Abstract
We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.
Highlights
Because it is highly desirable to construct wavelets within a class of analytically representable functions, compactly supported sibling frames with interorthogonality attract a considerable amount of attention, recently.In 1997, Ron and Shen completed the structure of the affine system, which can be factored during a multiresolution analysis construction
The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions
In [3], Han gave his investigation of symmetric tight framelet filter banks with a minimum number of generators and systematically studied them with three high-pass filters which are derived from the oblique extension principle
Summary
Because it is highly desirable to construct wavelets within a class of analytically representable functions, compactly supported sibling frames with interorthogonality attract a considerable amount of attention, recently. In 2000, compactly supported tight frames that correspond to refinable functions were studied and a constructive proof was given by Chui and He [2]. In 2005, Averbuch et al [6] obtained tight and sibling frames originated from discrete splines, in which, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. In 2007, a new type of pseudo-splines was introduced to construct symmetric or antisymmetric tight framelets with desired approximation orders by Dong and Shen [8]. They provided various constructions of wavelets and framelets.
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