Refinable Functions and Cascade Algorithms in Weighted Spaces with Infinitely Supported Masks

Refinable functions and cascade algorithms play a fundamental role in wavelet analysis, which is useful in many applications. In this paper we shall study several properties of refinable functions, cascade algorithms and wavelets, associated with infinitely supported masks, in the weighted subspaces $L_{2, p, \gamma}(\RR)$ of $L_2(\RR)$, where $1\le p \le \infty$, $\gamma\ge 0$ and $f\in L_{2, p, \gamma}(\RR)$ means \[ \| f\|_{L_{2, p, \gamma}(\RR)}:=\left \| \sum_{k\in \ZZ} \Big| \widehat{ e^{\gamma|\cdot|} f}(\cdot+2\pi k)\Big|^2\right\|_{L_p(\TT)}^{1/2}<\infty. \] In particular, $\|f\|_{L_{2,1,\gamma}(\RR)}=\| f e^{\gamma|\cdot|}\|_{L_2(\RR)}$ and $\|f\|_{L_{2,\infty,0}(\RR)}=\| \sum_{k\in \ZZ} |\hat f(\cdot+2\pi k)|^2||^{1/2}_{L_\infty(\TT)}$. For a mask $\hat a\in C^\beta(\TT)$ with $\beta>0$ and $\hat a(0)=1$, we prove that the cascade algorithm associated with the mask $\hat a$ converges in the space $L_{2, \infty, 0}(\RR)$ if and only if $\nu_2(\hat a)>0$, where the quantity $\nu_2(\hat a)$ will be defined in this paper and plays an important role in our study of refinable functions and cascade algorithms with infinitely supported masks. In particular, if the shifts of a refinable function $\phi$, satisfying $\hat \phi(2\cdot)=\hat a\hat \phi$, are stable in $L_2(\RR)$, then we must have $\nu_2(\hat a)>0$ and therefore the cascade algorithm associated with mask $\hat a$ converges in the space $L_{2, \infty, 0}(\RR)$. Based on this result, we are able to settle several problems on refinable functions, cascade algorithms and wavelets associated with infinitely supported masks. As an application of the characterization of the convergence of a cascade algorithm in the space $L_{2, \infty, 0}(\RR)$, we are able to show that for a mask $\hat a$ having exponential decay of order $r>0$, the cascade algorithm associated with mask $\hat a$ converges in the weighted space $L_{2, 1, \gamma}(\RR)$ for $0<\gamma<2r$ if and only if $\nu_2(\hat a)>0$. Consequently, if a mask $\hat a$ has exponential decay of order $r>0$ and $\nu_2(\hat a)>0$, then its standard refinable function $\phi$, defined by $\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi)$, must have exponential decay of order $2r$ in $L_2(\RR)$; that is, $\|\phi\|^2_{L_{2, 1, \gamma}(\RR)}=\int_\RR |\phi(x)|^2 e^{2\gamma |x|}\, dx<\infty$ for all $0<\gamma<2r$. As another application of the characterization of the convergence of a cascade algorithm in the space $L_{2, \infty, 0}(\RR)$, we characterize biorthogonal wavelets and Riesz wavelets in $L_2(\RR)$, which are derived from refinable functions and whose involved wavelet filters are in the class $C^\beta(\TT)$ for some $\beta>0$. We shall also investigate some properties of the quantity $\nu_2(\hat a)$ and discuss how to estimate $\nu_2(\hat a)$. Examples using fractional splines and the Butterworth filters will be given to illustrate the results in this paper.

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