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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2003, Volume 243, Pages 230–236
(Mi tm430)
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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 $(1+(x/\sigma )^{2n})^{-1}$.
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, Trudy Mat. Inst. Steklova, 243, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 230–236; Proc. Steklov Inst. Math., 243 (2003), 220–226
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https://www.mathnet.ru/eng/tm430 https://www.mathnet.ru/eng/tm/v243/p230
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