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Properties of Chebyshev Generalized Rational Fractions in \(L_1\)

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Abstract

It is shown that, under a natural constraint, a set of generalized rational fractions in an atomless \(L_1\)-space is a Chebyshev set with continuous metric projection only if this set is convex. Hence this set is not a uniqueness set in \(L_1\), and therefore, some \(x\in L_1\) has at least two nearest points in this set. As a result, it is shown that the set of classical algebraic fractions \( \mathscr{R} _{n,m}\) (consisting of ratios of algebraic polynomials of degree \(\le n\), \(\le m\), respectively) is not a Chebyshev set in \( L_1[a,b]\), and therefore, there exists a function \(x\in L_1[a,b]\) with at least two nearest points in \( \mathscr{R} _{n,m}\). This result solves one long-standing problem in rational approximation.

DOI 10.1134/S1061920822040161

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Notes

  1. Here “a.e.” stands for “almost everywhere”.

  2. Given an arbitrary nontrivial Abelian group, we say that a nonempty topological space is acyclic if all its reduced Čech homology groups over \(A\). For more details on acyclic sets, see, for example, [2, Sec. 6].

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Funding

This research was carried out with the financial support of the Russian Science Foundation 22-21-00204.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991, Russia

    I. G. Tsar’kov

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991, Russia

    I. G. Tsar’kov

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  1. I. G. Tsar’kov
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Correspondence to I. G. Tsar’kov.

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Tsar’kov, I.G. Properties of Chebyshev Generalized Rational Fractions in \(L_1\). Russ. J. Math. Phys. 29, 583–587 (2022). https://doi.org/10.1134/S1061920822040161

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