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The ortho-widths of some classes of periodic functions of two variables with a given majorant of the mixed moduli of continuity. (English. Russian original) Zbl 1007.42004
Izv. Math. 64, No. 1, 121-141 (2000); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 1, 123-144 (2000).
For a class of functions \(F\subset L_p\) the \(M\)th ortho-width is defined by \[ d_M^\perp(F,L_p)=\inf_{\{u_i(x)\}_{i=1}^M}\sup_{f\in F}\|f(x)-\sum_{i=1}^M(f(\cdot),u_i(\cdot))u_i(x)\|_p, \] where the infimum is taken over the orthonormal systems \(\{u_i(x)\}_{i=1}^M\) of bounded functions. The author considers classes of periodic functions of two real variables in \(L_p([-\pi,\pi]^2)\) with a given majorant of the mixed moduli of continuity. Namely, let \(\Omega\) be a given function of the mixed modulus of continuity type, then the class \(H_q^\Omega\) consists of all functions \(f\in L_q\) for which \(\int_{-\pi}^\pi f(x_1,x_2) dx_j=0,\)   \(j=1,2,\) and whose modulus of continuity satisfies \(\Omega(f;(t_1,t_2))_q\leq\Omega(t_1,t_2).\) In this paper, it is assumed that \(\Omega(0,0)=\Omega(t_1,0)=\Omega(0,t_2) =0\) and \[ \Omega(t_1,t_2)={t_1^r\over(\log{1/t_1})^{b_1}}\cdot{t_2^r\over(\log{1/t_2})^{b_2}}. \] In 8 theorems, the author obtains sharp ordinal estimates for the widths \(d_M^\perp(H_q^\Omega,L_p),\) as well as for related values, for various combinations of \(p,q,r,b_1,\) and \(b_2.\) Here anisotropic effects appear, discovered first by Telyakovskij in the case of the Nikols’kij classes, namely, the best crosses for \(H_q^\Omega\) are sometimes “improper”, that is, those associated with other such classes.

MSC:
42A10 Trigonometric approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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