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Mat. Sb., 2009, Volume 200, Number 11, Pages 61–108 (Mi msb4502)  

This article is cited in 12 scientific papers (total in 12 papers)

A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval

A. Yu. Trynin

Saratov State University named after N. G. Chernyshevsky

Abstract: Classes of functions in the space of continuous functions $f$ defined on the interval $[0,\pi]$ and vanishing at its end-points are described for which there is pointwise and approximate uniform convergence of Lagrange-type operators
$$ S_\lambda(f,x)=\sum_{k=0}^n\frac{y(x,\lambda)}{y'(x_{k,\lambda}) (x-x_{k,\lambda})}f(x_{k,\lambda}). $$
These operators involve the solutions $y(x,\lambda)$ of the Cauchy problem for the equation
$$ y"+(\lambda-q_\lambda(x))y=0 $$
where $q_\lambda\in V_{\rho_\lambda}[0,\pi]$ (here $V_{\rho_\lambda}[0,\pi]$ is the ball of radius $\rho_\lambda=o(\sqrt\lambda/\ln\lambda)$ in the space of functions of bounded variation vanishing at the origin, and $y(x_{k,\lambda})=0$). Several modifications of this operator are proposed, which allow an arbitrary continuous function on $[0,\pi]$ to be approximated uniformly.
Bibliography: 40 titles.

Keywords: sampling theorem, interpolation, uniform convergence, sinc approximation.

DOI: https://doi.org/10.4213/sm4502

Full text: PDF file (900 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:11, 1633–1679

Bibliographic databases:

UDC: 517.518.85
MSC: 41A05, 41A35
Received: 25.12.2007 and 03.08.2009

Citation: A. Yu. Trynin, “A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval”, Mat. Sb., 200:11 (2009), 61–108; Sb. Math., 200:11 (2009), 1633–1679

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    Cycle of papers

    This publication is cited in the following articles:
    1. A. Yu. Trynin, “On operators of interpolation with respect to solutions of a Cauchy problem and Lagrange–Jacobi polynomials”, Izv. Math., 75:6 (2011), 1215–1248  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. Yu. Trynin, “Differentsialnye svoistva nulei sobstvennykh funktsii zadachi Shturma–Liuvillya”, Ufimsk. matem. zhurn., 3:4 (2011), 133–143  mathnet  zmath
    3. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    4. A. Yu. Trynin, “On necessary and sufficient conditions for convergence of sinc-approximations”, St. Petersburg Math. J., 27:5 (2016), 825–840  mathnet  crossref  mathscinet  isi  elib
    5. A. Yu. Trynin, “On some properties of sinc approximations of continuous functions on the interval”, Ufa Math. J., 7:4 (2015), 111–126  mathnet  crossref  isi  elib
    6. A. Yu. Trynin, “Approximation of continuous on a segment functions with the help of linear combinations of sincs”, Russian Math. (Iz. VUZ), 60:3 (2016), 63–71  mathnet  crossref  isi
    7. A. Yu. Trynin, “Neobkhodimye i dostatochnye usloviya ravnomernoi na otrezke sink-approksimatsii funktsii ogranichennoi variatsii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:3 (2016), 288–298  mathnet  crossref  mathscinet  elib
    8. A. Ya. Umakhanov, I. I. Sharapudinov, “Interpolyatsiya funktsii summami Uittekera i ikh modifikatsiyami: usloviya ravnomernoi skhodimosti”, Vladikavk. matem. zhurn., 18:4 (2016), 61–70  mathnet
    9. A. Yu. Trynin, “Uniform convergence of Lagrange–Sturm–Liouville processes on one functional class”, Ufa Math. J., 10:2 (2018), 93–108  mathnet  crossref  isi
    10. A. Yu. Trynin, “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  mathnet  crossref  isi
    11. A. Yu. Trynin, “Skhodimost protsessov Lagranzha–Shturma–Liuvillya dlya nepreryvnykh funktsii ogranichennoi variatsii”, Vladikavk. matem. zhurn., 20:4 (2018), 76–91  mathnet  crossref
    12. A. Yu. Trynin, “Sufficient condition for convergence of Lagrange–Sturm–Liouville processes in terms of one-sided modulus of continuity”, Comput. Math. Math. Phys., 58:11 (2018), 1716–1727  mathnet  crossref  crossref  isi
  • Математический сборник Sbornik: Mathematics (from 1967)
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