ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

#### For citation:

Novikov V. V. Adjustment of Functions and Lagrange Interpolation Based on the Nodes Close to the Legendre Nodes. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2017, vol. 17, iss. 4, pp. 394-401. DOI: 10.18500/1816-9791-2017-17-4-394-401

Published online:
28.11.2017
Full text:
Language:
Russian
UDC:
517.51

# Adjustment of Functions and Lagrange Interpolation Based on the Nodes Close to the Legendre Nodes

Autors:
Novikov Vladimir Vasil’evich, Engels' Technology Institute, Branch of Saratov State Technical University
Abstract:

It is well known that the Lagrange interpolation of a continuous function based on the Chebyshev nodes may be divergent everywhere (for arbitrary nodes, almost everywhere) like the Fourier series of a summable function. On the other hand any measurable almost everywhere finite function can be “adjusted” in a set of arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises: does the class of continuous functions have a similar property with respect to any interpolation process? In the present paper we prove that there exists a matrix of nodes M γ arbitrarily close to the Legendre matrix with the following property: any function f ∈ C[−1,1] can be adjusted in a set of arbitrarily small measure such that the interpolation process of adjusted continuous function g based on the nodes M γ will be uniformly convergent to g on [a,b] ⊂ (−1,1).

Key words:
References:
1. Grünwald G. Uber Divergenzerscheinungen der Lagrangeschen Interpolationspolynome Stetiger Funktionen. Ann. Math., 1936, vol. 37, pp. 908–918.
2. Marcinkiewicz J. Sur la divergence des polynomes d’interpolation. Acta Litt. Sci. Szeged, 1936/37, vol. 8, pp. 131–135.
3. Erd˝ os P., Vertesi P. On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary system of nodes. Acta. Math. Acad. Sci. Hungar., 1980, vol. 36, iss. 1–2. pp. 71–89.
4. Menchoff D. Sur les séries de Fourier des fonctions continues. Rec. Math. (N.S), 1940, vol. 8(50), no. 3, pp. 493–518.
5. Bary N. K. A treatise on trigonometric series. Oxford, New York, Pergamon Press, 1964, vol. 1, 533 p.; vol. 2, 508 p. (Russ. ed. : Moscow, Fizmatlit, 1961. 936 p.)
6. Novikov V. V. Interpolyaciya tipa Lagranzha–Yakobi i analog usilennogo C-svojstva [In- terpolation of the Lagrange–Jacobi type and an analogue of the strengthened C-property]. Matematika. Mehanika [Mathematics. Mechanics]. Saratov, Saratov Univ. Press, 2007, iss. 9, pp. 66–68 (in Russian).
7. Nevai G. P. Zamechanija ob interpolirovanii [Remarks on interpolation]. Acta Math. Acad. Sci. Hungar., 1974, vol. 25, iss. 1–2, pp. 123–144 (in Russian).
8. Novikov V. V. A Criterion for Uniform Convergence of the Lagrange–Jacobi Interpolation Process. Math. Notes, 2006, vol. 79, no. 1, pp. 232–243. DOI: https://doi.org/10.18500/1816-9791-2015-15-4-418-422.
9. Privalov A. A. A criterion for uniform convergence of Lagrange interpolation processes. Soviet Math. (Iz. VUZ), 1986, vol. 30, no. 5, pp. 65–77.
10. Szeg˝ o G. Orthogonal Polynomials. Providence, Rhode Island, AMS, 1939. 440 p. (Russ.ed. : Moscow, Fizmatlit, 1962. 500 p.)
Short text (in English):