On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix ${\mathcal L}_n^{(\alpha_n,\beta_n)}$ of Jacobi nodes

2020 ◽  
Vol 84 (6) ◽  
pp. 1224-1249
Author(s):  
A. Yu. Trynin
Keyword(s):  
Bounded Variation ◽  
Uniform Approximation ◽  
Lagrange Interpolation ◽  
Functions Of Bounded Variation ◽  
Approximation Of Functions ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials

scholarly journals Bivariate uniform approximation via bivariate Lagrange interpolation polynomials

2014 ◽  
Vol 23 (1) ◽  
pp. 7-13
Author(s):  
DAN BARBOSU ◽  
◽  
OVIDIU T. POP ◽  
Keyword(s):  
Uniform Convergence ◽  
Uniform Approximation ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Formula ◽  
Interpolation Theory ◽  
Science Publisher ◽  
Lagrange Interpolation Formula ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials

In the present note, we extend some univariate uniform approximation results by means of Lagrange interpolating polynomials [Ivan, M., Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca (2004)] to the bivariate case. It is well known that generally, in the univariate case, the sequence of Lagrange interpolation polynomials does’t converges to the approximated function. This fact was first observed by G. Faber (see [9]), which constructed an example when the sequence of Lagrange interpolation polynomials diverges. The result of G. Faber was more generalized by I. Muntean (see [12]). M. Ivan established first sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials associated to a univariate real valued function. First, we represent the remainder term of bivariate Lagrange interpolation formula in terms of bivariate divided difference. Using this representation we establish sufficient conditions for the uniform convergence of the sequence of bivariate Lagrange interpolation polynomials to the approximated function.

scholarly journals On Lagrange interpolation with equally spaced nodes

2000 ◽  
Vol 62 (3) ◽  
pp. 357-368 ◽  
Author(s):  
Michael Revers
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
End Points ◽  
Quantitative Version ◽  
Equidistant Nodes ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials ◽  
Exact Rate Of Convergence

A well-known result due to S.N. Bernstein is that sequence of Lagrange interpolation polynomials for |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In this paper we present a quantitative version concerning the divergence behaviour of the Lagrange interpolants for |x|3 at equidistant nodes. Furthermore, we present the exact rate of convergence for the interpolatory parabolas at the point zero.

Approximation of functions of bounded variation by integrated Meyer-König and Zeller operators of finite type

2012 ◽  
Vol 49 (2) ◽  
pp. 254-268
Author(s):  
Tiberiu Trif
Keyword(s):  
Rate Of Convergence ◽  
Finite Type ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Approximation Of Functions ◽  
Durrmeyer Operators

I. Gavrea and T. Trif [Rend. Circ. Mat. Palermo (2) Suppl. 76 (2005), 375–394] introduced a class of Meyer-König-Zeller-Durrmeyer operators “of finite type” and investigated the rate of convergence of these operators for continuous functions. In the present paper we study the approximation of functions of bounded variation by means of these operators.

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