Geodesic
Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 230-236
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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 $$ and the norm of this polynomial in the space $L_2$ with the weight $(1+(x/\sigma )^{2n})^{-1}$. 
@article{TM_2003_243_a15,
     author = {G. A. Kalyabin},
     title = {Extrapolations with the {Least} {Norms} in the {Sobolev} {Spaces} $W_2^n$ on the {Half-Axis} and the {Whole} {Axis}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     year = {2003},
     volume = {243},
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     url = {http://geodesic.mathdoc.fr/item/TM_2003_243_a15/}
}
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G. A. Kalyabin. Extrapolations with the Least Norms in the Sobolev Spaces $W_2^n$ on the Half-Axis and the Whole Axis. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximations, and differential equations, Tome 243 (2003), pp. 230-236. http://geodesic.mathdoc.fr/item/TM_2003_243_a15/

[1] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR

[2] Kalyabin G. A., “Nailuchshie operatory prodolzheniya dlya sobolevskikh prostranstv na polupryamoi”, Funkts. analiz i ego pril., 36:2 (2002), 28–37 | MR | Zbl

[3] Besov O. V., Kalyabin G. A., “Spaces of differentiable functions”, Function spaces, differential operators and nonlinear analysis, The Hans Triebel anniversary volume, Birkhäuser, Basel, 2003, 3–21 | MR | Zbl

[4] Dvait G. B., Tablitsy integralov i drugie matematicheskie formuly, Nauka, M., 1978

[5] Kalyabin G. A., “On extrapolation with minimal norms in Bernstein classes”, Proc. Razmadze Math. Inst., 119, 1999, 85–92 | MR | Zbl

[6] Kalyabin G. A., “Verkhnie otsenki koeffitsientov algebraicheskikh mnogochlenov cherez ikh $L_p$-normy na intervalakh”, Tr. MIAN, 227, 1999, 152–161 | MR | Zbl

[7] Kalyabin G. A., “Asimptotika naimenshikh sobstvennykh znachenii matrits tipa Gilberta”, Funkts. analiz i ego pril., 35:1 (2001), 80–84 | MR | Zbl

[8] Kalyabin G. A., “O tochnykh konstantakh v neravenstvakh Kolmogorova dlya prostranstv Soboleva $W_2^n(\mathbb{R}_+)$”, Dokl. RAN, 388:2 (2003), 159–161 | MR | Zbl

[9] Kalyabin G. A., “Tochnye konstanty v neravenstvakh dlya promezhutochnykh proizvodnykh (sluchai Gabushina)”, Funkts. analiz i ego pril., 38:3 (2004), 29–38 | MR | Zbl