Geodesic
Comparison of Linear Differential Operators
Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 426-440.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we study the question of the existence of the inequality $$ \|Q(D)f\|_{L_q}\le\gamma_0\|P(D)f\|_{L_p}, $$ where $P$ and $Q$ are algebraic polynomials, $D=d/dx$, and $\gamma_0$ is independent of the function $f$. We obtain criteria (necessary and simultaneously sufficient conditions) for the existence of such inequalities for functions on the circle, on the whole line, and on the semiaxis. Besides, for the semiaxis, we obtain an inequality for $q=\infty$ and any $p\ge1$ with the smallest constant $\gamma_0$.
Keywords: linear differential operator, Kolmogorov multiplicative inequality, Fourier series, Schoenberg spline, Hölder's inequality, Minkowski's inequality.
Mots-clés : algebraic polynomial 
@article{MZM_2007_82_3_a10,
     author = {R. M. Trigub},
     title = {Comparison of {Linear} {Differential} {Operators}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {426--440},
     publisher = {mathdoc},
     volume = {82},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a10/}
}
TY  - JOUR
AU  - R. M. Trigub
TI  - Comparison of Linear Differential Operators
JO  - Matematičeskie zametki
PY  - 2007
SP  - 426
EP  - 440
VL  - 82
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a10/
LA  - ru
ID  - MZM_2007_82_3_a10
ER  - 
%0 Journal Article
%A R. M. Trigub
%T Comparison of Linear Differential Operators
%J Matematičeskie zametki
%D 2007
%P 426-440
%V 82
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a10/
%G ru
%F MZM_2007_82_3_a10
R. M. Trigub. Comparison of Linear Differential Operators. Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 426-440. http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a10/

[1] V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, S. A. Pichugov, Neravenstva dlya proizvodnykh i ikh prilozheniya, Naukova dumka, Kiev, 2003

[2] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka. Fizmalit, M., 1996 | MR | Zbl

[3] R. M. Trigub, E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers, Dordrecht, 2004 | MR | Zbl

[4] R. M. Trigub, “Multiplikatory Fure i $K$-funktsionaly prostranstv gladkikh funktsii”, Ukr. matem. visnyk, 2:2 (2005), 236–280 | MR

[5] A. Sharma, J. Tzimbalario, “Landau-type inequalities for some linear differential operators”, Illinois J. Math., 20:3 (1976), 443–455 | MR | Zbl

[6] Nguen Tkhi Tkheu Khoa, “Nekotorye ekstremalnye zadachi na klassakh funktsii, zadavaemykh lineinymi differentsialnymi operatorami”, Matem. sb., 180:10 (1989), 1355–1395 | MR | Zbl

[7] R. M. Trigub, “Multiplikatory Fure i tochnye neravenstva dlya differentsialnykh operatorov”, Izv. Tulskogo gos. un-ta. Ser. Matem. Mekh. Inform., 11:1 (2005), 219–228

[8] V. N. Gabushin, “Neravenstva dlya norm funktsii i ikh proizvodnykh v metrikakh $L_p$”, Matem. zametki, 1:3 (1967), 291–298 | Zbl

[9] B. R. Cramer, V. I. Gurariy, E. R. Tsekanovskii, “Results as by-Products”, Alabama J. Math., 27:1–2 (2003), 1–8