A survey on mean convergence of interpolatory processes

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

“…While there are very many results on mean convergence of Lagrange interpolation, the vast majority of these results deal with interpolation at zeros of orthogonal polynomials and their close cousins}at least in terms of sufficient conditions for mean convergence}see [3,5,6,9]. In a recent paper [2], the author used distribution functions to treat general interpolation arrays contained in a compact set.…”

“…The essential feature is that a single condition, namely (2), is sufficient for mean convergence of Lagrange interpolation in L p for at least one p > 0: This should be compared to results surveyed in [3,5,6,9], where amongst other things, the interpolation points are assumed to be zeros of orthogonal polynomials associated with weights satisfying a number of conditions. The price one pays for the simplicity of (2) is that invariably p51 or even p5 1 2 ; and p and r are different in (I) and (II).…”

On mean convergence of Hermite-Fejér interpolation

“…The interested reader may see the survey article by ]. Szabados and P. V~rtesi [10] or book [9]. Recently A. g. Varma and J. Prasad [13] studied the rate of mean convergence of Hermite-Fej6r interpolation.…”

Mean convergence of hermite interpolation revisited

“…[1,7,9,[12][13][14][15] and the references therein]). However, most of these papers concentrated on the sufficient conditions for the convergence, except perhaps [6,7] where both necessary and sufficient conditions are established for Lagrange interpolation and Hermite interpolation of the second order, respectively, and [13] where conditions that are almost necessary and sufficient are provided for the Hermite interpolation of higher order.…”