Pointwise convergence of bounded cascade sequences

Abstract:

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function $\phi _0$, a cascade sequence $(\phi _n)_{n=0}^{\infty }$ is constructed by the iteration $\phi _n=C_a\phi _{n-1}, n=1, 2, \dots ,$ where $C_a$ is defined by $C_ag=\sum _{\alpha \in \mathbb Z}a(\alpha )g(2\cdot -\alpha ), g\in L_p(\mathbb {R}).$ In this paper, under a condition that the sequence $(\phi _n)_{n=0}^\infty$ is bounded in $L_\infty (\mathbb {R})$, we prove that the following three statements are equivalent: (i) $(\phi _n)_{n=0}^{\infty }$ converges $\textrm {a.e.}\ x\in \mathbb {R}$. (ii) For $\textrm {a.e.}\ x\in \mathbb {R}$, there exist a positive constant $c$ and a constant $\gamma \in (0,1)$ such that $|\phi _{n+1}(x)-\phi _n(x)|\leq c\gamma ^n \forall n=1,2, \dots .$ (iii) For some $p\in [1, \infty ), (\phi _n)_{n=0}^{\infty }$ converges in $L_p(\mathbb {R})$. An example is presented to illustrate our result.