Abstract:
For functions from the Sobolev space $\overset\circ{W}{}^n_\infty[0;1]$ and an arbitrary point $a\in(0;1)$, the best estimates are obtained in the inequality $|f(a)|\leq A_{n,0,\infty}(a)\cdot \|f^{(n)}\|_{L_\infty[0;1]}$. 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 $\overset\circ{W}{}^n_\infty[0;1]$ in $L_\infty[0;1]$ are found.
Keywords:estimates of derivatives, Sobolev spaces, embedding theorems, approximation by polynomials, Peano kernel.
UDC:517.984, 517.518.82
Presented:B. S. Kashin Received: 21.11.2023 Revised: 11.01.2024 Accepted: 09.02.2024
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