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A class of generalized pseudosplines
Journal of Inequalities and Applications volume 2014, Article number: 359 (2014)
Abstract
In this paper, a class of refinable functions is given by smoothening pseudosplines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed.
1 Introduction
We denote by ℤ and ℝ the set of integers and real numbers, respectively. Let {L}_{p}(\mathbb{R}) stand for the classical Lebesgue space
with the norm {\parallel f\parallel}_{p}={({\int}_{\mathbb{R}}{f(t)}^{p}\phantom{\rule{0.2em}{0ex}}dt)}^{\frac{1}{p}} and {L}_{\mathrm{\infty}}(\mathbb{R}) consisting of all Lebesgue measurable and bounded functions on ℝ. Similarly, the discrete space {\ell}_{p}(\mathbb{Z}):=\{\{{a}_{n}\},{\sum}_{n}{{a}_{n}}^{p}<+\mathrm{\infty},n\in \mathbb{Z}\} with {\parallel \{{a}_{n}\}\parallel}_{p}={({\sum}_{n}{{a}_{n}}^{p})}^{\frac{1}{p}}. As usual, given f\in {L}_{1}(\mathbb{R})\cap {L}_{2}(\mathbb{R}), its Fourier transform is defined by
on ℝ. The Fourier transform of a function in {L}_{2}(\mathbb{R}) is understood as the unitary extension. We write h=f\ast g for the convolution h(x)={\int}_{\mathbb{R}}f(xt)g(t)\phantom{\rule{0.2em}{0ex}}dt, defined for any pair of functions f and g such that the integral exists almost everywhere. Clearly, \stackrel{\u02c6}{h}(\omega )=\stackrel{\u02c6}{f}(\omega )\stackrel{\u02c6}{g}(\omega ) in the frequency domain, when all the Fourier transforms exist in that formula. Given g\in {L}_{2}(\mathbb{R}), \{g(xk),k\in \mathbb{Z}\} is called a Riesz basis of its linearly generating space, if for each \{{\lambda}_{k}\}\in {\ell}_{2}(\mathbb{Z}) there exist two positive constants A and B such that
The numbers A, B are called lower Riesz bound and upper Riesz bound, respectively.
Multiresolution analysis provides a classical method to construct wavelets.
Definition 1 A multiresolution analysis of {L}_{2}(\mathbb{R}) means a sequence of closed linear subspaces {V}_{j} of {L}_{2}(\mathbb{R}) which satisfies

(i)
{V}_{j}\subset {V}_{j+1}, j\in \mathbb{Z},

(ii)
f(x)\in {V}_{j} if and only if f(2x)\in {V}_{j+1},

(iii)
\overline{{\bigcup}_{j\in \mathbb{Z}}{V}_{j}}={L}_{2}(\mathbb{R}) and {\bigcap}_{j\in \mathbb{Z}}{V}_{j}=\{0\},

(iv)
there exists a function \varphi \in {L}_{2}(\mathbb{R}) such that \{\varphi (xk),k\in \mathbb{Z}\} forms a Riesz basis of {V}_{0}.
The function ϕ in Definition 1 is said to be a scaling function, if it satisfies
for some sequence \{{a}_{k}\}\in {\ell}_{2}(\mathbb{Z}). Define the Fourier series \stackrel{\u02c6}{c} of a sequence \{{c}_{k}\}\in {\ell}_{2}(\mathbb{Z}) by
Then the refinement equation (1.2) becomes
The function \stackrel{\u02c6}{a} is called the refinement mask of ϕ. The pseudospline of Type I was first introduced in [1] to construct tight framelets. The pseudospline of Type II was first studied by Dong and Shen in [2]. There have been many developments in the theory of pseudosplines over the past ten years [3, 4]. Its applications in image denoising and image inpainting are also very extensive [5, 6]. The pseudospline is defined by its refinement mask. The refinement mask of a pseudospline of Type I with order (m,\ell ) is given by
and the refinement of a pseudospline of Type II with order (m,\ell ) is given by
The mask of Type I is obtained by taking the square root of the mask of Type II using the FejérRiesz lemma [7], i.e. {}_{2}\stackrel{\u02c6}{a}(\xi )={{}_{1}\stackrel{\u02c6}{a}(\xi )}^{2}. The corresponding pseudospline can be defined in terms of their Fourier transform, i.e.
In order to smoothen the pseudospline, one can use the convolution method. Take the smoothed pseudospline
where {\chi}_{[\frac{1}{2},\frac{1}{2}]} denotes the characteristic function of interval [\frac{1}{2},\frac{1}{2}] and n\u2a7em. This is equivalent to
Thus the symbol of {\varphi}_{n,m,\ell} becomes
Therefore, we define the smoothed pseudospline by its refinement mask for Type I:
and for Type II:
where r\u2a7e2m. When r=2m, it is the pseudospline. When r\ne 2m, it can be considered as an extension of pseudospline. Define the translated form of the Type II by
Then we get the differential relation
This inherits the property of a Bspline.
Remark 1 One may think that smoothing the pseudosplines by convolving them with Bsplines seems unnecessary since one can simply increase m of the original pseudosplines. However, by increasing m, we cannot get the differential relation (1.6), which is important for the construction of divergence free wavelets and curl free wavelets in the analysis of incompressible turbulent flows [8, 9].
Remark 2 Similar to the definition of (1.5), we can define a smoothed dual pseudospline by its refinement mask,
as an extension of dual pseudosplines in [3] and get the corresponding wavelets.
Remark 3 In addition, one can assume n\in \mathbb{R} in (1.5) and (1.7), as a generalization of fractional splines in [10].
2 Some lemmas
This section gives some lemmas that will be used to prove several results of this paper. We start with some results from [2].
Lemma 1 [2]
For given nonnegative integers m, j, ℓ,
This lemma will be used in Section 4 in order to prove the Riesz basis property of wavelets. Define {P}_{m,\ell}(y):={\sum}_{j=0}^{\ell}\left(\begin{array}{c}m+\ell \\ j\end{array}\right){y}^{j}{(1y)}^{\ell j}, {R}_{m,\ell}(y):={(1y)}^{m}{P}_{m,\ell}(y) and {R}_{r,m,\ell}={(1y)}^{\frac{r}{2}}{P}_{m,\ell}(y) where y={sin}^{2}(\xi /2), r, m, ℓ are nonnegative integers and r\u2a7e2m. Then one can find that
and the following lemma holds.
Lemma 2 [2]
For nonnegative integers m and ℓ with \ell \u2a7dm1, let {P}_{m,\ell} and {R}_{m,\ell} be the polynomials defined above. Then

(1)
{P}_{m,\ell}(y)={\sum}_{j=0}^{\ell}\left(\begin{array}{c}m1+j\\ j\end{array}\right){y}^{j};

(2)
{R}_{m,\ell}^{\prime}(y)=(m+\ell )\left(\begin{array}{c}m+\ell 1\\ \ell \end{array}\right){y}^{\ell}{(1y)}^{m1}.
With the two lemmas in hand, the basic property of the polynomial {R}_{r,m,\ell}, which will be used in Section 4, is given.
Lemma 3 For nonnegative integers r, m and ℓ,

(1)
define Q(y):={R}_{r,m,\ell}(y)+{R}_{r,m,\ell}(1y); then
\underset{y\in [0,1]}{min}Q(y)=Q\left(\frac{1}{2}\right)={2}^{1\frac{r}{2}\ell}\sum _{j=0}^{\ell}\left(\begin{array}{c}m+\ell \\ j\end{array}\right); 
(2)
define S(y):={R}_{r,m,\ell}^{2}(y)+{R}_{r,m,\ell}^{2}(1y); then
\underset{y\in [0,1]}{min}S(y)=S\left(\frac{1}{2}\right)={2}^{1r2\ell}{\left(\sum _{j=0}^{\ell}\left(\begin{array}{c}m+\ell \\ j\end{array}\right)\right)}^{2}.
Proof Since {R}_{r,m,\ell}(y)={(1y)}^{\frac{r}{2}m}{R}_{m,\ell}(y), its derivative is
So the derivative is {Q}^{\prime}(y)={R}_{r,m,\ell}^{\prime}(y)+{R}_{r,m,\ell}^{\prime}(1y)=I+\mathit{II}, where
and
Now, we compute them, respectively. For I, by using (1) of Lemma 2, one has
For II, by using (2) of Lemma 2, one has
For j=0,\dots ,\ell, since {y}^{\frac{r}{2}1}{(1y)}^{j}\u2a7d{(1y)}^{\frac{r}{2}1}{y}^{j} for all y\in [0,\frac{1}{2}] and {y}^{\frac{r}{2}1}{(1y)}^{j}\u2a7e{(1y)}^{\frac{r}{2}1}{y}^{j} for all y\in [\frac{1}{2},1], one has
This means Q(y) reaches its minimum value at the point y=1/2. Furthermore,
This completes the proof of (1). For (2) of this lemma, since
we have {S}^{\prime}(y)=\mathit{III}+\mathit{IV}, where
and
For III, by (1) of Lemma 2, we have
For IV, by (2) of Lemma 2, we have
Since
and for every j,
we have
This means S(y) reaches its minimum at point y=1/2. Furthermore, we have
This completes the lemma. □
3 Regularity and stability of scaling function
In this section, we discuss the regularity and stability of a scaling function generated by the refinement mask of a smoothed pseudospline. Let
Then the decay of \stackrel{\u02c6}{\varphi} can be characterized by \stackrel{\u02c6}{a} as stated in the following theorem.
Theorem 1 [2]
Let \stackrel{\u02c6}{a}(\xi ) be a refinement mask of the refinable function ϕ of the form
Suppose that
Then \stackrel{\u02c6}{\varphi}(\xi )\u2a7dC{(1+\xi )}^{n+\kappa}, with \kappa =log(\mathcal{L}(\frac{2\pi}{3}))/log2, and this decay is optimal.
In order to use this lemma, one needs to consider the polynomial corresponding to \mathcal{L}(\xi ). In fact, Dong and Shen give an important proposition to estimate it in the following proposition.
Proposition 1 [2]
Let {P}_{m,\ell}(y) be defined as in Section 2, where m, ℓ are nonnegative integers with \ell \u2a7dm1. Then
Combing Theorem 1 and Proposition 1, we have the following theorem, which characterizes the regularity of a smoothed pseudospline.
Theorem 2 Let _{2}ϕ be the smoothed pseudospline of Type II with order r, m, ℓ, then
where \kappa =log({P}_{m,\ell}(\frac{3}{4}))/log2. Consequently, {}_{2}\varphi \in {C}^{{\alpha}_{2}\u03f5} where {\alpha}_{2}=r\kappa 1. Furthermore, let _{1}ϕ be the smoothed pseudospline of Type I with order n, m, ℓ. Then
Consequently, {}_{1}\varphi \in {C}^{{\alpha}_{1}\u03f5} with {\alpha}_{1}=n\frac{\kappa}{2}1.
Proof Notice that \mathcal{L}(\xi ) in Theorem 1 is exactly {P}_{m,\ell}({sin}^{2}(\frac{\xi}{2})) and 4y(1y)={sin}^{2}(\xi ); one can easily prove this theorem by Theorem 1 and Proposition 1. □
This theorem shows {}_{k}\varphi \in {L}_{2}(\mathbb{R}) for k=1,2. Since r\u2a7e2m, the regularity of ϕ is better than a pseudospline but the support is longer. For r=6, m=2, \ell =1 the smoothed pseudospline {}_{2}\varphi _{r,m,\ell} is shown in Figure 1.
Now, we consider the stability of the smoothed pseudospline. When ϕ is compactly supported in {L}_{2}(\mathbb{R}), it was shown by Jia and Micchelli [11] that the upper Riesz bound in (1.1) always exists. Furthermore, they assert that the existence of a lower Riesz bound is equivalent to
where 0 denotes the zero sequence in {\ell}_{2}(\mathbb{Z}). Since a smoothed pseudospline is compactly supported and belongs to {L}_{2}(\mathbb{R}) for k=1,2, the stability is equivalent to (3.1).
Theorem 3 Smoothed pseudosplines are stable.
Proof By the definition of refinement mask, for each fixed \frac{r}{2}\u2a7em\u2a7e1 and for any 0\u2a7d\ell \u2a7dm1, {cos}^{2m}(\xi /2)\u2a7d{}_{2}{\stackrel{\u02c6}{a}}_{r,m,\ell}(\xi ) holds for all \xi \in \mathbb{R}. Therefore, we have
where {B}_{r} stands for the Bspline with order r. Since {B}_{r} is stable, the vector {({\stackrel{\u02c6}{B}}_{r}(\xi +2k\pi ))}_{k\in \mathbb{Z}}\ne \mathbf{0}. Hence {({\stackrel{\u02c6}{\varphi}}_{r,m,\ell}(\xi +2k\pi ))}_{k\in \mathbb{Z}}\ne \mathbf{0}.
For a smoothed pseudospline of Type I, since {}_{2}\stackrel{\u02c6}{a}_{2n,m,\ell}(\xi )={{}_{1}{\stackrel{\u02c6}{a}}_{n,m,\ell}}^{2}={}_{1}{\stackrel{\u02c6}{a}}_{n,m,\ell}(\xi ){\cdot}_{1}{\stackrel{\u02c6}{a}}_{n,m,\ell}(\xi ), one has
Therefore, the set of zeros of {}_{1}\stackrel{\u02c6}{\varphi}_{n,m,\ell}(\xi ) is contained in that of {}_{2}\stackrel{\u02c6}{\varphi}_{2n,m,\ell}(\xi ) and this guarantees the stability of {}_{1}\varphi (\xi ). □
This theorem shows the stability of a smoothed pseudospline. From the definition of a Riesz basis, one can find that the translate of a smoothed pseudospline is also linearly independent.
4 Riesz wavelets
Since all smoothed pseudosplines are compactly supported, refinable, stable in {L}_{2}(\mathbb{R}), the sequence of spaces {({V}_{n})}_{n\in \mathbb{Z}} defined via Definition 1 forms an MRA. The corresponding wavelets can be constructed by the classical method. Define
and X(\psi ):=\{{\psi}_{n,k}={2}^{n/2}\psi ({2}^{n}k),n,k\in \mathbb{Z}\}. Then X(\psi ) is a Riesz basis. To prove this, the following theorem is needed.
Theorem 4 [2]
Let \stackrel{\u02c6}{a}(\xi ) be a finitely supported refinement mask of a refinable function \varphi \in {L}_{2}(\mathbb{R}) with \stackrel{\u02c6}{a}(0)=1 and \stackrel{\u02c6}{a}(\pi )=0, such that \stackrel{\u02c6}{a} can be factorized into the form
where ℒ is the Fourier series of a finitely supported sequence with \mathcal{L}(\pi )\ne 0. Suppose that
Define \stackrel{\u02c6}{\psi}(2\xi ):={e}^{i\xi}\overline{\stackrel{\u02c6}{a}(\xi +\pi )}\stackrel{\u02c6}{\varphi}(\xi ) and \tilde{\mathcal{L}}:=\frac{\mathcal{L}(\xi )}{{\stackrel{\u02c6}{a}(\xi )}^{2}+{\stackrel{\u02c6}{a}(\xi +\pi )}^{2}}. Assume that {\parallel \mathcal{L}(\xi )\parallel}_{{L}_{\mathrm{\infty}}(\mathbb{R})}<{2}^{n1} and {\parallel \tilde{\mathcal{L}}(\xi )\parallel}_{{L}_{\mathrm{\infty}}(\mathbb{R})}<{2}^{n1}. Then X(\psi ) is a Riesz basis for {L}_{2}(\mathbb{R}).
From the above theorem, the key step is to estimate the upper Riesz bound of \mathcal{L}(\xi ) and \tilde{\mathcal{L}}(\xi ). Notice that
One has {}_{1}\mathcal{L}(\xi )={({P}_{m,\ell}({sin}^{2}(\xi /2)))}^{\frac{1}{2}}, {\mathcal{L}}_{2}(\xi )={P}_{m,\ell}({sin}^{2}(\xi /2)) and
Thus, we have the following theorem.
Theorem 5 Let _{ k }ϕ, k=1,2 be the smoothed pseudospline of Types I and II with order (r,n,m,\ell ). The refinement masks _{ k }a are given in (1.3) and (1.4). Define
Then X(\psi ) forms a Riesz basis for {L}_{2}(\mathbb{R}).
Proof By (1) of Lemma 3, one obtains
Applying Lemma 1, one obtains \parallel {}_{1}{\tilde{\mathcal{L}}}_{\mathrm{\infty}}\parallel \u2a7d{2}^{n1}<{2}^{n\frac{1}{2}}. Similarly, one can get
Notice that
Hence, {}_{k}\mathcal{L}(\xi )\u2a7d{}_{k}\tilde{\mathcal{L}}(\xi ) for k=1,2. By using Theorem 4, one gets the desired result. □
By definition, the wavelets are also in {L}_{2}(\mathbb{R}) and have the same regularity as the scaling function. Still, the support is longer than for pseudospline wavelets. For r=6, m=2, \ell =1, the smoothed pseudospline {}_{2}\psi _{6,2,1} is shown in Figure 2.
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Acknowledgements
This study was partially supported by the Joint Funds of the National Natural Science Foundation of China (Grant No. U1204103). The authors would like to express their thanks to the reviewers for their helpful comments and suggestions.
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Zhuang, Z., Yang, J. A class of generalized pseudosplines. J Inequal Appl 2014, 359 (2014). https://doi.org/10.1186/1029242X2014359
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DOI: https://doi.org/10.1186/1029242X2014359
Keywords
 pseudospline
 wavelets
 scaling function