The Lebesgue Function and Lebesgue Constant of Lagrange Interpolation for Erdoős Weights

Boundedness and Uniform Numerical Approximation of the Weighted Hilbert Transform on the Real Line

“…Weaker versions of this result have been proved in [5,36] and L p analogues can be found for example in [8]. …”

“…The quadrature formulae we investigate are so-called interpolatory product methods, given by the following approach: For a given admissible weight w, we denote by and denote by x n;n ðw 2 Þ < x nÀ1;n ðw 2 Þ < Á Á Á < x 2;n ðw 2 Þ < x 1;n ðw 2 Þ their n real simple zeros. Moreover, following [36,5], we define x 0;n to be a point with the property that jp n ðx 0;n Þwðx 0;n Þj ¼ kp n wk L 1 ðIÞ and set x nþ1;n ¼ Àx 0;n : Here and in the following, we shall often use the abbreviated notation p n for p n ðw 2 ; ÁÞ: It is instructive to note that in [5,36], it was shown that AEx 0;n are close to the largest and smallest zeros of p n ðw 2 ; ÁÞ:…”

Interpolatory product quadratures for Cauchy principal value integrals with Freud weights

“…Looking at the second part of the sum, we consider several cases. We may assume, much as in [3] that x ≥ 0.…”

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

“…The point of this is that in many of the classical situations where mean convergence of Lagrange interpolation has been studied, notably at the zeros of orthogonal polynomials for the many generalisations of Jacobi weights [10], [11], [13], [16], [18] or the exponential weights on [ 1; 1] [7] or the Freud or Erdös weights on R [1], [8] all the information that is needed to estimate is readily available: namely lower bounds on 0 n (t j ), upper and lower bounds on j , and upper bounds on n and `j. For those weights, one may scale the interpolation points to [ 1; 1] and may choose L = 1 and (1.7) will be satis…ed so all one needs to check is whether is bounded independently of n and whether (1.17) is satis…ed with appropriate : This should in principle allow one to unify a wide range of results.…”