Mean convergence of extended Lagrange interpolation for exponential weights (Q1810775)

As the authors note, in this paper they complete their investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the real line. The authors call these weights strongly admissible. The authors obtain some sufficient and necessary conditions for the convergence of the Lagrange interpolation polynomial to the approximating function in weighted \(L^p\left({\mathbb R}\right)\)-norm. The article also contains an extended summary of recent related work on \({\mathbb R}\) and \(\left[-1,1\right]\) by the authors as well as Szabados, Vértesi, Lubinsky and Matjila. The authors emphasize several important fundamental ideas, applied in the proofs, that were developed by Erdös, Turán, Askey, Freud, Nevai, Szabados, Vértesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle. These ideas and methods are clearly described in the article during the proofs of the results. This style is much more convenient than just references to the results necessary. An extended bibliography reflects several results obtained in this area. The article may be very useful for specialists in interpolation and approximation theory as well as in numerical analysis.