A class of generalized pseudo-splines

Journal of Inequalities and Applications, Sep 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.

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A class of generalized pseudo-splines

Journal of Inequalities and Applications A class of generalized pseudo-splines Zhitao Zhuang 0 Jianwei Yang 0 0 College of Mathematics and Information Science, North China University of Water Resources and Electric Power , Zhengzhou, 450011 , P.R. China 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. - Lp(R) := f , f (t) p dt < +∞ on R. The Fourier transform of a function in L(R) is understood as the unitary extension. We write h = f ∗ g for the convolution h(x) = R f (x – t)g(t) dt, defined for any pair of functions f and g such that the integral exists almost everywhere. Clearly, hˆ (ω) = fˆ(ω)gˆ(ω) in the frequency domain, when all the Fourier transforms exist in that formula. Given g ∈ L(R), {g(x – k), k ∈ Z} is called a Riesz basis of its linearly generating space, if for each {λk} ∈ (Z) there exist two positive constants A and B such that k∈Z k∈Z ≤ B The numbers A, B are called lower Riesz bound and upper Riesz bound, respectively. Multiresolution analysis provides a classical method to construct wavelets. Definition  A multiresolution analysis of L(R) means a sequence of closed linear subspaces Vj of L(R) which satisfies Z (i) Vj ⊂ Vj+, j ∈ , (ii) f (x) ∈ Vj if and only if f (x) ∈ Vj+, (iii) j∈Z Vj = L(R) and j∈Z Vj = {}, (iv) there exists a function φ ∈ L(R) such that {φ(x – k), k ∈ Z} forms a Riesz basis of V. k∈Z Then the refinement equation (.) becomes R φˆ (ξ ) = aˆ(ξ /)φˆ (ξ /), ξ ∈ . φn,m, = φm, ∗ χ[–  ,  ] ∗ · · · ∗ χ[–  ,  ], In order to smoothen the pseudo-spline, one can use the convolution method. Take the smoothed pseudo-spline The function aˆ is called the refinement mask of φ. The pseudo-spline of Type I was first introduced in [] to construct tight framelets. The pseudo-spline of Type II was first studied by Dong and Shen in []. There have been many developments in the theory of pseudosplines over the past ten years [, ]. Its applications in image denoising and image inpainting are also very extensive [, ]. The pseudo-spline is defined by its refinement mask. The refinement mask of a pseudo-spline of Type I with order (m, ) is given by aˆ (ξ )  := aˆ m, (ξ )  := cosm(ξ /) and the refinement of a pseudo-spline of Type II with order (m, ) is given by The mask of Type I is obtained by taking the square root of the mask of Type II using the Fejér-Riesz lemma [], i.e.  aˆ(ξ ) = |aˆ (ξ )|. The corresponding pseudo-spline can be defined in terms of their Fourier transform, i.e. where χ[–  ,  ] denotes the characteristic function of interval [–  ,  ] and n ≥ m. This is equivalent to aˆ n,m, (ξ ) = φˆn,m, (ξ )/φˆn,m, (ξ ) = aˆ m, (ξ ) cos(ξ /) n–m. Therefore, we define the smoothed pseudo-spline by its refinement mask for Type I: and for Type II: where r ≥ m. When r = m, it is the pseudo-spline. When r = m, it can be considered as an extension of pseudo-spline. Define the translated form of the Type II by T φˆr,m, (ξ ) := e–ir ξ φˆ r,m, (ξ ). Then we get the differential relation T φr+,m, (x) = T φr,m, (x) – T φr,m, (x – ). This inherits the property of a B-spline. Remark  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 pseudosplines. However, by increasing m, we cannot get the differential relation (.), which is important for the construction of divergence free wavelets and curl free wavelets in the analysis of incompressible turbulent flows [, ]. Remark  Similar to the definition of (.), we can define a smoothed dual pseudo-spline by its refinement mask, as an extension of dual pseudo-splines in [] and get the corresponding wavelets. 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  [] For given nonnegative integers m, j, , for all m ≥  and  ≤ ≤ m – . Rm, sin(ξ /) = aˆ m, (ξ ) and Rr,m, sin(ξ /) = aˆ r,m, (ξ ) and the following lemma holds. Lemma  [] For nonnegative integers m and polynomials defined above. Then () Pm, (y) = j= m–j+j yj; () Rm, (y) = –(m + ) m+ – y ( – y)m–. ≤ m – , let Pm, and Rm, be the With the two lemmas in hand, the basic property of the polynomial Rr,m, , which will be used in Section , is given. Lemma  For nonnegative integers r, m and , () define Q(y) := Rr,m, (y) + Rr,m, ( – y); then = – r – () define S(y) := Rr,m, (y) + Rr,m, ( – y); then  = –r–  . I = II = ( – y) r –mRm, (y) – y r –mRm, ( – y). Now, we compute them, respectively. For I, by using () of Lemma , one has I = For II, by using () of Lemma , one has II = (m + ) for all y ∈ [  , ], one has Q (y) = I + II ≤ , y ∈ [,  ]; ≥ , y ∈ [  , ]. Q(/) = Rr,m, (/) = – r Pm, (/) = – r – This completes the proof of (). For () of this lemma, since S(y) = Rr,m, (y) + Rr,m, ( – y) = ( – y)r–mRm, (y) + yr–mRm, ( – y),   we have S (y) = III + IV , where For III, by () of Lemma , we have III = (r – m) yr–Pm, ( – y) – ( – y)r–Pm, (y) = (r – m) yr–( – y) +j – y +j( – y)r– y  (r–)( – y)j + ( – y)  (r–)yj y  (r–)( – y)j – ( – y)  (r–)yj . = (r – m) For IV , by () of Lemma , we have = Rm, (y) and for every j, yr–( – y) +j – y +j( – y)r– S (y) = III + IV S(/) =  – r Pm, (y)  = –r–  . 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 pseudo-spline. Let aˆ (ξ ) = cosn(ξ /) L(ξ ) , ξ ∈ [–π , π ]. In order to use this lemma, one needs to consider the polynomial corresponding to L(ξ ). In fact, Dong and Shen give an important proposition to estimate it in the following proposition. Proposition  [] Let Pm, (y) be defined as in Section , where m, are nonnegative integers with ≤ m – . Then Pm, (y) ≤ Pm, Pm, (y)Pm, y( – y) ≤ Combing Theorem  and Proposition , we have the following theorem, which characterizes the regularity of a smoothed pseudo-spline. φˆ(ξ ) ≤ C  + |ξ | –n+ κ . where κ = log(Pm, (  ))/ log . Consequently, φ ∈ Cα– where α = r – κ – . Furthermore, let φ be the smoothed pseudo-spline of Type I with order n, m, . Then Consequently, φ ∈ Cα– with α = n – κ – . Proof Notice that |L(ξ )| in Theorem  is exactly Pm, (sin( ξ )) and y( – y) = sin(ξ ); one can easily prove this theorem by Theorem  and Proposition . This theorem shows kφ ∈ L(R) for k = , . Since r ≥ m, the regularity of φ is better than a pseudo-spline but the support is longer. For r = , m = , =  the smoothed pseudo-spline φr,m, is shown in Figure . Now, we consider the stability of the smoothed pseudo-spline. When φ is compactly supported in L(R), it was shown by Jia and Micchelli [] that the upper Riesz bound in (.) always exists. Furthermore, they assert that the existence of a lower Riesz bound is equivalent to where  denotes the zero sequence in (Z). Since a smoothed pseudo-spline is compactly supported and belongs to L(R) for k = , , the stability is equivalent to (.). Theorem  Smoothed pseudo-splines are stable. Proof By the definition of refinement mask, for each fixed r ≥ m ≥  and for any  ≤ m – , cosm(ξ /) ≤  aˆr,m, (ξ ) holds for all ξ ∈ R. Therefore, we have Therefore, the set of zeros of φˆn,m, (ξ ) is contained in that of φˆn,m, (ξ ) and this guarantees the stability of φ(ξ ). 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 Since all smoothed pseudo-splines are compactly supported, refinable, stable in L(R), the sequence of spaces (Vn)n∈Z defined via Definition  forms an MRA. The corresponding wavelets can be constructed by the classical method. Define Theorem  [] Let aˆ (ξ ) be a finitely supported refinement mask of a refinable function φ ∈ L(R) with aˆ () =  and aˆ (π ) = , such that aˆ can be factorized into the form L(ξ) Define ψˆ (ξ ) := e–iξ aˆ (ξ + π )φˆ (ξ ) and L˜ := |aˆ(ξ)|+|aˆ(ξ+π)| . Assume that L(ξ ) L∞(R) < n– and L˜(ξ ) L∞(R) < n–. Then X(ψ ) is a Riesz basis for L(R). From the above theorem, the key step is to estimate the upper Riesz bound of |L(ξ )| and |L˜(ξ )|. Notice that aˆ n,m, (ξ )  = aˆ n,m, (ξ ) = cosn(ξ /)Pm, sin(ξ /) .  One has |L(ξ )| = (Pm, (sin(ξ /)))  , |L(ξ )| = Pm, (sin(ξ /)) and |L˜| = |L˜| = Thus, we have the following theorem. Theorem  Let kφ, k = ,  be the smoothed pseudo-spline of Types I and II with order (r, n, m, ). The refinement masks ka are given in (.) and (.). Define Proof By () of Lemma , one obtains L˜ L∞(R) = sup (Pm, (y))  y∈[,] Rn,m, (y) + Rn,m, ( – y) ≤ miny∈[,](Rn,m, (y) + Rn,m, ( – y)) L˜ L∞(R) ≤ r– < r–  . Figure 2 The corresponding wavelets. By definition, the wavelets are also in L(R) and have the same regularity as the scaling Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed to each part of this work equally and read and approved the final manuscript. 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. 1. Daubechies , I, Han, B, Ron, A, Shen , Z: Framelets: MRA-based constructions of wavelet frames . Appl. Comput. Harmon. Anal . 14 ( 1 ), 1 - 46 ( 2003 ) 2. Dong , B, Shen , Z: Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal . 22 ( 1 ), 78 - 104 ( 2007 ) 3. Dong , B, Dyn , N, Hormann , K: Properties of dual pseudo-splines . Appl. Comput. Harmon. Anal . 29 ( 1 ), 104 - 110 ( 2010 ) 4. Shen , Y , Li , S, Mo, Q: Complex wavelets and framelets from pseudo splines . J. Fourier Anal. Appl . 16 ( 6 ), 885 - 900 ( 2010 ) 5. Cai , J-F, Shen, Z: Framelet based deconvolution . J . Comput. 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Zhitao Zhuang, Jianwei Yang. A class of generalized pseudo-splines, Journal of Inequalities and Applications, 2014, 359,