Two-sided estimates for the norms of some functionals in approximation theory (Q2751667)

Let \(C^r[a,b]\) be the Banach space of all real-valued functions on \([a,b]\) having continuous derivatives up to order \(r \in \mathbb N\), normed by \(\|f\|=\sum _{s=0}^{r-1} |f^{(s)}(a)|+ \|f^{(r)}\|_\infty.\) Consider a matrix of nodes \(a\leq x_n^1 < \dots < x_n^{i_n} \leq b\), \(n\geq 1,\) and a matrix of weights \( \{a^{ks} : n\leq 1\), \(1\leq k \leq i_n\), \(0\leq s\leq r\},\) where \(({i_n}) \) is a strictly increasing sequence of natural numbers. Define a continuous linear functional \(D_n\) on \(C^r[a,b]\) by \(D_n(f) = \sum_{s=0}^r \sum_{k=1}^{i_n} a_n^{ks} f^{(s)}(x_n^k).\) Using the two-side estimate NEWLINE\[NEWLINE M\biggl|\sum_{k=1}^{i_n}a_n^{kr}\biggr|\leq \|D_n\|\leq m^r \sum_{s=0}^r\sum _{k=1}^{i_n} |a_n^{ks}|NEWLINE\]NEWLINE the author proves that if the sequences \((\sum_{k=1}^{i_n}|a_n^{ks}|), 0\leq s\leq r, \) are bounded then the sequence \((D_n)\) is pointwise convergent to a continuous linear functional \(A\) on \(C^r[a,b]\). If the sequence \((\sum_{k=1}^{i_n}|a_n^{kr}|)\) is unbounded then \(\limsup |D_n(f)|= \infty\), for every \(f\) in a superdense subset of \(C^r[a,b]\) [see \textit{I. Muntean} and \textit{S. Cobzaş}, J. Approximation Theory 31, 138-153 (1981; Zbl 0478.41030]. Taking \(A(f) = \int_a^b f(x) dx, \) one obtains a result of \textit{H. Bandemer} [Wiss. Z. Tech. Hochsch. Karl-Marx-Stadt 8, 205-208 (1966; Zbl 0171.37104)].NEWLINENEWLINEFor the entire collection see [Zbl 0968.00041].