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 $p=1$?

Thanks in advance.


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