Convergence of modified Lagrange interpolation to \(L_p\)-functions based on the zeros of orthonormal polynomials with Freud weights
Let \(\alpha >0\), \(1<p< \infty\), \(W_Q (x)\) be the Freud weight, and let \(L^*(f) \) denote a modified Lagrange interpolation polynomial to a measurable function \(f\) satisfying \[ {\underset x\in \mathbb{R} {\text{ess sup}}} \bigl| f(x)\bigr | W_Q(x) \bigl(1 + | x| \bigr)^\alpha <\infty. \] The author proves that then with a suitable constant \(\Delta= \Delta (p, \alpha)\) \[ \lim_{n \to\infty} \int^\infty_{-\infty} \biggl[ \bigl| f(x)- L^*_n(f;x) \bigr| W_Q(x) \bigl(1+ | x| \bigr)^{-\Delta} \biggr]^p dx=0. \]
scientific article; zbMATH DE number 1197064