Lebesgue function type sums of Hermite interpolations (Q1333029)

Let \(X= \{x_{kn}\}\) be an infinite triangular interpolatory matrix, \(- 1\leq x_{nn}< x_{n-1,n}<\cdots< x_{1n}\leq 1\). For \(m\geq 1\) let \(h_{tknm}(X, x)\) be the polynomials of degree \(mn- 1\) verifying \[ h^{(p)}_{tknm} (X, x_{qn})= \delta_{tp} \delta_{tq},\quad t,p= 0,\dots, m-1,\quad k,q= 1,\dots, n \] (\(\delta\) is the Kronecker delta). The Hermite interpolatory polynomials are defined by \[ H_{nm}(f, X, x)= \sum^ n_{k= 1} f(x_{kn})\;h_{oknm}(X, x) \] and verify \[ H^{(t)}_{nm}(f, x_ k)= \delta_{ot} f(x_{kn}),\quad 1\leq k\leq n,\quad 0\leq t\leq m-1. \] \(H_{n1}\) are the classical Lagrange interpolation polynomials and \(H_{n2}\) are the classical Hermite- Fejér interpolation polynomials. The main result of this paper (Theorem 2.1) asserts that for arbitrary fixed \(X\), \(m\) and \(t\), \(0\leq t\leq m-1\), \(m-t\) odd, there exists a constant \(c= c(m,t)> 0\) such that for every \(\varepsilon> 0\) there exists sets \(H_ n= H(\varepsilon, X, m,t)\) with \(| H_ n|\leq \varepsilon\) such that \(\lambda_{tnm}(X, x)> \eta\cdot n^{-t} \log n\), for all \(x\in R\backslash H_ n\), where \(\eta= \eta(\varepsilon, m,t)> 0\) and \(\lambda_{tnm}(X, x)= \sum^ n_{k= 1} | h_{tknm}(X,x)|\). This delimitation is important in proving a.e. divergence of \(H_{nm}(f, x)\), for \(m\) odd, and for the study of mean convergence. Some open problems are posed.