Generalized probabilistic approximation characteristic based on Birkhoff orthogonality and related conclusions in $S_{\infty}$-norm

Year 2025, Volume: 8 Issue: 4, 228 - 240, 15.12.2025

Weiye Zhang , Chong Wang , Huan Li

https://doi.org/10.33205/cma.1807082

https://izlik.org/JA62HL56WE

Abstract

In this article, we generalize the definition of the probabilistic Gel'fand width from the Hilbert space to the strictly convex reflexive space by giving Birkhoff left orthogonal decomposition theorem. Meanwhile, a more natural definition of Gel'fand width in the classical setting is selected to make sure probabilistic and average Gel'fand widths will not lose their meaning, so that we can give the equality relation between the probabilistic Gel'fand width and the probabilistic linear width of the Hilbert space. Meanwhile, we use this relationship to continue the study of the Gel'fand widths of the univariate Sobolev space and the multivariate Sobolev space, especially in $S_{\infty}$-norm, and determine the exact order of probabilistic and average Gel'fand widths.

Keywords

$S_{\infty}$-Norm , Sobolev space , probabilistic and average Gelfand widths , Birkhoff orthogonality , strictly convex reflexive space

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