Pseudo-Butterworth refinable functions

Abstract

We introduce the pseudo-Butterworth refinable function with order $\It{(n;m,\ell)}$ which is defined by the pseudo-Butterworth mask ▷수식삽입◁ (원문을 참조하세요) with positive integers $\It{n,m}$ and nonnegative integer $\ell\leq m-1$. This family contains the pseudo-splines (when ->$\It{n=1}$) and the Butterworth refinable functions (when ->$\It{m=1}$). The pseudo-Butterworth refinable functions contain interpolatory refinable functions (when $\ell=m-1$) and provide a rich family of the refinable functions. The pseudo-Butterworth refinable functions are not compactly supported (if $\It{n\gt 1}$) but have exponential decay to compensate for the lack of compact support. This thesis gives a comprehensive analysis of the pseudo-Butterworth refinable functions. The Sobolev exponents of the pseudo-Butterworth refinable functions are computed in terms of the parameters and their dependance on the parameters are analyzed for the regularity of the refinable functions. We give the asymptotic analysis of the Sobolev exponents of the pseudo-Butterworth refinable functions as one of the parameters increases. We show that the pseudo-Butterworth refinable function can generate a Riesz wavelet and compute its approximation order for the proper projection. We prove that the pseudo-Butterworth refinable function converges to the Shannon refinable function as one of the parameters increases. We show that the pseudo-Butterworth refinable function decays exponentially. Finally, we consider another type of refinable functions coming from the Riesz factorization of the pseudo-Butterworth masks. We also give the corresponding analysis of the refinable functions such as regularity, asymptotic analysis of Sobolev exponents, wavelet constructions and approximation order, etc.