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),


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})}}
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 +
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 $$$\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}$$$.


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

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