# Discrete norm approximation of the $L^p$ norm for spline functions

In Theorem 5.2 in Lynche (1988) "A data reduction strategy for splines with applications to the approximation of functions and data", a bound for the difference between the $$(l_2,t)$$ and $$L^2$$ norms is given (Equation 5.8), where $$(l_2,t)$$ is the discrete weighted norm defined in Equation 5.4.

Could this result be extended to the general $$(l_p,t)$$ and $$L^p$$ $$(l_2,t)$$ norms for $$1\leq p\leq\infty$$? Or at least for $$p=1$$?