Pseudo-splines,wavelets and framelets

The first type of pseudo-splines were introduced in I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46; I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163–181] to construct tight framelets with desired approximation orders via the unitary extension principle of A. Ron, Z. Shen, Affine systems in $L_2({\mathbb{R}}^d)$: The analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997) 408–447]. 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, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996]) 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, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–204]). 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 I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46] 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 $L_2(\mathbb{R})$.