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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
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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.


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  • 1. S. Mandelbrojt, Séries adhérentes, régularisation des suites, applications, Gauthier-Villars, Paris, 1952 (French). MR 0051893
  • 2. E. M. Dyn′kin, Pseudoanalytic continuation of smooth functions. Uniform scale, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974) Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 40–73 (Russian). MR 0587795
  • 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, Mat. Sb. 199 (2008), no. 7, 41–62 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 7-8, 985–1007 (2008). MR 2488222, https://doi.org/10.1070/SM2008v199n07ABEH003950
  • 5. H. G. Dales and A. M. Davie, Quasianalytic Banach function algebras, J. Functional Analysis 13 (1973), 28–50. MR 0343038
  • 6. A. F. Leont′ev, \cyr Posledovatel′nosti polinomov iz èksponent, “Nauka”, Moscow, 1980 (Russian). MR 577300
  • 7. V. V. Andrievskiĭ, V. I. Belyĭ, and V. K. Dzyadyk, \cyr Konformnye invarianty v konstruktivnoĭ teorii funktsiĭ kompleksnogo peremennogo, “Naukova Dumka”, Kiev, 1998 (Russian, with Russian and Ukrainian summaries). MR 1738290
  • 8. Dieter Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). MR 604011
  • 9. A. F. Leont′ev, \cyr Ryady eksponent, Izdat. “Nauka”, Moscow, 1976 (Russian). MR 0584943
  • 10. V. K. Dzyadyk, \cyr Vvedenie v teoriyu ravnomernogo priblizheniya funktsiĭ polinomami, Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0612836
  • 11. I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. MR 0067952
    I. P. Natanson, Theory of functions of a real variable. Vol. II, Translated from the Russian by Leo F. Boron, Frederick Ungar Publishing Co., New York, 1961. MR 0148805
  • 12. B. V. Shabat, \cyr Vvedenie v kompleksnyĭ analiz., Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0584932
    B. W. Szabat, Wstęp do analizy zespolonej, Państwowe Wydawnictwo Naukowe, Warsaw, 1974 (Polish). Translated from the Russian by Krystyna Linder and Ryszard Kopiecki. MR 0584933
    B. V. Shabat, \cyr Vvedenie v kompleksnyĭ analiz. Chast′ I. Funktsii odnogo peremennogo., Izdat. “Nauka”, Moscow, 1976 (Russian). Second edition, revised and augmented. MR 0584934
    B. V. Shabat, \cyr Vvedenie v kompleksnyĭ analiz. Chast′ II. Funktsii neskol′kikh peremennykh., Izdat. “Nauka”, Moscow, 1976 (Russian). Second edition, revised and augmented. MR 0584935
  • 13. F. D. Gahov, \cyr Kraevye zadachi., Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0156162
    F. D. Gakhov, Boundary value problems, Translation edited by I. N. Sneddon, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. MR 0198152
  • 14. N. I. Muskhelishvili, \cyr Singulyarnye integral′nye uravneniya, Third, corrected and augmented edition, Izdat. “Nauka”, Moscow, 1968 (Russian). \cyr Granichnye zadachi teorii funktsiĭ i nekotorye ikh prilozheniya k matematicheskoĭ fizike. [Boundary value problems in the theory of function and some applications of them to mathematical physics]; With an appendix by B. Bojarski. MR 0355495
    N. I. Muskhelishvili, Singular integral equations, Wolters-Noordhoff Publishing, Groningen, 1972. Boundary problems of functions theory and their applications to mathematical physics; Revised translation from the Russian, edited by J. R. M. Radok; Reprinted. MR 0355494
  • 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 à l’étude des fonctions dérivables d’une variable réelle, Acta Math. 71 (1939), 317–358 (French). MR 0000848, https://doi.org/10.1007/BF02547758
  • 18. Henri Cartan, Sur les classes de fonctions définies par des inégalités portant sur leurs dérivées successives, Actual. Sci. Ind., no. 867, Hermann et Cie., Paris, 1940 (French). MR 0006352
  • 19. Mathematica. Encyclopaedic dictionary, Soviet. Encicloped., Moscow, 1988. (Russian)
  • 20. Thøger Bang, Om quasi-analytiske Funktioner, Thesis, University of Copenhagen,], 1946 (Danish). MR 0017782
  • 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, https://doi.org/10.2307/2315100
  • 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)

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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