Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation

• Autorzy: Valiullin Artur R. , Valiullin Albert R. , Solodov Alexei P.

Identyfikatory

DOI

10.1515/dema-2022-0019

• Języki publikacji: EN

Abstrakty

EN

Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors {ξn}∞n=1 is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑∞n=1ξ2n<∞ . In the present article, we relax this condition and show that the convergence is guaranteed for ξn=o(1√n) . This result is sharp because the convergence may fail to hold for ξn≍1√n .

Słowa kluczowe

EN

greedy expansion; prescribed coefficients; Hilbert space; greedy approximation; convergence

PL

ekspansja; współczynniki; przestrzeń Hilberta; aproksymacja; konwergencja

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 254--264
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Valiullin Artur R.

• Department of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia
• Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

• https://orcid.org/0000-0002-2971-7385

autor

Valiullin Albert R.

• Department of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia
• Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

• https://orcid.org/0000-0002-6010-2303

autor

Solodov Alexei P.

• Department of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, GSP-1, Moscow, 119991, Russia
• Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

• https://orcid.org/0000-0002-5910-2736

Bibliografia

• [1] V. V. Galatenko and E. D. Livshits, Generalized approximate weak greedy algorithms, Math. Notes 78 (2005), no. 2, 186–201, DOI: https://doi.org/10.4213/mzm2581.
• [2] R. Gribonval and M. Nielsen, Approximate weak greedy algorithms, Adv. Comput. Math. 14 (2001), no. 4, 361–378, DOI: https://doi.org/10.1023/A:1012255021470.
• [3] V. N. Temlyakov, Weak greedy algorithms, Adv. Comput. Math. 12 (2000), no. 2–3, 213–227, DOI: https://doi.org/10.1023/A:1018917218956.
• [4] J. H. Friedman and W. Stueuzle, Projection pursuit regression, J. Amer. Statist. Assoc. 76 (1981), 817–823, DOI: https://doi.org/10.1080/01621459.1981.10477729.
• [5] S. Mallat and Z. Zhang, Matching pursuit with time-frequency dictionaries, IEEE Trans. Signal Process 41 (1993), no. 12, 3397–3415, DOI: https://doi.org/10.1109/78.258082.
• [6] V. N. Temlyakov, Greedy algorithms with prescribed coefficients, J. Fourier Anal. Appl. 13 (2007), no. 1, 71–86, DOI: https://doi.org/10.1007/s00041-006-6033-x.
• [7] V. N. Temlyakov, Greedy expansions in Banach spaces, Adv. Comput. Math. 26 (2007), no. 4, 431–449, DOI: https://doi.org/10.1007/s10444-005-7452-y.
• [8] Ar. R. Valiullin, Al. R. Valiullin, and V. V. Galatenko, Greedy expansions with prescribed coefficients in Hilbert spaces, Int. J. Math. Math. Sci. 2018 (2018), 4867091, DOI: https://doi.org/10.1155/2018/4867091.
• [9] Ar. R. Valiullin and Al. R. Valiullin, Sharp conditions for the convergence of greedy expansions with prescribed coefficients, Open Math. 19 (2021), no. 1, 1–10, DOI: https://doi.org/10.1515/math-2021-0006.
• [10] K. Knopp, Theory and Application of Infinite Series, Blackie & Son, Glasgow, 1928, pp. 290–293.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).