Polynomial interpolation, an \(L\)-function, and pointwise approximation of continuous functions
Let \(\mathcal{P}_n\) be the set of algebraic polynomials of degree at most \(n\). A function \(f\in C[-1,1]\) satisfies the pointwise rapid convergence property (PRCP) if there exists a sequence of polynomials \(Q_n(f,x)\in \mathcal{P}_n\) such that \(\|f-Q_n\|_{C[-1,1]}\leq C(f)E_n(f)\), \(n\in \mathbb{N}\), and for every \(x\in [-1,1]\), \(\lim_{n\to \infty}|f(x)-Q_n(f,x)|/E_n(f)=0\), where \(E_n(f)=\min\{\|f-p\|_{C[-1,1]},\;p\in \mathcal{P}_n\}\). The paper is mainly devoted to find a class \(M\) of continuous functions which are not piecewise analytic and fulfill the PRCP. Such a problem has been technically linked to the asymptotical behavior of \(\Delta_{s,n}=f_s(x)-L_{2n}(f,x)\), as \(n\to \infty\), where \(L_{2n}\in \mathcal{P}_{2n}\) is the Lagrange polynomial interpolation to the function \(f_s(x)=|x|^s\), \(Re\,s>0\), \(s\neq 2,4,\dots\), at the Chebyshev nodes. The exponents are chosen as the zeros \(\{s_k\}_{k=1}^\infty\) of the \(L\)-function \(L(s,\chi)=\sum_{k=0}^\infty (-1)^k (2k+1)^{-s}\), with the character \(\chi\) modulo 4, to obtain that \(M=span\{g_k\}\) satisfies the PRCP, where \(g_k\) is one of the functions \(f_{s_k}\), \(Re\,f_{s_k}\), \(Im\,f_{s_k}\). A new representation for \(|L(s,\chi)|\), for complex \(s\) is obtained in terms of \(|\Delta_{s,n}|\), where \(x\neq 0\) and \(s\neq 2,4,\dots\).
scientific article; zbMATH DE number 5306484
scientific article