Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Estimation of intermediate derivatives and a Bang-type theorem. I


Author: R. A. Gaǐsin
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 1.
Journal: St. Petersburg Math. J. 27 (2016), 15-31
MSC (2010): Primary 26E10
DOI: https://doi.org/10.1090/spmj/1374
Published electronically: December 7, 2015
MathSciNet review: 3443264
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Certain estimates for intermediate derivatives on a quasismooth arc are proved and applied. For arcs of bounded slope, the corresponding results by Bang and Leont'ev are generalized.


References [Enhancements On Off] (What's this?)

  • 1. S. Mandelbrojt, Series adherentes, regularization des suites, applications, Gauthier-Villars, Paris, 1952. MR 0051893 (14:542f)
  • 2. E. M. Dyn'kin, Pseudoanalytic continuation of smooth function. Uniform scale, Math. Programming and Related Questions (Proc. VII Winter School, Drogobych, 1974), Theory of Function and Functional Analysis, Central Econom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 40-73 (Russian) MR 0587795 (58:28536)
  • 3. S. Mandelbrojt, Quasi-analytical cases of function, ONTI, Moscow-Leningrad, 1937. (Russian)
  • 4. A. M. Gaĭsin and I. G. Kinzyabulatov, A theorem of Levinson-Sjöberg type; applications, Math. Sb. 199 (2008), no. 7, 41-62; English transl., Sb. Math. 199 (2008), no. 7-8, 985-1007. MR 2488222 (2010b:30054)
  • 5. H. G. Dales and A. M. Davie, Quasianalytic Banach function algebras, J. Functional Analysis 13 (1973), no. 1, 28-50. MR 0343038 (49:7782)
  • 6. A. F. Leont'ev, Sequences of polynomials of exponentials, Nauka, Moskow, 1980. (Russian) MR 577300 (81m:30002)
  • 7. V. V. Andrievskiĭ, V. I. Belyĭ, and V. K. Dzyadyk, Conformal invariants in the constructive theory of functions of a complex variable, Naukova Dumka, Kiev, 1998. (Russian) MR 1738290 (2000i:30065)
  • 8. D. Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser, Basel, 1980. MR 604011 (82i:30055)
  • 9. A. F. Leont'ev, Exponential series, Nauka, Moscow, 1976. (Russian) MR 0584943 (58:28451)
  • 10. V. K. Dzyadyk, Introduction to the theory of uniform approximation of functions by polynomials, Nauka, Moscow, 1977. (Russian) MR 0612836 (58:29579)
  • 11. I. P. Natanson, Theory of functions of a real variable, Gosudarstv. Izdat. Tehn.-Teor. Lit., 1957; English transl., Frederick Ungar Publ. Co., New York, 1961. MR 0067952 (16:804c); MR 0148805 (26:63:6309)
  • 12. B. V. Shabat, Introduction to complex analysis. Pt. 1, Nauka, Moscow, 1976. (Russian) MR 0584934 (58:28443)
  • 13. F. D. Gakhov, Boundary value problems, Fizmatgiz, Moscow, 1963; English transl., Pergamon Press, Oxford, 1966. MR 0156162 (27:6094); MR 0198152 (33:6311)
  • 14. N. I. Muskhelishvili, Singular integral equations, Nauka, Moscow, 1968; English transl., Nordhoff Publ., Groningen, 1972. MR 0355495 (50:7969); MR 0355494 (50:7968)
  • 15. J. Hadamard, Sur le module maximum d'une fonction et de ses dérivées, C.R. Séances Soc. Math. France 41 (1914), 68-72.
  • 16. T. Carleman, Les functions quasi analytiques, Paris, 1926.
  • 17. A. Gorny, Contribution á l'étude des fonctions dérivables d'une variable réelle, Acta Math. 71 (1939), 317-358. MR 0000848 (1:137d)
  • 18. H. Cartan, Sur les classes de fonctions définies par des inégalités portan sur leurs dérivées successives, Actual. Sci. Ind., vol. 867, Hermann, Paris, 1940. MR 0006352 (3:292b)
  • 19. Mathematica. Encyclopaedic dictionary, Soviet. Encicloped., Moscow, 1988. (Russian)
  • 20. T. Bang, Om quasi-analytiske funktioner, Thesis, Copenhagen Univ., 1946. MR 0017782 (8:199g)
  • 21. R. L. Zeinstra, Müntz-Szász approximation on curves and area problems for zero sets, Thesis, Amsterdam Univ., 1985.
  • 22. P. J. Cohen, A simple proof of the Denjoy-Carleman theorem, Amer. Math. Monthly, 75 (1968), 26-31. MR 0225957 (37:1547)
  • 23. J. A. Siddiqi, Non-spanning sequences of exponentials on rectifiable plane arcs, Linear and Complex Analysis. Problem Book, Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 555-556.
  • 24. R. A. Gaĭsin, Existence criterion for regular minorant of non-quasianalicity of Carleman class, Nonlinear Analysis and Spectral Problems , Bash. Gos. Univ., Ufa, 2013, pp. 44-46. (Russian)
  • 25. -, Existence criterion for regular minorant, not subject to Bang condition, Fundamental Mathematics and Applications in Natural Sciences, Vol. I, Bash. Gos. Univ., Ufa, 2013, pp. 48-56. (Russian)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 26E10

Retrieve articles in all journals with MSC (2010): 26E10


Additional Information

R. A. Gaǐsin
Affiliation: Bashkir State University, ul. Zaki Validi 32, 450074 Ufa, Russia
Email: rashit.gajsin@mail.ru

DOI: https://doi.org/10.1090/spmj/1374
Keywords: Estimates for intermediate derivatives, quasismooth arc, quasianalytic classes of functions
Received by editor(s): April 1, 2014
Published electronically: December 7, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society