ARTICLE
TITLE

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

SUMMARY

For any q>p>0q > p > 0, ?>0,\omega > 0, d=2?,d \ge 2 \omega,  we obtain the following sharp inequality of various metrics?x?Lq(Id)=?f+c?Lq(I2?)?f+c?Lp(I2?)?x?Lp(Id) \|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)S_{\varphi}(\omega) of dd-periodic functions xx having zeros with given the sine-shaped 2\omega2\omega-periodic comparison function \varphi\varphi, where c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty] is such that \|x_{\pm}\|_{L_p(I_{d})} = \|(\varphi + c)_{\pm}\|_{L_p(I_{2\omega})}\,. \|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})}\|x_{+}\|_{L_p(I_{d})} / \|x_-\|_{L_p(I_{d})}.

PAGES
pp. 11 - 23
COLLECTIONS

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