Approximation of functions of variable smoothness by Fourier–Legendre sums

Assume that $0<\mu\leqslant 1$, and let $r\geqslant 1$ be an integer. Let $\Delta =\{a_1,\dots,a_l\}$, where the $a_i$ are points in the interval $(-1,1)$. The classes $S^rH^\mu_\Delta$ and $S^rH^\mu_\Delta(B)$ are introduced. These consist of functions with absolutely continuous $(r-1)$th derivative on $[-1,1]$ such that their $r$th and $(r+1)$th derivatives satisfy certain conditions outside the set $\Delta$. It is proved that for $0<\mu<1$ the Fourier–Legendre sums realize the best approximation in the classes $S^rH^\mu_\Delta(B)$. Using the Fourier–Legendre expansions, polynomials $\mathscr Y_{n+2r}$ of order $n+2r$ are constructed that possess the following property: for $0<\mu<1$ the $\nu$th derivative of the polynomial $\mathscr Y_{n+2r}$ approximates $f^{(\nu)}(x)$ $(f\in S^rH^\mu_\Delta)$ on $[-1,1]$ to within $O(n^{\nu+1-r-\mu})$, and the accuracy is of order $O(n^{\nu-r-\mu})$ outside $\Delta$.