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 [
4‐
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 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
f̂ 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.
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
\(H^{s}(\mathbb {R}^{d})\).
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.
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
$$\hat{\psi}^{(j)}_{p}\bigl(2^{-j}\xi \bigr)=m_{p}^{(j+1)}\bigl(2^{-j-1}\xi\bigr)\hat{\varphi}^{(j+1)}\bigl(2^{-j-1}\xi\bigr),\quad p=1,2,\ldots, 2^{d}-1. $$
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)
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:
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.
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).
Acknowledgements
The authors would like to thank the anonymous referees for their insightful comments and suggestions.
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