Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights

1996 ◽  
Vol 48 (4) ◽  
pp. 710-736 ◽  
Author(s):  
S. B. Damelin ◽  
D. S. Lubinsky
Keyword(s):  
Continuous Function ◽  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Polynomial ◽  
Necessary And Sufficient Conditions ◽  
Lagrange Interpolation Polynomial ◽  
Mean Convergence ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdös weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < ∞, Δ ∊ ℝ, k > 0. Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ: ℝ —> ℝ satisfyingit is necessary and sufficient that

Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights II

1996 ◽  
Vol 48 (4) ◽  
pp. 737-757 ◽  
Author(s):  
S. B. Damelin ◽  
D. S. Lubinsky
Keyword(s):  
Continuous Function ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Polynomial ◽  
Weighting Factor ◽  
Lagrange Interpolation Polynomial ◽  
Mean Convergence ◽  
Turán Theorem ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdős weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ:ℝ. —> ℝ satisfyingit is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.

Mean Convergence of Lagrange Interpolation for Exponential weights on [-1, 1]

1998 ◽  
Vol 50 (6) ◽  
pp. 1273-1297 ◽  
Author(s):  
D. S. Lubinsky
Keyword(s):  
Orthogonal Polynomials ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exponential Weights ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
Necessary And Sufficient

AbstractWe obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on [-1, 1], such asw(x) = exp(-(1 - x2)-α), α > 0orw(x) = exp(-expk(1 - x2)-α), k≥1, α > 0,where expk = exp(exp(. . . exp( ) . . .)) denotes the k-th iterated exponential.

On multi-dimensional annealing problems

1989 ◽  
Vol 105 (1) ◽  
pp. 177-184 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Deterministic Function ◽  
Image Position ◽  
Necessary And Sufficient

In [1] Chan and Williams considered a one-dimensional diffusion of the formwhere F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.

scholarly journals Mean convergence of Grünwald interpolation operators

2003 ◽  
Vol 2003 (33) ◽  
pp. 2083-2095
Author(s):  
Zhixiong Chen
Keyword(s):  
Orthogonal Polynomials ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Mean Convergence ◽  
Zeros Of Orthogonal Polynomials ◽  
General Weight ◽  
Interpolation Operators ◽  
Necessary And Sufficient

We investigate weightedLpmean convergence of Grünwald interpolation operators based on the zeros of orthogonal polynomials with respect to a general weight and generalizedJacobiweights. We give necessary and sufficient conditions for such convergence for all continuous functions.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

On the Hausdorff and Trigonometric Moment Problems

1961 ◽  
Vol 13 ◽  
pp. 454-461
Author(s):  
P. G. Rooney
Keyword(s):  
Moment Problem ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Functions Of Bounded Variation ◽  
Moment Problems ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that(1)For various collections K this problem has been solved—see (3, Chapter III)By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that(2)For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

Coefficient Regions for Univalent Trinomials

1980 ◽  
Vol 32 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Q. I. Rahman ◽  
J. Waniurski
Keyword(s):  
Unit Disk ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient

The problem of determining necessary and sufficient conditions bearing upon the numbers a2 and a3 in order that the polynomial z + a2z2 + a3z3 be univalent in the unit disk |z| < 1 was solved by Brannan ([3], [4]) and by Cowling and Royster [6], at about the same time. For his investigation Brannan used the following result due to Dieudonné [7] and the well-known Cohn rule [9].THEOREM A (Dieudonné criterion). The polynomial1is univalent in |z| < 1 if and only if for every Θ in [0, π/2] the associated polynomial2does not vanish in |z| < 1. For Θ = 0, (2) is to be interpreted as the derivative of (1).The procedure of Cowling and Royster was based on the observation that is univalent in |z| < 1 if and only if for all α such that 0 ≧ |α| ≧ 1, α ≠ 1 the functionis regular in the unit disk.

Traces of Matrices of Zeros and Ones

1960 ◽  
Vol 12 ◽  
pp. 463-476 ◽  
Author(s):  
H. J. Ryser
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Critical Survey ◽  
Combinatorial Properties ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Sum Vector

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj. We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class1.1consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).

Unique Extension and Product Measures

1967 ◽  
Vol 19 ◽  
pp. 757-763 ◽  
Author(s):  
Norman Y. Luther
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Necessary And Sufficient Conditions ◽  
Product Measure ◽  
Unique Extension ◽  
Product Measures ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Finite Measures

Following (2) we say that a measure μ on a ring is semifinite ifClearly every σ-finite measure is semifinite, but the converse fails.In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.

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