×

Convergence of the Lagrange-Sturm-Liouville processes for continuous functions of bounded variation. (Russian. English summary) Zbl 1463.41008

Summary: The uniform convergence within an interval \((a,b)\subset [0,\pi]\) of Lagrange processes in eigenfunctions \(L_n^{SL}(f,x)=\sum\nolimits_{k=1}^nf(x_{k,n})\frac{U_n(x)}{U_n'(x_{k,n})(x-x_{k,n})}\) of the Sturm-Liouville problem is established. (Here \(0<x_{1,n}<x_{2,n}<\dots<x_{n,n}<\pi\) denote the zeros of the eigenfunction \(U_n\) of the Sturm-Liouville problem.) A continuous functions \(f\) on \([0,\pi]\) which is of bounded variation on \((a,b)\subset [0,\pi]\) can be uniformly approximated within the interval \((a,b)\subset [0,\pi]\). A criterion for uniform convergence within an interval \((a,b)\) of the constructed interpolation processes is obtained in terms of the maximum of the sum of the moduli of divided differences of the function \(f\). Outside of the interval \((a, b)\), the Lagrange interpolation process may diverge. The boundedness in the totality of the Lagrange fundamental functions constructed from eigenfunctions of the Sturm-Liouville problem is established. The case of the regular Sturm-Liouville problem with a continuous potential of bounded variation is also considered. The boundary conditions for the third kind Sturm-Liouville problem without Dirichlet conditions are studied. In the presence of service functions for calculating the eigenfunctions of the regular Sturm-Liouville problem, the Lagrange-Sturm-Liouville operator under study is easily implemented by computer technology.

MSC:

41A05 Interpolation in approximation theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
94A12 Signal theory (characterization, reconstruction, filtering, etc.)

Software:

MuST; Chebfun
PDFBibTeX XMLCite
Full Text: DOI MNR

References:

[1] Kramer H. P., “A generalized sampling theorem”, J. Math. Phus., 38 (1959), 68-72 · Zbl 0196.31702
[2] Natanson G. I., “Ob odnom interpolyatsionnom protsesse”, Uchen. zapiski Leningrad. ped. in-ta im. A. I. Gertsena, 166 (1958), 213-219 (in Russian)
[3] Novikov I. Ya., Stechkin S. B., “Basic Wavelet Theory”, Russian Mathematical Surveys, 53:6 (1998), 1159-1231 · Zbl 0955.42019
[4] Novikov I. Ya., Stechkin S. B., “Basic Constructions of Wavelets”, Fundamental and Applied Mathematics, 3:4 (1997), 999-1028 (in Russian) · Zbl 0936.42015
[5] Stenger F., Numerical Metods Based on Sinc and Analytic Functions, Springer, N. Y., 1993, 565 pp. · Zbl 0803.65141
[6] Dobeshi I., Ten Lectures on Wavelets, Society for Industrial and Appled Mathematics, Philadelphia, Pennsylvana, 1992
[7] Oren E. Livne, Achi E. Brandt, “MuST: The multilevel sinc transform”, SIAM J. Sci. Comput., 33:4 (2011), 1726-1738 · Zbl 1230.65146
[8] Coroianu L., Sorin G. Gal., “Localization results for the non-truncated max-product sampling operators based on Fejer and sinc-type kernels”, Demonstratio Mathematica, 49:1 (2016), 38-49 · Zbl 1347.41027
[9] Richardson M., Trefethen L., “A sinc function analogue of Chebfun”, SIAM J. Sci. Comput., 33:5 (2011), 2519-2535 · Zbl 1234.41001
[10] Khosrow M., Yaser R., Hamed S., “Numerical Solution for First Kind Fredholm Integral Equations by Using Sinc Collocation Method”, Int. J. Appl. Phy. Math., 6:3 (2016), 120-128
[11] Trynin A. Yu., “Necessary and Sufficient Conditions for the Uniform on a Segment Sinc-Approximations Functions of Bounded Variation”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 16:3 (2016), 288-298 (in Russian) · Zbl 1355.42001
[12] Trynin A. Yu., Sklyarov V. P., “Error of sinc approximation of analytic functions on an interval”, Sampling Theory in Signal and Image Processing, 7:3 (2008), 263-270 · Zbl 1182.65020
[13] Marwa M. Tharwat, “Sinc approximation of eigenvalues of Sturm-Liouville problems with a Gaussian multiplier”, Calcolo: a Quarterly on Numerical Analysis and Theory of Computation, 51:3 (2014), 465-484 · Zbl 1317.34175
[14] Trynin A. Yu., “On the Estimation of the Approximation of Analytic Functions by an Interpolation Operator with Respect to Syncs”, Mathematics. Mechanics, 7 (2005), 124-127 (in Russian)
[15] Zayed A. I., Schmeisser G., New Perspectives on Approximation and Sampling Theory, Applied and Numerical Harmonic Analysis, Springer Int. Publ. Switzerland, N. Y.-Dordrecht-London, 2014 · Zbl 1302.00086
[16] Trynin A. Yu., “Estimates for the Lebesgue Functions and the Nevai Formula for the Sinc-Approximations of Continuous Functions on an Interval”, Siberian Mathematical Journal, 48:5 (2007), 929-938 · Zbl 1164.41307
[17] Trynin A. Yu., “Tests for Pointwise and Uniform Convergence of Sinc Approximations of continuous functions on a Closed Interval”, Sbornik: Mathematics, 198:10 (2007), 1517-1534 · Zbl 1138.41001
[18] Trynin A. Yu., “A Criterion for the Uniform Convergence of Sinc-Approximations on a Segment”, Russian Mathematics (Iz. VUZ), 52:6 (2008), 58-69 · Zbl 1210.42003
[19] Sklyarov V. P., “On the best uniform sinc-approximation on a finite interval”, East J. Approx., 14:2 (2008), 183-192 · Zbl 1219.41022
[20] Trynin A. Yu., “On Divergence of Sinc-Approximations Everywhere on \((0,\pi)\)”, St. Petersburg Mathematical Journal, 22 (2011), 683-701 · Zbl 1227.42003
[21] Trynin A. Yu., “On Some Properties of Sinc Approximations of Continuous Functions on the Interval”, Ufa Mathematical Journal, 7:4 (2015), 111-126
[22] Umakhanov A. Y., Sharapudinov I. I., “Interpolation of Functions by the Whittaker Sums and Their Modifications: Conditions for Uniform Convergence”, Vladikavkaz Math. J., 18 (2016), 61-70 · Zbl 1474.41007
[23] Umakhanov A. Y., Sharapudinov I. I., “Interpolation of Functions by Whittaker Sums and their Modifications: Conditions of Uniform Convergence”, Materialy 18th Intern. Saratov Winter School. «Modern Problems of the Theory of Functions and their Applications» (Saratov, 2016), 332-334 · Zbl 1474.41007
[24] Trynin A. Yu., “On Necessary and Sufficient Conditions for Convergence of Sinc-Approximations”, St. Petersburg Mathematical Journal, 27:5 (2016), 825-840 · Zbl 1347.41020
[25] Trynin A. Yu., “Approximation of Continuous on a Segment Functions with the Help of Linear Combinations of Sincs”, Russian Mathematics (Iz. VUZ), 60:3 (2016), 63-71 · Zbl 1346.42004
[26] Trynin A. Yu., “A Generalization of the Whittaker-Kotel”nikov-Shannon Sampling Theorem for Continuous Functions on a Closed Interval”, Sbornik: Mathematics, 200:11 (2009), 1633-1679 · Zbl 1194.41012
[27] Trynin A. Yu., “On Operators of Interpolation with Respect to Solutions of a Cauchy Problem and Lagrange-Jacobi Polynomials”, Izvestiya: Mathematics, 75:6 (2011), 1215-1248 · Zbl 1236.41003
[28] Trynin A. Yu., “On the Absence of Stability of interpolation in Eigenfunctions of the Sturm-Liouville Problem”, Russian Mathematics (Iz. VUZ), 44:9 (2000), 58-71 · Zbl 1004.41010
[29] Trynin A. Yu., “Differential Properties of Zeros of Eigenfunctions of the Sturm-Liouville Problem”, Ufa Mathematical Journal, 3:4 (2011), 130-140 · Zbl 1249.34095
[30] Trynin A. Yu., “On Inverse Nodal Problem for Sturm-Liouville Operator”, Ufa Mathematical Journal, 5:4 (2013), 112-124
[31] Trynin A. Yu., “The Divergence of Lagrange Interpolation Processes in Eigenfunctions of the Sturm-Liouville Problem”, Russian Mathematics (Iz. VUZ), 54:11 (2010), 66-76 · Zbl 1238.94020
[32] Trynin A. Yu., “Localization Principle for Lagrange-Sturm-Liouville Processes”, Mathematics. Mechanics, 8 (2005), 137-140 (in Russian)
[33] Trynin A. Yu., “On one Integral Sign of Convergence of Lagrange-Sturm-Liouville Processes”, Mathematics. Mechanics, 9 (2007), 94-97
[34] Trynin A. Yu., Teorema otschetov na otrezke i ee obobscheniya, LAP LAMBERT Acad. Publ., 2016, 488 pp. (in Russian)
[35] Golubov B. I., “Spherical Jump of a Function and the Bochner-Riesz Means of Conjugate Multiple Fourier Series and Fourier Integrals”, Mathematical Notes, 91:3-4 (2012), 479-486 · Zbl 1284.42020
[36] Dyachenko M. I., “On a Class of Summability Methods for Multiple Fourier Series”, Sbornik: Mathematics, 204:3 (2013), 307-322 · Zbl 1408.42002
[37] Maksimenko I. E., Skopina M. A., “Multidimensional Periodic Wavelets”, St. Petersburg Mathematical Journal, 15:2 (2004), 165-190 · Zbl 1056.42029
[38] Dyachenko M. I., “Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series”, Mathematical Notes, 76:5 (2004), 673-681 · Zbl 1076.42006
[39] Borisov D. I., Dmitriev S. V., “On the spectral stability of kinks in 2D Klein-Gordon model with parity-time-symmetric perturbation”, Stud. Appl. Math., 138:3 (2017), 317-342 · Zbl 1364.35305
[40] Golubov B. I., “Absolute Convergence of Multiple Fourier Series”, Mathematical Notes, 37:1 (1985), 8-15 · Zbl 0649.42008
[41] Borisov D. I., Znojil M., “On Eigenvalues of a \(\mathscr{PT} \)-Symmetric Operator in a thin Layer”, Sbornik: Mathematics, 208:2 (2017), 173-199 · Zbl 1371.35178
[42] 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) · Zbl 1326.26021
[43] Farkov Yu. A., “On the Best Linear Approximation of Holomorphic Functions”, Journal of Mathematical Sciences, 218:5 (2016), 678-698 · Zbl 1355.30035
[44] Sansone D., Ordinary Differential Equations, v. 1, 2, Foreign Languages Publ. House, M., 1953
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.