Properties Of Generalized Pseudo-Butterworth Refinable Functions

Pseudo-Butterworth refinable functions are an extension of pseudo-splines. This family of refinable functions contains various refinable functions, thus leads to large flexibilities in filter designs, wavelet and framelet constructions. Based on the pseudo-Butterworth masks, Luo et al introduce two parameters and propose the so-called generalized pseudo-Butterworth masks:When we choose different parameters, the generalized pseudo-Butterworth refinable functions can be transformed into various forms of refinable functions, including the first type of pseudo-splines, the second type of pseudo-splines, Butterworth, pseudo-Butterworth, dual pseudo-splines and fractional splines. Thus, the generalized pseudo-Butterworth refinable functions have a greater flexibility in practical applications. Consequently the analysis of the generalized pseudo-Butterworth refinable functions has important significances in filter and wavelet frames construction. The work of this paper has some degree of guiding significance in selecting parameters of the generalized pseudo-Butterworth refinable functions, so it will deepen the corresponding understanding of this family of refinable functions.This paper discusses some properties of the generalized pseudo-Butterworth refinable functions by using Fourier analysis methods, and the main content of this paper is as follows:(1) This paper proves that the sequence{V,n}n∈Z generated by the generalized pseudo-Butterworth refinable functions can form a multiresolution analysis for L2(R), The multiresolution analysis structure makes the fast implementation of wavelets decomposition and reconstruction possible. The approximation order of refinable functions are analyzed, and the results show that the maximum approximation order can be obtained when s=m+a+l, and this value is independent of m.(2) This paper also discusses the regularity of the generalized pseudo-Butterworth refinable functions and investigates the Sobolev exponent’s dependence on parameters. Then the asymptotic analysis of the Sobolev exponent as mâ†'∞or nâ†'∞is also presented. In addition, this paper proves that the refinable functions have polynomial decay property. Results show that when α=0and s<m+l, the generalized pseudo-Butterworth refinable functions have better smoothness compared with pseudo-Butterworth refinable functions.