Treffer: Necessary conditions of convergence of Hermite–Fejér interpolation polynomials for exponential weights: Necessary conditions of convergence of Hermite-Fejér interpolation polynomials for exponential weights

Let \(w_Q\) be an admissible weight on \(I=(-a,a),\) \(a=1\) or \(a=\infty\) see \textit{H.S. Jung} [J. Approximation Theory 120, 217--241 (2003; Zbl 1061.41010)], let \(H_n(w_Q^2;\cdot)\) be the Hermite-Fejér interpolation operator with respect to the zeros of orthogonal polynomials \(p_n(w_Q^2;x)\) with respect to \(w_Q^2.\) Among main results of the paper there are Theorems 2.3, 2.4: Theorem 2.3: if \(v:I\to \mathbb R^+\) is a measurable function satisfying the conditions \[ \lim_{x\to a} xv(x)Q^{-2/3}(x)/\log(1+| x| )=b, \tag{1} \] and \(b=\infty\) then there exists a continuous function \(f:I\to\mathbb R\) such that \[ \lim_{| x| \to a}| f(x)w_Q^2(x)Q(x)\log(1+| x| )| =0, \tag{2} \] and \[ \limsup_{n\to\infty}\| H_n(w_Q^2;f)w_Q^2v\|_{L_{\infty}(I)}=\infty. \] Theorem 2.4: if for every continuous function \(f\) defined on \(I\) satisfying (2) it holds that \[ \lim_{n\to\infty}\|(f-H_n(w_Q^2;f))w_Q^2v\|_{L_{\infty}(I)}=0 \] then it is necessary that (1) holds with \(b