Abstract—

For functions from the Sobolev space \(\overset{\circ}{W}{\,}_{\infty }^{n}[0;1]\) and an arbitrary point \(a \in (0;1)\), the best estimates are obtained in the inequality \({\text{|}}f(a){\text{|}} \leqslant {{A}_{{n,0,\infty }}}(a)\, \cdot \,{\text{||}}{{f}^{{(n)}}}{\text{|}}{{{\text{|}}}_{{{{L}_{\infty }}[0;1]}}}\). The connection of these estimates with the best approximations of splines of a special type by polynomials in \({{L}_{1}}[0;1]\) and with the Peano kernel is established. Exact constants of the embedding of the space \(\overset{\circ}{W}{\,}_{\infty }^{n}[0;1]\) in \({{L}_{\infty }}[0;1]\) are found.

中文翻译：

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具有统一范数的 Sobolev 空间中函数的精确估计

摘要-

对于来自 Sobolev 空间\(\overset{\circ}{W}{\,}_{\infty }^{n}[0;1]\)和任意点\(a \in (0;1 )\)，最佳估计值通过不等式\({\text{|}}f(a){\text{|}} \leqslant {{A}_{{n,0,\infty }}}获得(a)\, \cdot \,{\text{||}}{{f}^{{(n)}}}{\text{|}}{{{\text{|}}}_{{ {{L}_{\infty }}[0;1]}}}\)。这些估计与\({{L}_{1}}[0;1]\)中的多项式特殊类型样条的最佳近似值以及 Peano 核之间的联系已建立。\({{L}_{\infty)中空间\(\overset{\circ}{W}{\,}_{\infty }^{n}[0;1]\)嵌入的精确常量}}[0;1]\)被发现。