A class of generalized pseudo-splines
© Zhuang and Yang; licensee Springer 2014
Received: 14 July 2014
Accepted: 12 September 2014
Published: 24 September 2014
In this paper, a class of refinable functions is given by smoothening pseudo-splines 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.
The numbers A, B are called lower Riesz bound and upper Riesz bound, respectively.
Multiresolution analysis provides a classical method to construct wavelets.
if and only if ,
there exists a function such that forms a Riesz basis of .
This inherits the property of a B-spline.
Remark 1 One may think that smoothing the pseudo-splines by convolving them with B-splines seems unnecessary since one can simply increase m of the original pseudo-splines. 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].
as an extension of dual pseudo-splines in  and get the corresponding wavelets.
Remark 3 In addition, one can assume in (1.5) and (1.7), as a generalization of fractional splines in .
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 .
Lemma 1 
and the following lemma holds.
Lemma 2 
With the two lemmas in hand, the basic property of the polynomial , which will be used in Section 4, is given.
- (1)define ; then
- (2)define ; then
This completes the lemma. □
3 Regularity and stability of scaling function
Then the decay of can be characterized by as stated in the following theorem.
Theorem 1 
Then , with , and this decay is optimal.
In order to use this lemma, one needs to consider the polynomial corresponding to . In fact, Dong and Shen give an important proposition to estimate it in the following proposition.
Proposition 1 
Combing Theorem 1 and Proposition 1, we have the following theorem, which characterizes the regularity of a smoothed pseudo-spline.
Consequently, with .
Proof Notice that in Theorem 1 is exactly and ; one can easily prove this theorem by Theorem 1 and Proposition 1. □
where 0 denotes the zero sequence in . Since a smoothed pseudo-spline is compactly supported and belongs to for , the stability is equivalent to (3.1).
Theorem 3 Smoothed pseudo-splines are stable.
where stands for the B-spline with order r. Since is stable, the vector . Hence .
Therefore, the set of zeros of is contained in that of and this guarantees the stability of . □
This theorem shows the stability of a smoothed pseudo-spline. From the definition of a Riesz basis, one can find that the translate of a smoothed pseudo-spline is also linearly independent.
4 Riesz wavelets
and . Then is a Riesz basis. To prove this, the following theorem is needed.
Theorem 4 
Define and . Assume that and . Then is a Riesz basis for .
Thus, we have the following theorem.
Then forms a Riesz basis for .
Hence, for . By using Theorem 4, one gets the desired result. □
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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