Widths of functional classes defined by majorants of generalized moduli of smoothness in the spaces ${\mathcal S}^{p}$

  • F. Abdullayev Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz republic; Mersin University, Mersin, Turkey
  • Anatolii Serdyuk Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
  • A. Shidlich Institute of mathematics, NAS of Ukraine
Keywords: Kolmogorov width, Bernstein width, best approximation, generelized module of smoothness, Jackson-type inequality

Abstract

UDC 517.5

We obtain exact Jackson-type inequalities in terms of best approximations and averaged values of generalized moduli of smoothness in spaces ${\mathcal S}^p$. For classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness, the Kolmogorov, Bernstein, linear, and projective widths in the spaces ${\mathcal S}^p$ are found.

References

F. Abdullayev, S. Chaichenko, A. Shidlich, Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces, Math. Inequal. Appl. 24 (2021), no. 2, 323--336, https://doi.org/10.7153/mia-2021-24-23 DOI: https://doi.org/10.7153/mia-2021-24-23

F. G. Abdullayev, P. O¨ zkartepe, V. V. Savchuk , A. L. Shidlich, Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces ${mathcal S}^p$ , Filomat, 33 , № 5, 1471 – 1484 (2019), https://doi.org/10.2298/fil1905471a DOI: https://doi.org/10.2298/FIL1905471A

N. Ajnulloev, Znachenie poperechnikov nekotory`kh klassov differencziruemy`kh funkczij v $L_2$, Dokl. AN TadzhSSR, 29 , № 8, 415 – 418 (1984).

V. F. Babenko, S. V. Konareva, Neravenstva tipa Dzheksona –Stechkina dlya approksimaczii e`lementov gil`bertova prostranstva, Ukr. mat. zhurn., 70 , № 9, 1155 – 1165 (2018).

J. Boman, H. S. Shapiro, Comparison theorems for a generalized modulus of continuity, Ark. Mat., 9 , 91 – 116 (1971), https://doi.org/10.1007/BF02383639 DOI: https://doi.org/10.1007/BF02383639

J. Boman, Equivalence of generalized moduli of continuity, Ark. Mat., 18 , 73 – 100 (1980)Б https://doi.org/10.1007/BF02384682 DOI: https://doi.org/10.1007/BF02384682

S. B. Vakarchuk, Neravenstva tipa Dzheksona i tochny`e znacheniya poperechnikov klassov funkczij v prostranstvakh ${mathcal S}^p, 1 leq p < infty$ , Ukr. mat. zhurn., 56 , № 5. 595 – 605 (2004).

S. B. Vakarchuk, A. N. Shhitov, O nekotory`kh e`kstremal`ny`kh zadachakh teorii approksimaczii funkczij v prostranstvakh ${mathcal S}^p, 1 leq p < infty$ , Ukr. mat. zhurn., 57 , № 11, 1458 – 1466 (2005).

S. B. Vakarchuk, Neravenstva tipa Dzheksona s obobshhenny`m modulem neprery`vnosti i tochny`e znacheniya $n$-poperechnikov klassov $(psi , beta )$-differencziruemy`kh funkczij v $L_2$. I , Ukr. mat. zhurn., 68 , № 6, 723 – 745 (2016).

S. N. Vasil`ev, Neravenstvo Dzheksona –Stechkina v $ L_2[ pi , pi ]$, Tr. In-ta matematiki i mekhaniki UrO RAN, 7 , № 1, 75 – 84 (2001).

V. R. Vojczekhivs`kij, Nerivnosti tipu Dzheksona pri nablizhenni funkczij z prostoru ${mathcal S}^p$ sumami Zigmunda, Praczi In-tu matematiki NAN Ukrayini, 35 , 33 – 46 (2002).

M. G. Esmaganbetov, Poperechniki klassov iz $L_2[0, 2pi ]$ i minimizacziya tochny`kh konstant v neravenstvakh tipa Dzheksona, Mat. zametki, 66 , № 6, 816 – 820 (1999). DOI: https://doi.org/10.4213/mzm1117

A. I. Kozko, A. V. Rozhdestvenskij, O neravenstve Dzheksona v $L_2$ s obobshhenny`m modulem neprery`vnosti, Mat. sb., 195 , № 8, 3 – 46 (2004). DOI: https://doi.org/10.4213/sm838

A. Pinkus, $n$-Widths in approximation theory, Springer-Verlag (1985)Б 291 pp. ISBN: 3-540-13638-X, https://doi.org/10.1007/978-3-642-69894-1 DOI: https://doi.org/10.1007/978-3-642-69894-1

V. V. Savchuk, A. L. Shidlich, Approximation of functions of several variables by linear methods in the space ${mathcal S}^p$ , Acta Sci. Math. (Szeged), 80 , № 3-4, 477 – 489 (2014)Б https://doi.org/10.14232/actasm-012-837-8 DOI: https://doi.org/10.14232/actasm-012-837-8

A. S. Serdyuk, Poperechniki v prostori ${mathcal S}^p$ klasiv funkczij, shho oznachayut`sya modulyami neperervnosti yikhnikh $psi$ -pokhidnikh, Praczi In-tu matematiki NAN Ukrayini, 46 , 229 – 248 (2003).

A. I. Stepanecz, Approksimaczionny`e kharakteristiki prostranstv ${mathcal S}^p_{varphi}$ , Ukr. mat. zhurn., 53 , № 3, 392 – 416 (2001).

A. I. Stepanets, Methods of approximation theory, VSP, Leiden, Boston (2005)Б https://doi.org/10.1515/9783110195286 DOI: https://doi.org/10.1515/9783110195286

A. I. Stepanecz, Zadachi teorii priblizhenij v linejny`kh prostranstvakh, Ukr. mat. zhurn, 58 , № 1, 47 – 92 (2006).

A. I. Stepanecz, A. S. Serdyuk, Pryamy`e i obratny`e teoremy` priblizheniya funkczij v prostranstve ${mathcal S}^p$ , Ukr. mat. zhurn., 54 , №1, 106 – 124 (2002).

M. D. Sterlin, Tochny`e postoyanny`e v obratny`kh teoremakh teorii priblizhenij, Dokl. AN SSSR, 202 , № 3, 545 – 547 (1972).

L. V. Tajkov, Neravenstva, soderzhashhie nailuchshie priblizheniya i modul` neprery`vnosti funkczij iz $L_2$, Mat. zametki, 20 , № 3, 433 – 438 (1976).

L. V. Tajkov, Strukturny`e i konstruktivny`e kharakteristiki funkczij iz $L_2$ , Mat. zametki, 25 , № 2, 217 – 223 (1979).

V. M. Tikhomirov, Nekotory`e voprosy` teorii priblizhenij, Izd-vo Mosk. gos. un-ta, Moskva (1976).

M. F. Timan, Approksimacziya i svojstva periodicheskikh funkczij, Nauk. dumka, Kiev (2009).

N. I. Cherny`kh, O nailuchshem priblizhenii periodicheskikh funkczij trigonometricheskimi polinomami v $L_2 $, Mat. zametki, 2 , № 2, 513 – 522 (1967).

V. V. Shalaev, O poperechnikakh v $L_2$ klassov differencziruemy`kh funkczij, opredelyaemy`kh modulyami neprery`vnosti vy`sshikh poryadkov, Ukr. mat. zhurn., 43 , № 1, 125 – 129 (1991).

H. S. Shapiro, A Tauberian theorem related to approximation theory, Acta Math., 120 , 279 – 292 (1968), https://doi.org/10.1007/BF02394612 DOI: https://doi.org/10.1007/BF02394612

Kh. Yussef, O nailuchshikh priblizheniyakh funkczij i znacheniyakh poperechnikov klassov funkczij v $L_2$ , Primenenie funkczional`nogo analiza v teorii priblizhenij: Sb. nauch.tr. Tver. gos. un-ta, 100 – 114 (1988).

Kh. Yussef, Poperechniki klassov funkczij v prostranstve$ L_2(0, 2pi )$, Primenenie funkczional`nogo analiza v teorii priblizhenij: Sb. nauch. tr. Kalinin. gos. un-ta, 167 – 175 (1990).

Published
18.06.2021
How to Cite
AbdullayevF., SerdyukA., and ShidlichA. “Widths of Functional Classes Defined by Majorants of Generalized Moduli of Smoothness in the Spaces ${\mathcal S}^{p}$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 6, June 2021, pp. 723 -7, doi:10.37863/umzh.v73i6.6432.
Section
Research articles