Geodesic
On operators of interpolation with respect to solutions of a~Cauchy problem and Lagrange--Jacobi polynomials
Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1215-1248.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe classes of continuous functions for which one has pointwise and uniform convergence of certain Lagrange-type operators (constructed from solutions of a Cauchy problem) and the Lagrange–Jacobi interpolation polynomials ${\mathcal L}_n^{(\alpha_{n},\beta_{n})}(F,\cos\theta)$. We also obtain sufficient conditions for the equiconvergence of these interpolation processes.
Keywords: Lagrange operators, sampling theorem, theory of approximation of functions.
Mots-clés : interpolation processes 
@article{IM2_2011_75_6_a5,
     author = {A. Yu. Trynin},
     title = {On operators of interpolation with respect to solutions of {a~Cauchy} problem and {Lagrange--Jacobi} polynomials},
     journal = {Izvestiya. Mathematics },
     pages = {1215--1248},
     publisher = {mathdoc},
     volume = {75},
     number = {6},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a5/}
}
TY  - JOUR
AU  - A. Yu. Trynin
TI  - On operators of interpolation with respect to solutions of a~Cauchy problem and Lagrange--Jacobi polynomials
JO  - Izvestiya. Mathematics 
PY  - 2011
SP  - 1215
EP  - 1248
VL  - 75
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a5/
LA  - en
ID  - IM2_2011_75_6_a5
ER  - 
%0 Journal Article
%A A. Yu. Trynin
%T On operators of interpolation with respect to solutions of a~Cauchy problem and Lagrange--Jacobi polynomials
%J Izvestiya. Mathematics 
%D 2011
%P 1215-1248
%V 75
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a5/
%G en
%F IM2_2011_75_6_a5
A. Yu. Trynin. On operators of interpolation with respect to solutions of a~Cauchy problem and Lagrange--Jacobi polynomials. Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1215-1248. http://geodesic.mathdoc.fr/item/IM2_2011_75_6_a5/

[1] A. Yu. Trynin, “A generalization of the Whittaker–Kotel'nikov–Shannon sampling theorem for continuous functions on a closed interval”, Sb. Math., 200:11 (2009), 1633–1679 | DOI | MR | Zbl

[2] A. Yu. Trynin, “Tests for pointwise and uniform convergence of sinc approximations of continuous functions on a closed interval”, Sb. Math., 198:10 (2007), 1517–1534 | DOI | MR | Zbl

[3] A. Yu. Trynin, “Estimates for the Lebesgue functions and the Nevai formula for the $sinc$-approximations of continuous functions on an interval”, Siberian Math. J., 48:5 (2007), 929–938 | DOI | MR | Zbl

[4] A. Yu. Trynin, “A criterion for the uniform convergence of sinc-approximations on a segment”, Russian Math. (Iz. VUZ), 52:6 (2008), 58–69 | DOI | MR | Zbl

[5] P. K. Suetin, Klassicheskie ortogonalnye mnogochleny, Fizmatlit, M., 1976 | MR | Zbl

[6] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, RI, 1959 | MR | Zbl | Zbl

[7] Ya. L. Geronimus, “O skhodimosti interpolyatsionnogo protsessa Lagranzha s uzlami v kornyakh ortogonalnykh mnogochlenov”, Izv. AN SSSR. Ser. matem., 27:3 (1963), 529–560 | MR | Zbl

[8] S. A. Agakhanov, “Otsenka funktsii Lebega dlya interpolyatsionnogo protsessa po kornyam polinomov Yakobi”, Izv. vuzov. Matem., 1967, no. 11, 3–6 | MR | Zbl

[9] G. I. Natanson, “Dvustoronnyaya otsenka funktsii Lebega interpolyatsionnogo protsessa Lagranzha s uzlami Yakobi”, Izv. vuzov. Matem., 1967, no. 11, 67–74 | MR | Zbl

[10] D. L. Berman, “Printsip monotonii v teorii interpolyatsii funktsii deistvitelnogo peremennogo”, Izv. vuzov. Matem., 1972, no. 4, 10–17 | MR | Zbl

[11] G. P. Nevai, “Zamechaniya ob interpolirovanii”, Acta Math. Acad. Sci. Hungar., 25:1–2 (1974), 123–144 | DOI | MR | Zbl

[12] A. A. Kel'zon, “Interpolation of functions with bounded $p$-variation”, Russian Math. (Iz. VUZ), 22:5 (1978), 99–102 | MR | Zbl

[13] S. S. Pilipčuk, “Tests for the convergence of interpolation processes”, Russian Math. (Iz. VUZ), 23:12 (1979), 41–46 | MR | Zbl

[14] S. S. Pilipčuk, “Divergence of Lagrange interpolation processes on sets of second category”, Russian Math. (Iz. VUZ), 23:3 (1979), 30–36 | MR | Zbl

[15] S. S. Pilipchuk, “O raskhodimosti interpolyatsionnykh protsessov Lagranzha na schetnykh mnozhestvakh”, Izv. vuzov. Matem., 1980, no. 12, 38–44 | MR | Zbl

[16] A. A. Privalov, “The divergence of the Lagrange interpolation processes with respect to Jacobi nodes on a set of positive measure”, Siberian Math. J., 17:4 (1976), 630–648 | DOI | MR | Zbl | Zbl

[17] V. P. Sklyarov, Pryamye i obratnye teoremy teorii priblizhenii dlya nailuchshikh priblizhenii s vesom $(1-x)^\alpha(1+x)^\beta$, dep. v VINITI 5160-81

[18] A. A. Privalov, “Uniform convergence criteria for Lagrange interpolation processes”, Russian Math. (Iz. VUZ), 30:5 (1986), 65–77 | MR | Zbl | Zbl

[19] Ph. Hartman, Ordinary differential equations, Wiley, New York–London–Sydney, 1964 | MR | MR | Zbl | Zbl

[20] B. M. Levitan, I. S. Sargsjan, Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), 59, Kluwer Acad. Publ., Dordrecht, 1991 | MR | MR | Zbl | Zbl

[21] A. Yu. Trynin, “Absence of stability of interpolation with respect to eigenfunctions of the Sturm–Liouville problem”, Russian Math. (Iz. VUZ), 44:9 (2000), 58–71 | MR | Zbl

[22] I. P. Natanson, Constructive function theory, vol. I–III, Ungar Publ., New York, 1964–1965 | MR | MR | Zbl | Zbl