Vol 218, No 5 (2016)

We obtain formulas for the generalized functional norm associated with the two-weight integral quasi-norm. We describe a minimal generalized Banach function space containing a given quasi-Banach space defined by the two-weight integral quasi-norm.

We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packets of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is “compressed” in one universal optimality condition by using the concept of centered family of compacta.

This is a survey of some recent applications of Boolean-valued models of set theory to the study of order-bounded operators in vector lattices.

V. M. Tikhomirov expressed Kolmogorov widths of the class W^r := W_∞^r[−1, 1] in the space C := C[−1, 1] as a norm of special splines: d_N(W^r, C) = ‖xN − r, r‖C, N ≥ r; these splines were named Chebyshev splines. The function x_n,r is a perfect spline of order r with n knots. We study the asymptotic behavior of Chebyshev splines for r→∞and fixed n. We calculate the asymptotics of knots and the C-norm of x_n,r and prove that x_n,r/x_n,r(1) = T_n+r+o(1). As a corollary, we obtain that d_n+r(W^r, C)/d_r(W^r, C) ~ A_nr^−n/2 as r→∞.

Let Ω be an open subset of the complex plane ℂ and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H^∞ (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy–Sobolev-type in L^p-spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H^∞ (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the ε-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber’s approximations in connection with their applications in numerical analysis are mentioned.