Abstract:
Order estimates are obtained for the Kolmogorov n-widths of the intersection of two weighted Sobolev classes on an interval with the same smoothness for large n. The weights have a general form, and one of them is in a certain sense significantly less than the other. The constants in the order equality are independent of the weights. Order estimates are obtained for the Kolmogorov n-widths of the intersection of two weighted Sobolev classes Wrp1,g1[a,b] and Wrp2,g2[a,b] in the weighted Lebesgue space Lq,v[a,b] for large n. It is assumed that p1>p2. The weights g1, g2, and v have general form. The conditions on these functions are such that the order of the width in n is the same as for the unweighted Sobolev class Wrp1[a,b]. In addition, the weight g2 in a certain sense is considerably less than the weight g1. The constants in the order equality for the width depend only on p1, p2, q, and r. The upper estimate reduces to the use of our earlier result (2010) for one weighted Sobolev class. The lower estimate is derived by using the discretization method and estimating the width of the intersection of the p1- and p2-ellipsoids. Then a polyhedron of special form is inscribed in this set, and the required lower estimate is obtained for the width of the polyhedron under an appropriate choice of the parameters.
Keywords:
Kolmogorov widths, intersection of function classes.
Citation:
A. A. Vasil'eva, “Kolmogorov widths of the intersection of two weighted Sobolev classes on an interval with the same smoothness”, Trudy Inst. Mat. i Mekh. UrO RAN, 29, no. 4, 2023, 55–63
\Bibitem{Vas23}
\by A.~A.~Vasil'eva
\paper Kolmogorov widths of the intersection of two weighted Sobolev classes on an interval with the same smoothness
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2023
\vol 29
\issue 4
\pages 55--63
\mathnet{http://mi.mathnet.ru/timm2036}
\crossref{https://doi.org/10.21538/0134-4889-2023-29-4-55-63}
\elib{https://elibrary.ru/item.asp?id=54950395}
\edn{https://elibrary.ru/aintub}
Linking options:
https://www.mathnet.ru/eng/timm2036
https://www.mathnet.ru/eng/timm/v29/i4/p55
This publication is cited in the following 1 articles:
Yu. V. Malykhin, K. S. Ryutin, “Poperechniki i zhestkost bezuslovnykh mnozhestv i sluchainykh vektorov”, Izv. RAN. Ser. matem., 89:2 (2025), 45–59