Convergence of regularized greedy approximations
- Authors: Svetlov I.P.1
-
Affiliations:
- Lomonosov Moscow State University
- Issue: Vol 89, No 2 (2025)
- Pages: 114-127
- Section: Articles
- URL: https://bakhtiniada.ru/1607-0046/article/view/303949
- DOI: https://doi.org/10.4213/im9608
- ID: 303949
Cite item
Open Access
Access granted
Subscription Access
Abstract
We consider a new version of a greedy algorithm in biorthogonal systems in separable Banach spaces.We consider approximations of an element $f$ via $m$-term greedy sum, which isconstructed from the expansion by choosing the first$m$ greatest in absolute value coefficients.It is known that the greedy algorithm does not always converge to the original element.We prove a theorem showing that the new version of a greedy algorithm(called the regularized greedy algorithm) always converges to the original element in Efimov–Stechkin spaces. We also construct examples that show the significance of the conditions of the main theorem.
Keywords
About the authors
Iurii Petrovich Svetlov
Lomonosov Moscow State University
Author for correspondence.
Email: yuri.svetlov@math.msu.ru
without scientific degree, no status
References
- V. Temlyakov, Greedy approximation, Cambridge Monogr. Appl. Comput. Math., 20, Cambridge Univ. Press, Cambridge, 2011, xiv+418 pp.
- S. V. Konyagin, V. N. Temlyakov, “A remark on greedy approximation in Banach spaces”, East J. Approx., 5:3 (1999), 365–379
- P. Wojtaszczyk, “Greedy algorithms for general Biorthogonal systems”, J. Approx. Theory, 107:2 (2000), 293–314
- V. N. Temlyakov, “Greedy algorithm and $m$-term trigonometric approximation”, Constr. Approx., 14:4 (1998), 569–587
- Н. В. Ефимов, С. Б. Стечкин, “Аппроксимативная компактность и чебышевские множества”, Докл. АН СССР, 140:3 (1961), 522–524
- I. Singer, “Some remarks on approximative compactness”, Rev. Roumaine Math. Pures Appl., 9 (1964), 167–177
- A. R. Alimov, I. G. Tsar'kov, Geometric approximation theory, Springer Monogr. Math., Springer, Cham, 2021, xxi+508 pp.
- S. J. Dilworth, D. Kutzarova, V. N. Temlyakov, “Convergence of some greedy algorithms in Banach spaces”, J. Fourier Anal. Appl., 8:5 (2002), 489–506
- С. В. Конягин, И. Г. Царьков, “Пространства Ефимова–Стечкина”, Вестн. Моск. ун-та. Сер. 1. Матем., мех., 1986, № 5, 20–27
- А. Н. Колмогоров, С. В. Фомин, Элементы теории функций и функционального анализа, 6-е перераб. изд., Наука, М., 1989, 624 с.
- А. Р. Алимов, И. Г. Царьков, Основы геометрической теории приближений, Часть I. Приближение выпуклыми множествами, Изд. П. Ю. Мархотин, М., 2016, 120 с.
- Б. С. Кашин, А. А. Саакян, Ортогональные ряды, 2-е доп. изд., АФЦ, М., 1999, x+550 с.
Supplementary files
