Smoothness of functions and Fourier coefficients

Мұқаба

Дәйексөз келтіру

Толық мәтін

Ашық рұқсат Ашық рұқсат
Рұқсат жабық Рұқсат берілді
Рұқсат жабық Тек жазылушылар үшін

Аннотация

We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the $L_p$ modulus of smoothness, $1< p< \infty$, of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory.
Bibliography: 34 titles.

Авторлар туралы

Mikhail Dyachenko

Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Хат алмасуға жауапты Автор.
Email: dyach@mail.ru
Doctor of physico-mathematical sciences, Professor

Askhat Mukanov

Universitat Autònoma de Barcelona; Centre de Recerca Matemàtica; Kazakhstan Branch of Lomonosov Moscow State University

Email: mukanov.askhat@gmail.com

Sergei Tikhonov

Centre de Recerca Matemàtica; Institució Catalana de Recerca i Estudis Avançats; Universitat Autònoma de Barcelona

Email: stikhonov@crm.cat
Candidate of physico-mathematical sciences, no status

Әдебиет тізімі

  1. Н. К. Бари, Тригонометрические ряды, Физматгиз, М., 1961, 936 с.
  2. S. Tikhonov, “Trigonometric series of Nikol'skii classes”, Acta Math. Hungar., 114:1-2 (2007), 61–78
  3. G. G. Lorentz, “Fourier-Koeffizienten und Funktionenklassen”, Math. Z., 51:2 (1948), 135–149
  4. А. Ф. Тиман, М. Ф. Тиман, “Обобщенный модуль непрерывности и наилучшее приближение в среднем”, Докл. АН СССР, 71:1 (1950), 17–20
  5. D. Gorbachev, S. Tikhonov, “Moduli of smoothness and growth properties of Fourier transforms: two-sided estimates”, J. Approx. Theory, 164:9 (2012), 1283–1312
  6. А. А. Конюшков, “О классах Липшица”, Изв. АН СССР. Сер. матем., 21:3 (1957), 423–448
  7. S. Aljančic, “On the integral moduli of continuity in $L_p (1
  8. М. К. Потапов, M. Бериша, “Модули гладкости и коэффициенты Фурье периодических функций одного переменного”, Publ. Inst. Math. (Beograd) (N.S.), 26(40) (1979), 215–228
  9. S. Aljančic, M. Tomic, “Über den Stetigkeitsmodul von Fourier-Reihen mit monotonen Koeffizienten”, Math. Z., 88:3 (1965), 274–284
  10. R. Askey, “Smoothness conditions for Fourier series with monotone coefficients”, Acta Sci. Math. (Szeged), 28 (1967), 169–171
  11. R. P. Boas Jr., Integrability theorems for trigonometric transforms, Ergeb. Math. Grenzgeb., 38, Springer-Verlag New York Inc., New York, 1967, v+66 pp.
  12. Y. Kolomoitsev, S. Tikhonov, “Hardy–Littlewood and Ulyanov inequalities”, Mem. Amer. Math. Soc. (to appear)
  13. Б. В. Симонов, С. Ю. Тихонов, “Теоремы вложения в конструктивной теории приближений”, Матем. сб., 199:9 (2008), 107–148
  14. S. Tikhonov, “Trigonometric series with general monotone coefficients”, J. Math. Anal. Appl., 326:1 (2007), 721–735
  15. S. Tikhonov, “Best approximation and moduli of smoothness: computation and equivalence theorems”, J. Approx. Theory, 153:1 (2008), 19–39
  16. D. B. H. Cline, “Regularly varying rates of decrease for moduli of continuity and Fourier transforms of functions on $mathbb{R}^d$”, J. Math. Anal. Appl., 159:2 (1991), 507–519
  17. W. Trebels, “Estimates for moduli of continuity of functions given by their Fourier transform”, Constructive theory of functions of several variables (Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., 571, Springer, Berlin, 1977, 277–288
  18. J. Garcia-Cuerva, V. I. Kolyada, “Rearrangement estimates for Fourier transforms in $L^p$ and $H^p$ in terms of moduli of continuity”, Math. Nachr., 228 (2001), 123–144
  19. R. J. Le, S. P. Zhou, “A remark on “two-sided” monotonicity condition: an application to $L_p$ convergence”, Acta Math. Hungar., 113:1-2 (2006), 159–169
  20. L. Leindler, “Relations among Fourier coefficients and sum-functions”, Acta Math. Hungar., 104:1-2 (2004), 171–183
  21. C. Oehring, “Asymptotics of rearranged trigonometric and Walsh–Fourier coefficients of smooth functions”, J. Math. Anal. Appl., 164:2 (1992), 422–446
  22. B. Szal, “On the integral modulus of continuity of Fourier series”, Acta Math. Hungar., 131:1-2 (2011), 138–159
  23. Dansheng Yu, Songping Zhou, “On relations among Fourier coefficients and sum-functions”, Studia Sci. Math. Hungar., 45:3 (2008), 301–319
  24. E. Liflyand, S. Tikhonov, “A concept of general monotonicity and applications”, Math. Nachr., 284:8-9 (2011), 1083–1098
  25. Lei Feng, V. Totik, Song Ping Zhou, “Trigonometric series with a generalized monotonicity condition”, Acta Math. Sin. (Engl. Ser.), 30:8 (2014), 1289–1296
  26. M. I. Dyachenko, S. Yu. Tikhonov, “Smoothness and asymptotic properties of functions with general monotone Fourier coefficients”, J. Fourier Anal. Appl., 24:4 (2018), 1072–1097
  27. M. Dyachenko, E. Nursultanov, A. Kankenova, “On summability of Fourier coefficients of functions from Lebesgue space”, J. Math. Anal. Appl., 419:2 (2014), 959–971
  28. M. I. Dyachenko, E. D. Nursultanov, A. M. Zhantakbayeva, “Hardy–Littlewood type theorems”, Eurasian Math. J., 4:2 (2013), 140–143
  29. Z. Ditzian, V. H. Hristov, K. G. Ivanov, “Moduli of smoothness and $K$-functionals in $L_p$, $0
  30. А. Зигмунд, Тригонометрические ряды, т. I, II, Мир, М., 1965, 615 с., 537 с.
  31. R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993, x+449 pp.
  32. F. Dai, Z. Ditzian, S. Tikhonov, “Sharp Jackson inequalities”, J. Approx. Theory, 151:1 (2008), 86–112
  33. А. А. Конюшков, “Наилучшие приближения тригонометрическими полиномами и коэффициенты Фурье”, Матем. сб., 44(86):1 (1958), 53–84
  34. M. Dyachenko, A. Mukanov, S. Tikhonov, “Hardy–Littlewood theorems for trigonometric series with general monotone coefficients”, Studia Math. (to appear)

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© Дьяченко М.I., Муканов А.B., Тихонов С.Y., 2019

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