Ten lectures on wavelets
Stability and orthonormality of multivariate refinable functions
SIAM Journal on Mathematical Analysis
Smooth refinable functions provide good approximation orders
SIAM Journal on Mathematical Analysis
Multidimensional Interpolatory Subdivision Schemes
SIAM Journal on Numerical Analysis
SIAM Journal on Mathematical Analysis
Analysis and construction of optimal multivariate biorthogonal wavelets with compact support
SIAM Journal on Mathematical Analysis
The Sobolev regularity of refinable functions
Journal of Approximation Theory
Stationary Subdivision
Computing the Smoothness Exponent of a Symmetric Multivariate Refinable Function
SIAM Journal on Matrix Analysis and Applications
IEEE Transactions on Information Theory - Part 2
Quasi-interpolatory refinable functions and construction of biorthogonal wavelet systems
Advances in Computational Mathematics
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Pseudo-splines constitute a new class of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. Pseudo-splines were first introduced by Daubechies, Han, Ron and Shen in [Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14(1) (2003), 1-46] and Selenick in [Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10(2) (2001) 163-181], and their properties were extensively studied by Dong and Shen in [Pseudo-splines, wavelets and framelets, 2004, preprint]. It was further shown by Dong and Shen in [Linear independence of pseudo-splines, Proc. Amer. Math. Soc., to appear] that the shifts of an arbitrarily given pseudo-spline are linearly independent. This implies the existence of biorthogonal dual refinable functions (of pseudo-splines) with an arbitrarily prescribed regularity. However, except for B-splines, there is no explicit construction of biorthogonal dual refinable functions with any given regularity. This paper focuses on an implementable scheme to derive a dual refinable function with a prescribed regularity. This automatically gives a construction of smooth biorthogonal Riesz wavelets with one of them being a pseudo-spline. As an example, an explicit formula of biorthogonal dual refinable functions of the interpolatory refinable function is given.