Asymptotic Behavior of $M$-Band Scaling Functions of Daubechies Type

Abstract

This paper deals with the asymptotic behavior of $M$-band scaling functions ${}_N^M\varphi$ and $M$-band symbols ${}_N^{M}H$ as $M\to \infty$ for $N\ge 2$. This is followed by pointwise convergence, and $L^p$-convergence $(1\le p\le \infty )$ of ${}_N^M\varphi$, and the limit function $g$ of ${}_N^M\varphi$ as $M\to \infty$.