On Converse Marcinkiewicz—Zygmund Inequalities in L p ,p>1

Abstract

We obtain converse Marcinkiewicz-Zygmund inequalities such asfor polynomials P of degree n 1, under general conditions on the points ft j g n j=1 and on the function . The weights f j g n j=1 are appropriately chosen. We illustrate the results by applying them to extended Lagrange interpolation for exponential weights on [ 1; 1].

Marcinkiewicz–Zygmund inequalities

“…and this proves (12) by taking arbitrarily small. Observe that the polynomial p n (z) = (z n − 1)/(1 − z) has the property that |z−1|>R/n,|z|=1…”

“…If we put together (17) and (18), we find that for every ε there is an r such that #(Z(n) ∩ I (t + r)) (1 − ε)#{w j ∈ I (t)} = (1 − ε)t and this implies (12).…”

“…For the sake of completeness we give a characterization in Theorem 3 of the families of points that satisfy only the first inequality. Sequences that satisfy the second inequality with different weights have been studied in [12]. In our problem all points have equal weights.…”

On Mean Convergence of Lagrange Interpolation for General Arrays

“…Most of the work dealing with mean convergence of Lagrange interpolation for general arrays involves necessary conditions [6,9], since sufficient conditions are hard to come by. Some sufficient conditions for convergence of general arrays in L p , p>1, have been given in [3].…”

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

“…There is a vast literature dealing with necessary and sufficient conditions for weighted convergence of Lagrange interpolation for even Freud, Erdős, and exponential weights on (−1, 1). We refer the reader to [1][2][3][4][5][7][8][9]17,[19][20][21][22]24,26,[28][29][30][31] and the many references cited therein. Especially, some necessary conditions for weighted convergence of Lagrange interpolation with respect to these weights were given in [4,8,9,28,30].…”