Various widths of the class $H_p^r$ in the space~$L_q$

V. E. Maiorov

Izvestiya. Mathematics , Tome 13 (1979) no. 1, pp. 73-87.

Résumé

A method of reducing the computation of $n$-widths of compact sets of functions to the analogous problem for finite-dimensional compact sets is presented. Using this method the author obtains best possible (in the “power scale”) estimates for Kolmogorov, Aleksandrov and entropy $n$-widths of the class $H_p^r$ of functions $f(x)$, $x\in R^S$, that are $2\pi$-periodic in each variable, satisfy the inequality $$ \biggl\|\frac{\partial^{rs}}{\partial x_1^r\cdots\partial x_s^r}\biggr\|_{L_p}\leqslant1 $$ and have the property that any Fourier coefficients with at least one zero index must be equal to zero. Bibliography: 21 titles.

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