Analysis Mathematica

, Volume 1, Issue 1, pp 63–73 | Cite as

кВАжИАНАлИтИЧЕскОЕ пРОДОлжЕНИЕ И ДИОФАН тОВы пРИБлИжЕНИь

  • А. И. пАВлОВ
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Quasianalytic continuation and diophantine approximation

Abstract

A class of functionsf(z) with the following properties is constructed:
  1. a)

    f(z) is analytic in the half-plane Rez>0;

     
  2. b)

    all the points on the imaginary axis are singular points off(z);

     
  3. c)

    there exists, on the imaginary axis, an everywhere dense setS of the cardinality of the continuum such thatf(z) is quasianalytically extendable through every pointiy0 of this set along the ray\(\Gamma _{y_0 } = \{ z:z = x + iy_0 , - \infty< x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } 0\}\).

     

The construction of the set S uses the method of Diophantine approximation.

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лИтЕРАтУРА

  1. [1]
    T. Carleman,Les fonctions quasi analytiques, Gauthier-Villars (Paris, 1926).Google Scholar
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    A. Ф. лЕОНтьЕВ, ОБ ОДН ОМ ДОпОлНЕНИИ к тЕОРЕ МЕ АДАМАРА,ДОкл. АН сс сР,206 (1972), 1049–1051.Google Scholar
  3. [3]
    А. Ф.лЕОНтьЕВ, РьДы п Ол ИНОМОВ ДИРИхлЕ И Их ОБ ОБЩЕНИь, тРУДы МАтЕМ. И Н-тА ИМ. В. А. стЕклОВА АН сссР,39 (1951).Google Scholar
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    А. Ф. лЕОНтьЕВ, О ВОс стАНОВлЕНИИ ФУНкцИИ пО кОЁФФИцИЕНтАМ ЕЕ Р ьДА ДИРИхлЕ,ИжВ. АН сс сР, сЕРИь МАтЕМ.,35 (1971), 125–153.Google Scholar

Copyright information

© Akadémiai Kiadó 1975

Authors and Affiliations

  • А. И. пАВлОВ
    • 1
  1. 1.МАтЕМАтИЧЕск ИИ ИНстИтУт ИМ. В. А. стЕ клОВА АН сссРМОскВАсссР

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