Sharp Estimates for Deviations of Linear Approximation Methods for Periodic Functions by Linear Combinations of Moduli of Continuity of Different Order

Abstract

Estimates for deviations are established for a large class of linear methods of approximation of periodic functions by linear combinations of moduli of continuity of different orders. These estimates are sharp in the sense of constants in the uniform and integral metrics. In particular, the following assertion concerning approximation by splines is proved: Suppose that $n,r,\mu \in \mathbb{N},\;\mu \geqslant r + {\text{1,}}\;r$ is odd, $1 \leqslant p \leqslant \infty ,\;f \in W_p^{\left( r \right)}$. Then

$$\begin{gathered}\left\| {f - X_{n,r,\mu } \left( f \right)} \right\|_p \leqslant \left( {\frac{{\pi }}{n}} \right)^r \left\{ {\frac{{A_{r,0} }}{2}{\omega }_{1} \left( {f^{\left( r \right)} ,\frac{{\pi }}{n}} \right)_p + \sum\limits_{{\nu } = 1}^{\mu - r - 1} {A_{r,{\nu }} {\omega }_{\nu } \left( {f^{\left( r \right)} ,\frac{{\pi }}{n}} \right)_p } } \right\} \hfill \\ \quad \quad \quad \quad \quad \quad {\kern 1pt} \quad + \left( {\frac{{\pi }}{n}} \right)^r \left( {\frac{{K_r }}{{{\pi }^r }} - \sum\limits_{{\nu } = 1}^{\mu - r - 1} {2^{\nu } A_{r,{\nu }} } } \right)2^{r - \mu } {\omega }_{\mu - r} \left( {f^{\left( r \right)} ,\frac{{\pi }}{n}} \right)_p ; \hfill \\ \end{gathered}$$

moreover, for $p = 1,\infty$ it is impossible to decrease the constants on $W_p^{\left( r \right)}$. Here, $K_r = \frac{4}{{\pi }}\sum\limits_{l = 0}^\infty {\frac{{\left( { - 1} \right)^{l\left( {r + 1} \right)} }}{{\left( {2l + 1} \right)^{r + 1} }},A_{r,{\nu }} }$ are some explicitly constructed constants, ${\omega }_{\nu } \left( {f,h} \right)_p$ is the modulus of continuity of order r for the function f, and $X_{n,r,\mu }$ are explicitly constructed linear operators with the values in the space of periodic splines of degree $\mu$ of minimal defect with 2n equidistant interpolation points. This assertion implies the sharp Jackson-type inequality

$$\left\| {f - X_{n,r,\mu } \left( f \right)} \right\|_p \leqslant \frac{{K_r }}{{2n^r }}{\omega }_{1} \left( {f^{\left( r \right)} ,\frac{{\pi }}{n}} \right)_p .$$

. Bibliography: 17 titles.