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Tr. Mat. Inst. Steklova, 2003, Volume 243, Pages 230–236 (Mi tm430)  

This article is cited in 1 scientific paper (total in 1 paper)

Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis

G. A. Kalyabinab

a S. P. Korolyov Samara State Aerospace University
b Samara Academy of Humanities

Abstract: The spaces $W_2^n(\mathbb R_+)$ of functions with finite norms $\| f| W_2^n(\mathbb R_+)\|_{\sigma} := (\|f|L_2(\mathbb R_+)\|^2 +{\sigma}^{-2n} \|f^{(n)}|L_2(\mathbb R_+)\|^2)^{1/2}$, $\sigma>0$, are studied. Let $\Omega _{n,\sigma }$ and $\omega _{n,\sigma }$ be the maximum and minimum of $\|f|W_2^n(\mathbb R_+ )\|_{\sigma}$ under the condition $\sum _0^{n-1} |f^{(s)}(0)|^2 = 1$. It is proved that, as $n\to\infty$, the quantities $n^{-1}\ln \Omega _{n,\sigma}$ and $n^{-1} \ln \omega _{n,\sigma}$ tend to explicitly calculated limits that depend on the number $\sigma$. The behavior of similar quantities $\Omega ^*_{n,\sigma}$ and $\omega ^*_{n,\sigma}$ for the functions defined on the whole axis $\mathbb R$ instead of the half-axis $\mathbb R_+$ is analyzed. The results obtained can be applied to inequalities between the $l_2$-norm of the set of coefficients of an algebraic polynomial of degree $<n$ and the norm of this polynomial in the space $L_2$ with the weight $(1+(x/\sigma )^{2n})^{-1}$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 243, 220–226

Bibliographic databases:
UDC: 517.518
Received in February 2003

Citation: G. A. Kalyabin, “Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis”, Function spaces, approximations, and differential equations, Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS, Tr. Mat. Inst. Steklova, 243, Nauka, MAIK Nauka/Inteperiodika, M., 2003, 230–236; Proc. Steklov Inst. Math., 243 (2003), 220–226

Citation in format AMSBIB
\Bibitem{Kal03}
\by G.~A.~Kalyabin
\paper Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis
\inbook Function spaces, approximations, and differential equations
\bookinfo Collected papers. Dedicated to the 70th birthday of Oleg Vladimirovich Besov, corresponding member of RAS
\serial Tr. Mat. Inst. Steklova
\yr 2003
\vol 243
\pages 230--236
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm430}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2049472}
\zmath{https://zbmath.org/?q=an:1073.41005}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 243
\pages 220--226


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    This publication is cited in the following articles:
    1. G. A. Kalyabin, “Some Problems for Sobolev Spaces on the Half-line”, Proc. Steklov Inst. Math., 255 (2006), 150–158  mathnet  crossref  mathscinet  elib
  •    . . .  Proceedings of the Steklov Institute of Mathematics
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