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Mat. Sb., 2000, Volume 191, Number 8, Pages 131–140 (Mi msb502)  

This article is cited in 11 scientific papers (total in 12 papers)

Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes

Yu. N. Subbotina, S. A. Telyakovskiib

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Associated with each continuous function $f$ of period 1 is the periodic spline $s_{r,n}(f)$ that has degree $r$, defect 1, nodes at the points $x_i=i/n$, $i=0,1,…,n-1$ and that interpolates $f$ at these points for $r$ odd and at the mid-points of the intervals $[x_i,x_{i+1}]$ for $r$ even.
For the corresponding Lebesgue constants $L_{r,n}$, that is the norms of the operators $f(x)\to s_{r,n}(f)$ from $C$ to $C$, the asymptotic formula
$$ L_{r,n}=\frac2\pi\log\min(r,n)+O(1), $$
is established, which holds uniformly in $r$ and $n$.

DOI: https://doi.org/10.4213/sm502

Full text: PDF file (204 kB)
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English version:
Sbornik: Mathematics, 2000, 191:8, 1233–1242

Bibliographic databases:

Document Type: Article
UDC: 517.518.8
MSC: 41A15, 41A05
Received: 07.10.1999

Citation: Yu. N. Subbotin, S. A. Telyakovskii, “Asymptotic behaviour of the Lebesgue constants of periodic interpolation splines with equidistant nodes”, Mat. Sb., 191:8 (2000), 131–140; Sb. Math., 191:8 (2000), 1233–1242

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. N. Subbotin, S. A. Telyakovskii, “Norms on $L$ of Periodic Interpolation Splines with Equidistant Nodes”, Math. Notes, 74:1 (2003), 100–109  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. K. V. Kostousov, V. T. Shevaldin, “Approximation by local trigonometric splines”, Math. Notes, 77:3 (2005), 326–334  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. V. A. Kim, “Exact Lebesgue Constants for Interpolatory $\mathscr L$-Splines of Third Order”, Math. Notes, 84:1 (2008), 55–63  mathnet  crossref  crossref  mathscinet  isi  elib
    4. V. A. Kim, “Sharp Lebesgue constants for bounded cubic interpolation $\mathcal L$-splines”, Siberian Math. J., 51:2 (2010), 267–276  mathnet  crossref  mathscinet  isi  elib  elib
    5. I. A. Shakirov, “A complete description of the Lebesgue functions for classical Lagrange interpolation polynomials”, Russian Math. (Iz. VUZ), 55:10 (2011), 70–77  mathnet  crossref  mathscinet  elib
    6. “Yurii Nikolaevich Subbotin. (K semidesyatipyatiletiyu so dnya rozhdeniya)”, Tr. IMM UrO RAN, 17, no. 3, 2011, 8–13  mathnet
    7. I. A. Shakirov, “Lebesgue functions corresponding to a family of Lagrange interpolation polynomials”, Russian Math. (Iz. VUZ), 57:7 (2013), 66–76  mathnet  crossref
    8. Yu. S. Volkov, Yu. N. Subbotin, “50 years to Schoenberg's problem on the convergence of spline interpolation”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 222–237  mathnet  crossref  mathscinet  isi  elib
    9. E. V. Strelkova, V. T. Shevaldin, “On Lebesgue constants of local parabolic splines”, Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 192–198  mathnet  crossref  mathscinet  isi  elib
    10. E. V. Strelkova, V. T. Shevaldin, “On uniform Lebesgue constants of local exponential splines with equidistant knots”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 206–217  mathnet  crossref  mathscinet  elib
    11. V. T. Shevaldin, O. Ya. Shevaldina, “Upper bounds for uniform Lebesgue constants of interpolational periodic sourcewise representable splines”, Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 175–181  mathnet  crossref  mathscinet  elib
    12. S. I. Novikov, “Lebesgue constants for some interpolational ${\mathcal L}$-splines”, Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 136–144  mathnet  crossref  crossref  mathscinet  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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