Geodesic
Function correction and Lagrange -- Jacobi type interpolation
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 1, pp. 24-35.

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It is well-known that the Lagrange interpolation 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 an arbitrarily small measure such that its Fourier series will be uniformly convergent. The question arises whether the class of continuous functions has a similar property with respect to any interpolation process. In the present paper, we prove that there exists the matrix of nodes $\mathfrak{M}_\gamma$ arbitrarily close to the Jacoby matrix $\mathfrak{M}^{(\alpha,\beta)}$, $\alpha,\beta>-1$ with the following property: any function $f\in{C[-1,1]}$ can be adjusted in a set of an arbitrarily small measure such that interpolation process of adjusted continuous function $g$ based on the nodes $\mathfrak{M}_\gamma$ will be uniformly convergent to $g$ on $[a,b]\subset(-1,1)$. 
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V. V. Novikov. Function correction and Lagrange -- Jacobi type interpolation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 1, pp. 24-35. http://geodesic.mathdoc.fr/item/ISU_2023_23_1_a2/

[1] Grünwald G., “Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome Stetiger Funktionen”, Annals of Mathematics, 37:4 (1936), 908–918 (in German) | DOI | MR

[2] Marcinkiewicz J., “Sur la divergence des polynomes d'interpolation”, Acta litterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae : Sectio scientiarum mathematicarum, 8 (1937), 131–135 (in French)

[3] Erdős P., Vértesi P., “On the almost everywhere divergence of Lagrange interpolatory polynomials for arbitrary system of nodes”, Acta Mathematica Academiae Scientiarum Hungaricae, 36:1–2 (1980), 71–89 | DOI | MR

[4] Menchoff D., “Sur les séries de Fourier des fonctions continues”, Recueil Mathématique (Nouvelle série), 8(50):3 (1940) (in French) | MR

[5] Bary N. K., A Treatise on Trigonometric Series, v. 2, Pergamon Press, Oxford–New-York, 1964, 508 pp. | MR

[6] Natanson G. I., “Two-sided estimate for the Lebesgue function of the Lagrange interpolation process with Jacobi nodes”, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1967, no. 11, 67–74 (in Russian)

[7] Privalov A. A., “A criterion for uniform convergence of Lagrange interpolation processes”, Soviet Math. (Iz. VUZ), 30:5 (1986), 65–77 | MR

[8] Nevai G. P., “Notes on interpolation”, Acta Mathematica Academiae Scientiarum Hungaricae, 25:1–2 (1974), 123–144 (in Russian) | DOI | MR

[9] Szegő G., Orthogonal Polynomials, AMS, Providence, Rhode Island, 1939, 440 pp. | MR

[10] Novikov V. V., “Adjustment of functions and Lagrange interpolation based on the nodes close to the Jacobi nodes”, Contemporary Problems of Function Theory and Their Applications, Nauchnaya kniga, Saratov, 2020, 277–280 (in Russian)