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This article is cited in 8 scientific papers (total in 8 papers)
Comparison of Linear Differential Operators
R. M. Trigub Donetsk National University
Abstract:
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, algebraic polynomial, Kolmogorov multiplicative inequality, Fourier series, Schoenberg spline, Hölder's inequality, Minkowski's inequality.
Received: 27.02.2006 Revised: 16.10.2006
Citation:
R. M. Trigub, “Comparison of Linear Differential Operators”, Mat. Zametki, 82:3 (2007), 426–440; Math. Notes, 82:3 (2007), 380–394
Linking options:
https://www.mathnet.ru/eng/mzm3854https://doi.org/10.4213/mzm3854 https://www.mathnet.ru/eng/mzm/v82/i3/p426
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Abstract page: | 605 | Full-text PDF : | 334 | References: | 65 | First page: | 7 |
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