Exact Estimates of Functions in Sobolev Spaces with Uniform Norm

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.