Sharp inequalities of various metrics on the classes of functions with given comparison function

T.V. Alexandrova (Oles Honchar Dnipro National University)
V.A. Kofanov (Oles Honchar Dnipro National University), https://orcid.org/0000-0003-0392-2257

Abstract


For any $$$q > p > 0$$$, $$$\omega > 0,$$$ $$$d \ge 2 \omega,$$$  we obtain the following sharp inequality of various metrics
$$
\|x\|_{L_q(I_{d})} \le \frac{\|\varphi +
c\|_{L_q(I_{2\omega})}}{\|\varphi + c \|_{L_p(I_{2\omega})}}
\|x\|_{L_p(I_{d})}
$$
on the set $$$S_{\varphi}(\omega)$$$ of $$$d$$$-periodic functions $$$x$$$ having zeros with given the sine-shaped $$$2\omega$$$-periodic comparison function $$$\varphi$$$, where $$$c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$$$ is such that
$$
\|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi +
c)_{\pm}\|_{L_p(I_{2\omega})}\,.
$$
In particular, we  obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $$$\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$$$.

Keywords


inequality of various metrics; a class of functions with given comparison function; Sobolev class of functions; polynomial; spline

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References


Babenko V.F., Kofanov V.A., Pichugov S.A. "Comparison of rearrangement and Kolmogorov-Nagy type inequalities for periodic functions", Approximation theory: A volume dedicated to Blagovest Sendov (B. Bojanov, ed.), Darba, Sofia, 2002; pp. 24-53.

Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280. doi:10.1007/BF02791137

Kofanov V.A. "Exact upper bounds of norms of functions and their derivatives on the classes of functions with given comparison function", Ukrainian Math. J., 2011; 63(7): pp. 969-984. (in Russian) doi:10.1007/s11253-011-0567-z

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, 1992; 304 p. (in Russian)

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Izbr. tr. Matematika, mekhanika, Nauka, Moscow, 1985; pp. 252-263. (in Russian)

Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Tikhomirov V.M. "Set widths in functional spaces and theory of the best approximations", Uspekhi mat. nauk, 1960; 15(3): pp. 81-120. (in Russian) doi:10.1070/RM1960v015n03ABEH004093




DOI: https://doi.org/10.15421/242102

  

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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
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