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On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

D.S. Lubinsky
2002 Journal of Approximation Theory  
THE RESULT 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  ...  51: ð2Þ 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  ... 
doi:10.1006/jath.2002.3698 fatcat:hkxpqhapqrevhnvpzggxgspjl4

On Mean Convergence of Lagrange Interpolation for General Arrays

D.S. Lubinsky
2000 Journal of Approximation Theory  
For n 1, let fxjng n j=1 be n distinct points in a compact set K  ...  Acknowledgement The author would like to thank Paul Nevai for helping to improve the formulation of the result of this manuscript.  ...  Most of the work dealing with mean convergence of Lagrange interpolation for general arrays involves necessary conditions [6] , [9] , since su cient conditions are hard to come by.  ... 
doi:10.1006/jath.1999.3448 fatcat:3z2mmuajhzcbfi4cz7p766fwlu

Marcinkiewicz–Zygmund inequalities and the numerical approximation of singular integrals for exponential weights: methods, results and open problems, some new, some old

S.B. Damelin
2003 Journal of Complexity  
In this short article, we explore some methods, results and open problems dealing with weighted Marcinkiewicz-Zygmund inequalities as well as the numerical approximation of integrals for exponential weights  ...  on the real line and on finite intervals of the line.  ...  Acknowledgments The author thanks the organizers of this workshop for the invitation to write this article and for the excellent conference.  ... 
doi:10.1016/s0885-064x(03)00027-x fatcat:lelo5gwpgfelthtzolorfsjwhi

Necessary conditions for weighted mean convergence of Lagrange interpolation for exponential weights

S.B. Damelin, H.S. Jung, K.H. Kwon
2001 Journal of Computational and Applied Mathematics  
Given a continuous real-valued function f which vanishes outside a ÿxed ÿnite interval, we establish necessary conditions for weighted mean convergence of Lagrange interpolation for a general class of  ...  even weights w which are of exponential decay on the real line or at the endpoints of (−1; 1). c  ...  Acknowledgements The ÿrst author was partially funded by the Centre for Applicable Analysis and Number Theory.  ... 
doi:10.1016/s0377-0427(00)00439-8 fatcat:vl2d5qxesjcyzngx4xygdccxxi

ON MEAN CONVERGENCE OF TRIGONOMETRIC INTERPOLANTS, AND THEIR UNIT CIRCLE ANALOGUES, FOR GENERAL ARRAYS

D.S. Lubinsky
2002 Analysis: International Mathematical Journal of Analysis and its Applications  
We also examine Lagrange interpolation on the unit circle. The results are analogues of our earlier ones for Lagrange interpolation on a real interval.  ...  Let X be a triangular array of interpolation points in a compact subset of [0; 2 ].  ...  Most positive results on mean convergence of Lagrange interpolation are closely linked to zeros of orthogonal polynomials, and are somewhat technical -see [6] , [8] , [12] , [13] .  ... 
doi:10.1524/anly.2002.22.1.97 fatcat:hsu2adbryfeexifjbryl2va5d4

Page 6046 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
The “good” nodes of weighted Lagrange interpolation on J := (—oo, 00) or (—1,1) are considered here.  ...  (H-EOTVO-NA; Budapest) On the “good” nodes of weighted Lagrange interpolation for exponential weights. Ann. Univ. Sci. Budapest. Eétvés Sect. Math. 44 (2001), 33-38 (2002).  ... 

Page 4953 of Mathematical Reviews Vol. , Issue 83m [page]

1983 Mathematical Reviews  
(*) for nd he ity ral er si) ns 2) ), Px 41 oR re 4953 APPROXIMATIONS AND EXPANSIONS summability of Cesaro C* with a>0 or the method (R,p,) of the weighted means of Riesz with p)+ --- +p,,0.  ...  Furthermore, a representation is given by means of finite differences of the function f and a theorem is established on the uniform convergence of sequences of such polynomials, assuming that / is a continuous  ... 

Page 3125 of Mathematical Reviews Vol. , Issue 2001E [page]

2001 Mathematical Reviews  
(SA-WITW-CAN; Johannesburg) On mean convergence of Lagrange interpolation for general arrays. (English summary) J. Approx. Theory 104 (2000), no. 2, 220-225.  ...  being both Nérlund methods and methods of weighted means.  ... 

On the theorem of Géza Grünwald and Józef Marcinkiewicz

László Szili, Péter Vértesi
2011 Banach Center Publications  
Try to get the G-M theorem for X (α,β) . B. Find those arrays X for which G-M theorem holds true. C. Extend the G-M theorem on infinite intervals using weighted norm. D.  ...  Actually, this theorem clearly shows the limitations of Lagrange interpolation even on these nodes.  ... 
doi:10.4064/bc95-0-12 fatcat:qpsizagogjbz7ozldrll4kmrwe

Page 6833 of Mathematical Reviews Vol. , Issue 2003i [page]

2003 Mathematical Reviews  
{For the entire collection see MR 2003e:41001.} 2003i:41004 41A10 41A25 Lubinsky, D.S. (1-GAIT; Atlanta, GA) On weighted mean convergence of Lagrange interpolation for general arrays.  ...  We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.” J.  ... 

Polynomial interpolation and approximation in Cd

T. Bloom, L. P. Bos, J.-P. Calvi, N. Levenberg
2012 Annales Polonici Mathematici  
Levenberg, Polynomial interpolation of holomorphic functions in C and C n , Rocky Mountain J. Math. 22 (1992), 441-470.  ...  We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N.  ...  The research of T. Bloom was supported in part by an NSERC of Canada grant. Polynomial interpolation and approximation in C d  ... 
doi:10.4064/ap106-0-5 fatcat:dapntr3ycnd45oqklriysgyxxq

On the Lebesgue Constant of Weighted Leja Points for Lagrange Interpolation on Unbounded Domains [article]

Peter Jantsch, Clayton G. Webster, Guannan Zhang
2017 arXiv   pre-print
This work focuses on weighted Lagrange interpolation on an unbounded domain, and analyzes the Lebesgue constant for a sequence of weighted Leja points.  ...  of interpolation nodes.  ...  In this work, we considered the properties of Leja points for weighted Lagrange interpolation on an unbounded domain.  ... 
arXiv:1606.07093v2 fatcat:lhsbe7bdyjd3td66srnmo42iu4

Equidistribution of Fekete Points on the Sphere

Jordi Marzo, Joaquim Ortega-Cerdà
2009 Constructive approximation  
They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere.  ...  The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently.  ...  We consider arrays of points on the sphere S d that determine the norm of polynomials, and also arrays of points where we are free to interpolate arbitrary values by polynomials, i.e. interpolating arrays  ... 
doi:10.1007/s00365-009-9051-5 fatcat:djzunlruavadvpwrdzu3sew2ky

Polynomial Interpolation and Approximation in C^d [article]

T. Bloom, L. P. Bos, J.-P. Calvi, N. Levenberg
2011 arXiv   pre-print
We update the state of the subject approximately 20 years after the publication of a previous article on this topic.  ...  This report is mostly a survey, with a sprinkling of assorted new results throughout.  ...  properties exists then it should come as a natural multivariate generalization of one of the many available expressions of univariate Lagrange-Hermite interpolation.  ... 
arXiv:1111.6418v1 fatcat:245vzlh7fbayxifwsgjq5ba6qi

MWRtools: a library for weighted residual method calculations

Yi-hung Lin, Hsiao-Yung Chang, Raymond A. Adomaitis
1999 Computers and Chemical Engineering  
We present a set of MATLAB functions developed for solving boundary-value problems using globally defined trial function expansions and weighted residual methods (MWR).  ...  This paper presents the results of our initial efforts to create computational modules that have a one-to-one correspondence with each step of implementing eigenfunction expansion, Galerkin projection,  ...  The differentiation arrays are based on the Lagrange interpolation polynomial with the given M points.  ... 
doi:10.1016/s0098-1354(99)00272-0 fatcat:fq6iirh2nfh2pmyizzed56ygui
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