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

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