We prove an extended version of asymptotic behavior of the orthonormal cardinal refinable functions from Blaschke products introduced by Contronei et al [2]. In fact, we show the orthonormal cardinal refinable function ${varphi}_{k,q}$ converges in $L^p(mathbb{R})$ ($2{leq}p{leq}{infty}$) to the Shannon refinable function as ${kappa}{ ightarrow}{infty}$ uniforml on a class $mathcal{Q}_{A,B}$ of real symmetric polynomials determined by positive constants $A{leq}B$.