Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

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Trynin A. Y., Kireeva E. D. The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange – Sturm – Liouville. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 1, pp. 51-63. DOI: 10.18500/1816-9791-2020-20-1-51-63, EDN: YRYAST

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
02.03.2020
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Language: 
Russian
Heading: 
Mathematics
Article type: 
Article
UDC: 
517.518.8
DOI: 
10.18500/1816-9791-2020-20-1-51-63
EDN: 
YRYAST

The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange – Sturm – Liouville

Autors: 
Trynin Aleksandr Yurievich, Saratov State University
Kireeva Ekaterina Dmitrievna, Saratov State University
Abstract: 

Let us say that the principle of localization holds at the class of functions F at point x0 ∈ [0, π] for the Lagrange –Sturm – Liouville interpolation process LSLn (f, x) if limn→∞ LSLn (f, x0) − LSLn (g, x0) = 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 Oδ(x0), δ > 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, π] 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 x0 ∈ [0, π]. in both cases depend solely on the values of the approximate function just in the neighborhood of this point x0 ∈ [0, π].

Key words: 
interpolation process
eigenfunctions
function approximation
localization principle
References: 
  1. Natanson G. I. About one interpolation process. Uchenye zapiski Leningradskogo pedagogicheskogo insituta [Scientific notes of the Leningrad Pedagogical Institute], 1958, vol. 166, pp. 213–219 (in Russian).
  2. Kashin B. S., Saakyan A. A. Ortogonal’nye ryady [Orthogonal series]. Moscow, Izd-vo AFTs, 1999. 550 p. (in Russian).
  3. Novikov I. Ya., Stechkin S. B. Basic wavelet theory. Russian Math. Surveys, 1998, vol. 53, iss. 6, pp. 1159–1231. DOI: https://doi.org/10.1070/rm1998v053n06ABEH000089
  4. Stenger F. Numerical Methods Based on Sinc and Analytic Functions. Springer Ser. Comput. Math., vol. 20, New York, Springer-Verlag, 1993. 565 p. DOI: https://doi.org/10.1007/978-1-4612-2706-9
  5. Dobeshi I. Desyat’ lektsiy po veivletam [Ten Wavelet Lectures]. Izhevsk, NITs “Regulyarnaya i khaoticheskaya dinamika”, 2001. 464 p. (in Russian).
  6. Butzer P. L. A retrospective on 60 years of approximation theory and associated fields. Journal of Approximation Theory, 2009, vol. 160, iss. 1–2, pp. 3–18. DOI: https://doi.org/10.1016/j.jat.2009.05.004
  7. Shmukler A. I., Shulman T. A. Certain properties of Kotel’nikov series. Soviet Math. (Iz. VUZ), 1974, vol. 18, iss. 3, pp. 81–90.
  8. Livne O. E., Brandt A. E. MuST: The multilevel sinc transform. SIAM J. Sci. Comput., 2011, vol. 33, iss. 4, pp. 1726–1738. DOI: https://doi.org/10.1137/100806904
  9. Krivoshein A., Skopina M. Multivariate Sampling-Type Approximation. Analysis and Applications, 2017, vol. 15, no. 4, pp. 521–542. DOI: https://doi.org/10.1142/S0219530516500147
  10. Kolomoitsev Yu., Skopina M. Around Kotelnikov – Shannon Formula. 2017 12th International Conference on Sampling Theory and Applications (SampTA 2017). IEEE, 2017, pp. 279–282. DOI: https://doi.org/10.1109/SAMPTA.2017.8024385
  11. Maleknejad K., Rostami Ya., Shahi Kalalagh H. Numerical solution for first kind Fredholm integral equations by using sinc collocation method. IJAPM, 2016, vol. 6, no. 3, pp. 120– 128. DOI: https://doi.org/10.17706/ijapm.2016.6.3.120-128
  12. Belichenko K. V., Sobolev V. M. Sinc approximation of data RFID methods. Matematika. Mekhanika [Mathematics. Mechanics]. Saratov, Izdatel’stvo Saratovskogo universiteta, 2017, iss. 19, pp. 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., 2014, vol. 55, iss. 6, pp. 1144–1160. DOI: https://doi.org/10.1134/S0037446614060184
  14. Coroianu L., Gal G. S. Localization results for the non-truncated max-product sampling operators based on Fejer and sinc-type kernels. Demonstratio Math., 2016, vol. 49, iss. 1, pp. 38–49. DOI: https://doi.org/10.1515/dema-2016-0005
  15. Richardson M., Trefethen L. A sinc function analogue of Chebfun. SIAM J. Sci. Comput., 2011, vol. 33, iss. 5, pp. 2519–2535. DOI: https://doi.org/10.1137/110825947
  16. Tharwat M. M. Sinc approximation of eigenvalues of Sturm – Liouville problems with a Gaussian multiplier. Calcolo, 2014, vol. 51, iss. 3, pp. 465–484. DOI: https://doi.org/10.1007/s10092-013-0095-3
  17. Alquran M. T., Al-Khaled K. Numerical comparison of methods for solving systems of conservation laws of mixed type. Int. Journal of Math. Analysis, 2011, vol. 5, no. 1, pp. 35–47.
  18. Sklyarov V. P. On the best uniform sinc-approximation on a finite interval. East J. Approx., 2008, vol. 14, iss. 2, pp. 183–192.
  19. Mohsen A., El-Gamel M. A Sinc-collocation method for the linear Fredholm integrodifferential equations. Z. Angew. Math. Phys., 2007, vol. 58, iss. 3, p. 380–390. DOI: https://doi.org/10.1007/s00033-006-5124-5
  20. Umakhanov A. Ya., Sharapudinov I. I. Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence. Vladikavkazskiy matematicheskiy zhurnal [Vladikavkaz Mathematical Journal], 2016, vol. 18, no. 4, pp. 61–70 (in Russian).
  21. Trynin A. Yu. On some properties of sinc approximations of continuous functions on the interval. Ufa Math. J., 2015, vol. 7, iss. 4, pp. 111–126. DOI: https://doi.org/10.13108/2015-7-4-111
  22. Trynin A. Yu. On necessary and sufficient conditions for convergence of sinc approximations. St. Petersburg Math. J., 2016, vol. 27, iss. 5, pp. 825–840. DOI: https://doi.org/10.1090/spmj/1419
  23. Trynin A. Yu. Approximation of continuous on a segment functions with the help of linear combinations of sincs. Russian Math. (Iz. VUZ), 2016, vol. 60, iss. 3, pp. 63–71. DOI: https://doi.org/10.3103/S1066369X16030087
  24. Trynin A. Yu. The divergence of Lagrange interpolation processes in eigenfunctions of the Sturm – Liouville problem. Russian Math. (Iz. VUZ), 2010, vol. 54, iss. 11, pp. 66–76. DOI: https://doi.org/10.3103/S1066369X10110071
  25. Trynin A. Yu. On the absence of stability of interpolation in eigenfunctions of the Sturm – Liouville problem. Russian Math. (Iz. VUZ), 2000, vol. 44, iss. 9, pp. 58–71.
  26. Mosina K. B. The Dini – Lipschitz principle for the Lagrange – Sturm – Liouville interpolation process. Matematika. Mekhanika [Mathematics. Mechanics]. Saratov, Izdatel’stvo Saratovskogo universiteta, 2013, iss. 15, pp. 56–59 (in Russian).
  27. Mosina K. B. Nevai formula for the Lagrange – Sturm – Liouville interpolation process. Trudy Matematicheskogo tsentra imeni N. I. Lobachevskogo [Works of the N. I. Lobachevsky Mathematical Center], 2013, vol. 46, pp. 316–318 (in Russian).
  28. Turashvili K. B. Asymptotic formulas for the eigenfunctions and eigenvalues of the Sturm – Liouville problem. Matematika. Mekhanika [Mathematics. Mechanics]. Saratov, Izdatel’stvo Saratovskogo universiteta, 2012, iss. 14, pp. 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. Trudy Matematicheskogo tsentra imeni N. I. Lobachevskogo [Works of the N. I. Lobachevsky Mathematical Center], 2011, vol. 44, pp. 347–350 (in Russian).
  30. Turashvili K. B. On the interpolation analogue of the integral sign of Dini. Matematika. Mekhanika [Mathematics. Mechanics]. Saratov, Izdatel’stvo Saratovskogo universiteta, 2010, iss. 12, pp. 94–98 (in Russian).
  31. Trynin A. Yu. Uniform convergence of Lagrange – Sturm – Liouville processes on one functional class. Ufa Math. J., 2018, vol. 10, iss. 2, pp. 93–108. DOI: https://doi.org/10.13108/2018-10-2-93
  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), 2018, vol. 62, iss. 8, pp. 51–63. DOI: https://doi.org/10.3103/S1066369X1808008X
  33. Trynin A. Yu. Teorema otschetov na otrezke i ee obobscheniya: Teorema diskretizatsii dlia sink approksimatsii i ee obobshchenie [Sample theorem on a segment and its generalization: Discretization theorem for sinc approximation and its generalization]. LAP LAMBERT Academic Publishing RU, 2016. 488 p. (in Russian).
  34. Trynin A. Yu. On inverse nodal problem for Sturm – Liouville operator. Ufa Math. J., 2013, vol. 5, iss. 4, pp. 112–124. DOI: https://doi.org/10.13108/2013-5-4-112
  35. Privalov A. A. Teoriya interpolirovaniya funktsii [Function Interpolation Theory]. Saratov, Izdatel’stvo Saratovskogo universiteta, 1990. 230 p. (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, 2012, vol. 91, iss. 4, pp. 479– 486. DOI: https://doi.org/10.1134/S0001434612030212
  37. Golubov B. I. Absolute convergence of multiple Fourier series. Math. Notes, 1985, vol. 31, iss. 1, pp. 8–15. DOI: https://doi.org/10.1007/BF01652507
  38. Dyachenko M. I. On a class of summability methods for multiple Fourier series. Sb. Math., 2013, vol. 204, iss. 3, pp. 307–322. DOI: https://doi.org/10.1070/SM2013v204n03ABEH004302
  39. Skopina M. A., Maksimenko I. E. Multidimensional periodic wavelets. St. Petersburg Math. J., 2004, vol. 15, iss. 2, pp. 165–190. DOI: https://doi.org/10.1090/S1061-0022-04-00808-8
  40. Dyachenko M. I. Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series. Math. Notes, 2004, vol. 76, iss. 5, pp. 673–681. DOI: https://doi.org/10.1023/B:MATN.0000049666.00784.9d
  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., 2013, vol. 13, iss. 2, pt. 1, pp. 3–8 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2013-13-2-1-3-8
  42. Sansone Dzh. Obyknovennye differentsial’nye uravneniya [Ordinary Differential Equations: in 2 vols.]. Moscow, Izdatel’stvo inostrannoi literatury, vol. 1, 1953, 336 p.; vol. 2, 1954, 428 p. (in Russian).
  43. Egorova I. A. On the principle of localization in the theory of interpolation. Uchenye zapiski Leningradskogo pedagogicheskogo insituta [Scientific notes of the Leningrad Pedagogical Institute], 1949, vol. 86, pp. 317–335 (in Russian).
  44. Natanson I. P. Teoriya funktsiy veshchestvennoy peremennoy [Theory of functions of a real variable]. Moscow, Nauka, 1974. 480 p. (in Russian).
Received: 
31.10.2018
Accepted: 
15.12.2018
Published: 
02.03.2020
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