The principle of localization at the class of functions integrable in the Riemann for the processes of Lagrange--Sturm--Liouville

A. Yu. Trynin ; E. D. Kireeva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 51-63.

Résumé

Let us say that the principle of localization holds at the class of functions $F$ at point $x_0 \in [0, \pi]$ for the Lagrange–Sturm–Liouville interpolation process $L_n^{SL}(f,x)$ if $\lim_{n \rightarrow \infty}\left|L_n^{SL}(f, x_0)-L_n^{SL}(g,x_0)\right|=0$ follows from the fact that the condition $f(x)=g(x)$ is met for any two functions f and g belonging to F in some neighborhood $O_\delta(x_0)$, $\delta>0$. It is proved that the principle of localization at the class of Riemann integrable functions holds for interpolation processes built on the eigenfunctions of the regular Sturm–Liouville problem with a continuous potential of bounded variation. It is established that the principle of localization at the class of continuous on the segment $[0, \pi]$ functions holds for interpolation processes built on the eigenfunctions of the regular Sturm–Liouville problem with an optional continuous potential of bounded variation. We consider the case of boundary conditions of the third kind, from which the boundary conditions of the first kind are removed. Approximative properties of Lagrange–Sturm–Liouville operators at point $x_0\in [0, \pi] $ in both cases depend solely on the values of the approximate function just in the neighborhood of this point $x_0\in [0, \pi]$.

Bibliographie

[1] Natanson G. I., “About one interpolation process”, Scientific notes of the Leningrad Pedagogical Institute, 166 (1958), 213–219 (in Russian)

[2] Kashin B. S., Saakyan A. A., Orthogonal series, Izd-vo AFTs, M., 1999, 550 pp. (in Russian)

[3] Novikov I. Ya., Stechkin S. B., “Basic wavelet theory”, Russian Math. Surveys, 53:6 (1998), 1159–1231 | DOI | DOI

[4] Stenger F., Numerical Methods Based on Sinc and Analytic Functions, Springer Ser. Comput. Math., 20, Springer-Verlag, N. Y., 1993, 565 pp. | DOI

[5] Dobeshi I., Ten Wavelet Lectures, NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevsk, 2001, 464 pp. (in Russian)

[6] Butzer P. L., “A retrospective on 60 years of approximation theory and associated fields”, Journal of Approximation Theory, 160:1–2 (2009), 3–18 | DOI

[7] Shmukler A. I., Shulman T. A., “Certain properties of Kotel'nikov series”, Soviet Math. (Iz. VUZ), 18:3 (1974), 81–90

[8] Livne O. E., Brandt A., “MuST: The Multilevel Sinc Transform”, SIAM J. Sci. Comput., 33:4 (2011), 1726–1738 | DOI

[9] Krivoshein A., Skopina M., “Multivariate sampling-type approximation”, Analysis and Applications, 15:4 (2017), 521–542 | DOI

[10] Kolomoitsev Yu., Skopina M., “Around Kotelnikov – Shannon formula”, 12th International Conference on Sampling Theory and Applications, SampTA 2017, IEEE, 2017, 279–282 | DOI

[11] Maleknejad K., Rostami Ya., Shahi Kalalagh H., “Numerical solution for first kind Fredholm integral equations by using sinc collocation method”, IJAPM, 6:3 (2016), 120–128 | DOI

[12] Belichenko K. V., Sobolev V. M., “Sinc approximation of data RFID methods”, Mathematics. Mechanics, 19, Izdatel'stvo Saratovskogo universiteta, Saratov, 2017, 7–9 (in Russian)

[13] Shakirov I. A., “Influence of the choice of Lagrange interpolation nodes on the exact and approximate values of the Lebesgue constants”, Siberian Math. J., 55:6 (2014), 1144–1160 | DOI

[14] Coroianu L., Gal S. G., “Localization results for the non-truncated max-product sampling operators based on Fejer and sinc-type kernels”, Demonstratio Math., 49:1 (2016), 38–49 | DOI

[15] Richardson M., Trefethen L., “A sinc function analogue of Chebfun”, SIAM J. Sci. Comput., 33:5 (2011), 2519–2535 | DOI

[16] Tharwat M. M., “Sinc approximation of eigenvalues of Sturm – Liouville problems with a Gaussian multiplier”, Calcolo, 51:3 (2014), 465–484 | DOI

[17] Alquran M. T., Al-Khaled K. M., “Numerical comparison of methods for solving systems of conservation laws of mixed type”, Int. Journal of Math. Analysis, 5:1 (2011), 35–47

[18] Sklyarov V. P., “On the best uniform sinc-approximation on a finite interval”, East J. Approx., 14:2 (2008), 183–192

[19] Mohsen A., El-Gamel M., “A sinc-collocation method for the linear Fredholm integro-differential equations”, Z. Angew. Math. Phys., 58:3 (2007), 380–390 | DOI

[20] Umakhanov A. Ya., Sharapudinov I. I., “Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence”, Vladikavkaz Mathematical Journal, 18:4 (2016), 61–70 (in Russian)

[21] Trynin A. Yu., “On some properties of sinc approximations of continuous functions on the interval”, Ufa Math. J., 7:4 (2015), 111–126 | DOI

[22] Trynin A. Yu., “On necessary and sufficient conditions for convergence of sinc approximations”, St. Petersburg Math. J., 27:5 (2016), 825–840 | DOI

[23] Trynin A. Yu., “Approximation of continuous on a segment functions with the help of linear combinations of sincs”, Russian Math. (Iz. VUZ), 60:3 (2016), 63–71 | DOI

[24] Trynin A. Yu., “The divergence of Lagrange interpolation processes in eigenfunctions of the Sturm – Liouville problem”, Russian Math. (Iz. VUZ), 54:11 (2010), 66–76 | DOI

[25] Trynin A. Yu., “On the absence of stability of interpolation in eigenfunctions of the Sturm – Liouville problem”, Russian Math. (Iz. VUZ), 44:9 (2000), 58–71

[26] Mosina K. B., “The Dini – Lipschitz principle for the Lagrange – Sturm – Liouville interpolation process”, Mathematics. Mechanics, 15, Izdatel'stvo Saratovskogo universiteta, Saratov, 2013, 56–59 (in Russian)

[27] Mosina K. B., “Nevai formula for the Lagrange – Sturm – Liouville interpolation process”, Works of the N. I. Lobachevsky Mathematical Center, 46, 2013, 316–318 (in Russian)

[28] Turashvili K. B., “Asymptotic formulas for the eigenfunctions and eigenvalues of the Sturm – Liouville problem”, Mathematics. Mechanics, 14, Izdatel'stvo Saratovskogo universiteta, Saratov, 2012, 73–76 (in Russian)

[29] Turashvili K. B., “On the lack of stability of interpolation with respect to the eigenfunctions of the Sturm – Liouville problem”, Works of the N. I. Lobachevsky Mathematical Center, 44, 2011, 347–350 (in Russian)

[30] Turashvili K. B., “On the interpolation analogue of the integral sign of Dini”, Mathematics. Mechanics, 12, Izdatel'stvo Saratovskogo universiteta, Saratov, 2010, 94–98 (in Russian)

[31] Trynin A. Yu., “Uniform convergence of Lagrange – Sturm – Liouville processes on one functional class”, Ufa Math. J., 10:2 (2018), 93–108 | DOI

[32] Trynin A. Yu., “A criterion of convergence of Lagrange – Sturm – Liouville processes in terms of one-sided modulus of variation”, Russian Math. (Iz. VUZ), 62:8 (2018), 51–63 | DOI

[33] Trynin A. Yu., Sample theorem on a segment and its generalization: Discretization theorem for sinc approximation and its generalization, LAP LAMBERT Academic Publishing RU, 2016, 488 pp. (in Russian)

[34] Trynin A. Yu., “On inverse nodal problem for Sturm – Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124 | DOI

[35] Privalov A. A., Function Interpolation Theory, Izdatel'stvo Saratovskogo universiteta, Saratov, 1990, 230 pp. (in Russian)

[36] Golubov B. I., “Spherical Jump of a Function and the Bochner – Riesz Means of Conjugate Multiple Fourier Series and Fourier Integrals”, Math. Notes, 91:4 (2012), 479–486 | DOI | DOI

[37] Golubov B. I., “Absolute convergence of multiple Fourier series”, Math. Notes, 31:1 (1985), 8–15 | DOI

[38] Dyachenko M. I., “On a class of summability methods for multiple Fourier series”, Sb. Math., 204:3 (2013), 307–322 | DOI | DOI

[39] Skopina M. A., Maksimenko I. E., “Multidimensional periodic wavelets”, St. Petersburg Math. J., 15:2 (2004), 165–190 | DOI

[40] Dyachenko M. I., “Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series”, Math. Notes, 76:5 (2004), 673–681 | DOI | DOI

[41] Ivannikova T. A., Timashova E. V., Shabrov S. A., “On necessary conditions for a minimum of a quadratic functional with a Stieltjes integral and zero coefficient of the highest derivative on the part of the interval”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 13:2-1 (2013), 3–8 (in Russian) ; (1954), 428 с. | DOI

[42] Sansone Dzh., Ordinary Differential Equations, in 2 vols., v. 1, Izdatel'stvo inostrannoi literatury, M., 1953, 336 pp.; v. 2, 1954, 428 pp. (in Russian)

[43] Egorova I. A., “On the principle of localization in the theory of interpolation”, Scientific notes of the Leningrad Pedagogical Institute, 86 (1949), 317–335 (in Russian)

[44] Natanson I. P., Theory of functions of a real variable, Nauka, M., 1974, 480 pp. (in Russian)