xi/2)\hat{\varphi}^{(j+1)}(\xi/2)+ \sum_{p=1}^{2^{d}-1}H_{p}( \xi)m_{p}^{(j+1)}(\xi/2)\hat{\varphi}^{(j+1)}(\xi/2). \end{aligned}$$
It follows that
$$B(\xi/2)=G(\xi)m_{0}^{(j+1)}(\xi/2)+\sum _{p=1}^{2^{d}-1}H_{p}(\xi )m_{p}^{(j+1)}( \xi/2). $$
By the periodicity (\(2\pi\mathbb {Z}^{d}\)-periodic) of G and \(H_{p}\) we have
$$B(\xi/2+\gamma_{q}\pi)=G(\xi)m_{0}^{(j+1)}(\xi/2+ \gamma_{q}\pi)+\sum_{p=1}^{2^{d}-1}H_{p}( \xi)m_{p}^{(j+1)}(\xi/2+\gamma_{q}\pi) $$
for \(q=0,1,\dots,2^{d}-1\). This completes proof. □

3 Multivariate box spline

Now we give an example of multivariate box splines in a Sobolev space. Using them, we construct a wavelet in \(H^{s}(\mathbb {R}^{d})\).

Let D be the direction matrix of order \(d\times\sum^{d+1}_{i=1}m_{i}, m_{i}\in\mathbb {N}_{0}, \forall i\), whose column vectors consist of \((m_{1},m_{2},\dots,m_{d+1})\) copies of the following \(d+1\) column vectors: \((1,0,\dots,0)^{T}, (0,1,0,\dots,0)^{T},\dots,(0,0,\dots,1)^{T}\), and \((1,1,\dots,1)^{T}\).

Fix \(s\geq0\) and the natural numbers \((m_{1},m_{2},\dots,m_{d+1})\) such that
$$\bigl\{ m[D]:=\min\{m_{i}+m_{j}: i\neq j \text{ for all } i,j=1,2,\dots ,d+1 \} \bigr\} +\frac{1}{2}>s. $$
Let \(M_{m_{1},m_{2},\dots,m_{d+1}}\) be a multivariate box spline function defined in terms of the Fourier transform by
$$\widehat{M}_{m_{1},m_{2},\dots,m_{d+1}}(\xi)=\prod_{j=1}^{d+1} \biggl(\frac {1-e^{-i\langle k_{j},\xi\rangle}}{i\langle k_{j},\xi\rangle} \biggr)^{m_{j}},\quad k_{j}\in D, m_{j}\in\mathbb {N}_{0}, \forall j. $$
The multivariate box spline \(M_{m_{1},m_{2},\dots,m_{d+1}}\) belongs to \(C^{m[D]-1}\), where \(m[D]+1\) is the minimum number of columns that can be discarded from D to obtain a matrix of \(\operatorname{rank}< d\) (see [15]).
For
$$W_{m_{1},m_{2},\dots,m_{d+1}}^{(j)}(\xi):=\sum_{l\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2\pi l \Vert ^{2} \bigr)^{s} \bigl\vert \widehat{M}_{m_{1},m_{2},\dots,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2}, $$
it is known that there exist \(c,C\geq0\) such that
$$0\leq c\leq\sum_{l\in\mathbb {Z}^{d}} \bigl\vert \widehat{M}_{m_{1},m_{2},\dots ,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2}\leq C< \infty. $$
Considering \(\xi:=(\xi_{1},\xi_{2},\dots,\xi_{d})\) and \(l:=(l_{1},l_{2},\dots ,l_{d})\), we have
$$\begin{aligned} &\sum_{l\in\mathbb {Z}^{d}}\bigl(1+2^{2j} \Vert \xi+2\pi l \Vert ^{2}\bigr)^{s} \bigl\vert \widehat {M}_{m_{1},m_{2},\dots,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2} \\ &\quad =\sum_{(l_{1},l_{2},\dots,l_{d})\in\mathbb {Z}^{d}}\Biggl(1+2^{2j} \sum_{i=1}^{d} \vert \xi _{i}+2 \pi l_{i} \vert ^{2}\Biggr)^{s} \bigl\vert \widehat{M}_{m_{1},m_{2},\dots,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2}. \end{aligned}$$
(7)
By mathematical induction we know that, for positive real numbers \(x_{i}\), \(i=1,\dots,d\),
$$\begin{aligned} \Biggl(\sum_{`i=1}^{d}x_{i} \Biggr)^{m}\leq d^{m} \Biggl(\sum _{i=1}^{d}(x_{i})^{m} \Biggr),\quad x_{i}\in\mathbb {R}_{+}. \end{aligned}$$
(8)
From (7) and (8) we have
$$\begin{aligned} &\sum_{(l_{1},l_{2},\ldots,l_{d})\in\mathbb {Z}^{d}} \Biggl(1+2^{2j}\sum _{i=1}^{d} \vert \xi _{i}+2\pi l_{i} \vert ^{2} \Biggr)^{s} \bigl\vert \widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2} \\ &\quad\leq(d+1)^{s} \Biggl(c+2^{2js}\sum _{(l_{1},l_{2},\ldots,l_{d})\in\mathbb {Z}^{d}} \Biggl(\sum_{i=1}^{d} \vert \xi_{i}+2\pi l_{i} \vert ^{2s} \Biggr) \bigl\vert \widehat {M}_{m_{1},m_{2},\ldots,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2} \Biggr) \\ &\quad\leq(d+1)^{s} \Biggl(c+2^{2js}C'\sum _{(l_{1},l_{2},\ldots,l_{d})\in\mathbb {Z}^{d}} \Biggl(\sum_{i=1}^{d} \bigl\vert \widehat{M}_{m_{i}-s,m_{d+1}}(\xi+2\pi l) \bigr\vert ^{2} \Biggr) \Biggr) \\ &\quad\leq C_{j}< +\infty, \end{aligned}$$
where \(C',C_{j}>0\), and \(m_{i}-s,m_{d+1}\) is the ith term subtracted by s. Hence we have the following:

Lemma 3.1

There exists two constants \(c_{j}\) and \(C_{j}\) such that
$$0< c_{j}\leq W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j)}(\xi)\leq C_{j}< +\infty. $$
Now, for every \(j\in\mathbb {Z}\), we define
$$\begin{aligned} \hat{\varphi}^{(j)}(\xi)=\frac{\widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}}(\xi )}{\sqrt{W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j)}(\xi)}}. \end{aligned}$$
(9)
Now we find a \(2\pi\mathbb {Z}^{d}\)-periodic function \(m_{0}^{(j)}\in L^{2}(\mathbb {Z}^{d})\) for which the scaling relation (5) holds:
$$\varphi^{(j)}(2\xi)=m_{0}^{(j+1)}(\xi) \varphi^{(j+1)}(\xi). $$
From (9) we get
$$\begin{aligned} m_{0}^{(j+1)}(\xi)&=\frac{\widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}}(2\xi )}{\widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}}(\xi)}\sqrt{ \frac {W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j+1)}(\xi)}{W_{m_{1},m_{2},\ldots ,m_{d+1}}^{(j)}(2\xi)}} \\ &=\prod_{i=1}^{d+1} \biggl( \frac{1+e^{-i\langle k_{i},\xi\rangle}}{2} \biggr)^{m_{i}}\sqrt{\frac{W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j+1)}(\xi )}{W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j)}(2\xi)}}. \end{aligned}$$
Finally, let us construct wavelets associated with the scaling function \(\varphi^{(j)},j\in\mathbb {Z}\). We define the \(2\pi\mathbb {Z}^{d}\)-periodic functions \(m_{p}^{(j)}\), \(p=1,2,\ldots,2^{d-1}\), by
$$m_{p}^{(j)}(\xi)=e^{-i\langle\gamma_{p},\xi\rangle}\mathcal {L}_{p}^{(j)}(2\xi )\overline{m_{0}^{(j+1)}( \xi+\gamma_{p}\pi)}, $$
where the trigonometric polynomial \(\mathcal {L}_{p}^{(j)}\) is to be chosen such that \(m_{p}^{(j)}\) satisfies (6) for all p.

Proposition 3.2

Suppose \(\varphi^{(j)}\) is a scaling function for an MRA \(V_{j}, j\in\mathbb {Z}\), of \(H^{s}(\mathbb {R}^{d})\) and \(m_{0}^{(j)}\) is the associated low pass filter. Then the distributions \(2^{j/2}\psi ^{(j)}(2^{j}x-k), j\in\mathbb {Z}, k\in\mathbb {Z}^{d}\), are an orthonormal basis for \(H^{s}(\mathbb {R}^{d})\) if and only if
$$\hat{\psi}_{p}^{(j)}(2\xi)=e^{-i\langle\gamma_{p},\xi\rangle}\mathcal {L}_{p}^{(j)}(2\xi)\overline{m_{0}^{(j+1)}( \xi+\gamma_{p}\pi)}\hat{\varphi}^{(j+1)}(\xi),\quad \forall p=1,2, \ldots,2^{d-1}, $$
a.e. on \(\mathbb {R}^{d}\) for some \(2\pi\mathbb {Z}^{d}\)-periodic function \(\mathcal {L}_{p}^{(j)}\) such that
$$\bigl\vert \mathcal {L}_{p}^{(j)}(\xi) \bigr\vert =1,\quad \forall p=1,2,\ldots,2^{d-1}, \textit{a.e. } \xi\in\mathbb {T}^{d}. $$

Proof

From Proposition 2.1 we get
$$\begin{aligned} \sum_{k\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2k\pi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{\psi}_{p}^{(j)}(\xi+2k\pi) \bigr\vert ^{2}=1,\quad \forall p=1,2,\ldots,2^{d-1}. \end{aligned}$$
(10)
Now, we have only to verify the density condition
$$\lim_{j\rightarrow+\infty} \bigl\vert \varphi^{(j)} \bigl(2^{-j}\xi\bigr) \bigr\vert =\bigl(1+ \Vert \xi \Vert ^{2}\bigr)^{-s/2}. $$
By definition,
$$W_{m_{1},m_{2},\ldots,m_{d+1}}^{(j)}\bigl(2^{-j}\xi\bigr)=\sum _{k\in\mathbb {Z}^{d}}\bigl(1+ \bigl\Vert \xi+2^{j+1}\pi k \bigr\Vert ^{2}\bigr)^{s} \bigl\vert \widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}} \bigl(2^{-j}\xi +2\pi k\bigr) \bigr\vert ^{2}. $$
The term associated with \(k=0\) converges to \((1+\|\xi\|^{2})^{s}\). Using the estimates
$$\bigl\vert \widehat{M}_{m_{1},m_{2},\ldots,m_{d+1}}\bigl(2^{-j}\xi+2\pi k\bigr) \bigr\vert = \Biggl\vert \prod_{j'=1}^{d+1} \biggl(\frac{1-e^{-i\langle k_{j'},2^{-j}\xi\rangle }}{i\langle k_{j'},2^{-j}\xi+2\pi k\rangle} \biggr)^{m_{j'}} \Biggr\vert $$
for \(\xi=(\xi_{1},\xi_{2},\ldots,\xi_{d})\) and
$$\begin{aligned} & \Biggl\vert \prod_{j'=1}^{d+1} \biggl( \frac{1-e^{-i\langle k_{j'},2^{-j}\xi \rangle}}{i\langle k_{j'},2^{-j}\xi+2\pi k\rangle} \biggr)^{m_{j'}} \Biggr\vert \\ & \quad\leq \Biggl(\prod_{j'=1}^{d+1} \biggl\vert \frac{\sin(2^{-(j+1)}\xi _{j'})}{2^{-j-1}\xi_{j'}+\pi k} \biggr\vert ^{m_{j'}} \Biggr) \biggl( \biggl\vert \frac {\sin(2^{-(j+1)}\sum^{d}_{j'=1}\xi_{j'})}{2^{-j-1}\sum^{d}_{j'=1}\xi _{j'}+d\pi k} \biggr\vert ^{m_{d+1}} \biggr) \\ &\quad \leq\frac{2^{-(j+1)(\sum^{d+1}_{j'=1}m_{j'})} (\prod_{j'=1}^{d} \vert \xi_{j'} \vert ^{m_{j'}} ) ( \vert \sum^{d}_{j'=1}\xi _{j'} \vert ^{m_{d+1}} )}{ \vert k \vert ^{\sum^{d+1}_{j'=1}m_{j'}}} \end{aligned}$$
for \(2^{-(j+1)} (\prod_{j'=1}^{d}|\xi_{j'}|^{m_{j'}} ) ( \vert \sum^{d}_{j'=1}\xi_{j'} \vert ^{m_{d+1}} )<1\) and \(k=0\), we see that, as \(j\rightarrow+\infty\), the sum of the other terms converges to 0. The conclusion follows easily.
If \(\psi_{p}^{(j)}\) is an orthonormal wavelet, then the orthonormality of \(\{2^{j/2}\psi_{p}^{(j)}(2^{j}\cdot-k): j\in\mathbb {Z}, k\in\mathbb {Z}^{d}, p=1,2,\ldots,2^{d-1}\}\) gives us
$$\begin{aligned} 1={}&\sum_{k\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2k\pi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{\psi}_{p}^{(j)}(\xi+2k\pi) \bigr\vert ^{2} \\ ={}&\sum_{k\in\mathbb {Z}^{d}} \bigl(1+2^{2j} \Vert \xi+2k \pi \Vert ^{2} \bigr)^{s} \bigl\vert \mathcal {L}_{p}^{(j)}(\xi) \bigr\vert ^{2} \bigl\vert \hat{\varphi}^{(j+1)}(\xi/2+k\pi) \bigr\vert ^{2} \\ &{} \times \bigl\vert {m_{0}^{(j+1)}(\xi/2+k\pi+ \gamma_{p}\pi)} \bigr\vert ^{2} \\ ={}& \bigl\vert \mathcal {L}_{p}^{(j)}(\xi) \bigr\vert ^{2} \Biggl(\sum_{l\in\mathbb {Z}^{d}} \bigl(1+2^{2(j+1)} \Vert \xi/2 +2l\pi \Vert ^{2} \bigr)^{s} \bigl\vert \hat{\varphi}^{(j+1)}(\xi/2+2l\pi ) \bigr\vert ^{2} \\ &{} \times\sum_{q=1}^{2^{d}-1} \bigl\vert {m_{0}^{(j+1)}(\xi/2+\gamma_{q}\pi)} \bigr\vert ^{2}+\sum_{l\in\mathbb {Z}^{d}} \bigl(1+2^{2(j+1)} \Vert \xi/2 +2l\pi+\gamma_{q}\pi \Vert ^{2} \bigr)^{s} \\ & {}\times \bigl\vert \hat{\varphi}^{(j+1)}(\xi/2+2l\pi+ \gamma_{q}\pi ) \bigr\vert ^{2} \bigl\vert {m_{0}^{(j+1)}(\xi/2)} \bigr\vert ^{2} \Biggr) \\ ={} &\bigl\vert \mathcal {L}_{p}^{(j)}(\xi) \bigr\vert ^{2} \Biggl(\sum_{q=0}^{2^{d}-1} \bigl\vert {m_{0}^{(j+1)}(\xi /2+\gamma_{q}\pi)} \bigr\vert ^{2} \Biggr)= \bigl\vert \mathcal {L}_{p}^{(j)}( \xi) \bigr\vert ^{2},\quad p=1,2,\ldots,2^{d-1}, \end{aligned}$$
for a.e. \(\xi\in\mathbb {T}^{d}\) and \(\gamma_{q},\gamma_{p}\in E_{d}\), which finishes our proof. □

4 Conclusion

In this paper, we have successfully generalized MRA over higher-dimensional Sobolev spaces by giving orthonormality and density conditions. Further, we constructed nonseparable orthonormal wavelets in a higher-dimensional Sobolev space by using multivariate box splines. The main obstacle in constructing wavelets is constructing low-pass and high-pass filters with the help of multivariate box splines, which satisfy the condition of orthonormality in \(H^{s}(\mathbb {R}^{d})\) for every scale j (because the \(H^{s}\)-norm is not dilation invariant).

Declarations

Acknowledgements

The authors would like to thank the anonymous referees for their insightful comments and suggestions.

Funding

The work of the corresponding author is supported by the Senior Research Fellowship of University Grants Commission (UGC), India (Grant no. SRF/AA/139/F-236/2013-14).

Authors’ contributions

Both authors contributed equally to the manuscript. Both authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Kirori Mal College, University of Delhi, New Delhi, India
(2)
Department of Mathematics, University of Delhi, New Delhi, India

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