Pointwise convergence of bounded cascade sequences

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 ${\rm a.e.} x\in \mathbb{R}$. (ii) For ${\rm a.e.} x\in \mathbb{R}$, there exist a positive constant $c$ and a constant $\gamma\in (0,1)$ such that $\vert\phi_{n+1}(x)-\phi_n(x)\vert\leq c\gamma^n\,\,\forall n=1,2, \dots.$ (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.