Refinable Functions and Cascade Algorithms in Weighted Spaces with Hölder Continuous Masks
Bin Han · SIAM Journal on Mathematical Analysis · 2008
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 Hölder continuous masks, in the weighted subspaces $L_{2, p, \gamma}(\mathbb R)$ of $L_2(\mathbb R)$, where $1\le p \le \infty$, $\gamma\ge 0$ and $f\in L_{2, p, \gamma}(\mathbb R)$ means $ \| f\|_{L_{2, p, \gamma}(\mathbb R)}:= \| \sum_{k\in \mathbb Z} | \widehat{ e^{\gamma|\cdot|} f}(\cdot+2\pi k)|^2\|_{L_p(\mathbb T)}^{1/2}0$ and $\hat a(0)=1$ (that is, $\hat a$ is a Hölder continuous mask with Hölder exponent $\beta$), we prove that the cascade algorithm associated with the mask $\hat a$ converges in the space $L_{2, \infty, 0}(\mathbb R)$ if and only if $ u_2(\hat a)>0$, where the quantity $ u_2(\hat a)$ will be defined in this paper and plays an important role in our study of refinable functions and cascade algorithms with Hölder continuous masks. In particular, if the shifts of a refinable function $\phi$, satisfying $\hat \phi(2\cdot)=\hat a\hat \phi$, are stable in $L_2(\mathbb R)$, then we must have $ u_2(\hat a)>0$, and therefore the cascade algorithm associated with mask $\hat a$ converges in the space $L_{2, \infty, 0}(\mathbb R)$. Based on this result, we are able to settle several problems on refinable functions, cascade algorithms, and wavelets associated with masks having infinitely many nonzero Fourier coefficients. As an application of the characterization of the convergence of a cascade algorithm in the space $L_{2, \infty, 0}(\mathbb R)$, 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 $L_{2, 1, \gamma}(\mathbb R)$ for $00$. Consequently, if a mask $\hat a$ has exponential decay of order $r>0$ and $ u_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 $L_2(\mathbb R)$; that is, $\|\phi\|^2_{L_{2, 1, \gamma}(\mathbb R)}=\int_\mathbb R |\phi(x)|^2 e^{2\gamma |x|}\, dx<\infty$ for all $0<\gamma<2r$. As another application of the characterization of the convergence of a cascade algorithm in the space $L_{2, \infty, 0}(\mathbb R)$, we completely characterize biorthogonal wavelets and Riesz wavelets in $L_2(\mathbb R)$, which are derived from refinable functions and whose involved wavelet filters in the frequency domain are Hölder continuous. We shall also investigate some basic properties of the quantity $ u_2(\hat a)$ and discuss how to calculate and estimate $ u_2(\hat a)$. Examples using fractional splines and the Butterworth filters will be given to illustrate the results in this paper.