Pseudo-splines, wavelets and framelets. (English) Zbl 1282.42035
Summary: The first type of pseudo-splines were introduced in [I. Daubechies, B. Han, A. Ron and Z. Shen, Appl. Comput. Harmon. Anal. 14, No. 1, 1–46 (2003; Zbl 1035.42031); I. W. Selenick, Appl. Comput. Harmon. Anal. 10, No. 2, 163–181 (2001; Zbl 0972.42025)] to construct tight framelets with desired approximation orders via the unitary extension principle of [A. Ron and Z. Shen, J. Funct. Anal. 148, No. 2, 408–447 (1997; Zbl 0891.42018)]]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provide a rich family of refinable functions. B-splines are one of the special classes of pseudo-splines; orthogonal refinable functions (whose shifts form an orthonormal system given in [I. Daubechies, Commun. Pure Appl. Math. 41, No. 7, 909–996 (1988; Zbl 0644.42026)]) are another class of pseudo-splines; and so are the interpolatory refinable functions (which are the Lagrange interpolatory functions at \(\mathbb Z\) and were first discussed in [S. Dubuc, J. Math. Anal. Appl. 114, 185–204 (1986; Zbl 0615.65005)]). The other pseudo-splines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type. This gives a wide range of choices of refinable functions that meets various demands for balancing the approximation power, the length of the support, and the regularity in applications. This paper will give a regularity analysis of pseudo-splines of the both types and provide various constructions of wavelets and framelets. It is easy to see that the regularity of the first type of pseudo-splines is between B-spline and orthogonal refinable function of the same order. However, there is no precise regularity estimate for pseudo-splines in general. In this paper, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity of the corresponding wavelets and framelets. The asymptotical regularity analysis, as the order of the pseudo-splines goes to infinity, is also provided. Furthermore, we show that in all tight frame systems constructed from pseudo-splines by methods provided both in [Zbl 1035.42031] and this paper, there is one tight framelet from the generating set of the tight frame system whose dilations and shifts already form a Riesz basis for \(L_2(\mathbb R)\).
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 65T60 | Numerical methods for wavelets |
| 65D07 | Numerical computation using splines |
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