Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

  • V. A. Kofanov

Abstract

For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem $$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$ on the set of functions $S^k_{\varphi}$ such that$\varphi ^{(i)}$ is the comparison function for $x^{(i)},\; i = 0, 1, . . . , k$, and (in the case $k = 0$) $L(x)_p \leq L(\varphi)_p$, where $$L(x)_p := \sup \left\{\left(\int^b_a|x(t)|^p dt \right)^{1/p}\; :\; a, b \in \textbf{R},\; |x(t)| > 0,\; t \in (a, b) \right\}$$ In particular, we solve this extremal problem for Sobolev classes and for bounded sets of the spaces of trigonometric polynomials and splines.
Published
25.07.2011
How to Cite
KofanovV. A. “Sharp Upper Bounds of Norms of Functions and Their Derivatives on Classes of Functions With Given Comparison Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, no. 7, July 2011, pp. 969-84, http://umj.imath.kiev.ua/index.php/umj/article/view/2778.
Section
Research articles