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Integral Identities for Polyanalytic Functions

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Topics in Classical and Modern Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

In this paper we overview some results from the theory of polyanalytic functions. We consider the problem of the mean value of polyanalytic functions of certain types. Similar problems were studied by M.O. Reade, M.B. Balk, and V.V. Volchkov. The cases of polygonal and circular domains with a polynomial weight are studied. We consider the results that are versions of the classical Cauchy, Morera, and Fedoroff theorems for polyanalitic functions under some conditions.

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References

  1. A. Daghighi, S.G. Krantz, Local maximum modulus property for polyanalytic functions. Complex Anal. Oper. Theory 10(2), 401–408 (2016)

    Article  MathSciNet  Google Scholar 

  2. V. Danchenko, Cauchy and Poisson formulas for polyanalytic functions and their applications. (Russian). Izv. Vyssh. Uchebn. Zaved. Mat. 60(1), 15–26 (2016)

    Google Scholar 

  3. S. Kakutani, M. Nagumo, On the functional equation \(\sum _{\nu =0}^{n-1}f(z+e^{2\nu \pi i}\xi )+nf(n)\). Zenkoku Sugaku Danwakai 66, 10–12 (1935)

    Google Scholar 

  4. H.R. Malonek, Selected topics in hypercomplex function theory. Clifford algebras and potential theory. Univ. Joensuu Dept. Math. Rep. Ser. 7, 111–150 (2004)

    Google Scholar 

  5. D. Pompeiu, Sur une classe de fonctions d’une variable complexe. Rendi Circ Mat. Palermo 33(1), 108–113 (1912)

    Article  Google Scholar 

  6. I.I. Privalov, Subharmonic Functions (M.-L., ONTI NKTP USSR, 1937), p. 199

    Google Scholar 

  7. T. Ramsey, Y. Weit, Mean values and classes of harmonic functions. Math. Proc. Camb. Philos. Soc. 96, 501–505 (1984)

    Article  MathSciNet  Google Scholar 

  8. M.O. Reade, On areolar monogenic functions. Bull. Am. Math. Soc. 53, 98–103 (1947)

    Article  MathSciNet  Google Scholar 

  9. M.O. Reade, A theorem of Fedoroff. Duke Math. J. 18(1), 105–109 (1951)

    Google Scholar 

  10. O.D. Trofimenko, A mean-value theorem for polyanalytic functions. Proc. Inst. Appl. Math. Mech. 17, 194–196 (2008)

    Google Scholar 

  11. O.D. Trofymenko, Generalization of the mean value theorem for polyanalytic functions in the case of a circle or disk. Visnyk DonNU, Ser. A: Nat. Sci. 1, 28–31 (2009)

    Google Scholar 

  12. O.D. Trofymenko, Analog of the mean-value theorem for polynomials of special form. Ukr. Math. J. 63(5), 815–826 (2011)

    Article  MathSciNet  Google Scholar 

  13. O.D. Trofymenko, Two-radii theorem for solutions of some mean value equations. Mat. Stud. 40(2), 137–143 (2013)

    Google Scholar 

  14. O.D. Trofymenko, Convolution equations and mean-value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane. Ukr. Mat. Visn., 14(2), 279–294 (2017)

    Google Scholar 

  15. V.V. Volchkov, Morera type theorems on the unit disc. Anal. Math. 20, 49–63 (1994)

    Article  MathSciNet  Google Scholar 

  16. V.V. Volchkov, New mean-value theorems for polyanalytic functions (Russian). Mat. Zametki 56(3), 20–28, 157 (1994); translation in Math. Notes 56(3–4) (1994) 1995, 889–895

    Google Scholar 

  17. V.V. Volchkov, Integral Geometry and Convolution Equations (Kluwer Academic, Dordrecht, 2003), p. 454

    Google Scholar 

  18. V.V. Volchkov, V.V. Volchkov, Offbeat Integral Geometry on Symmetric Spaces (Birkhäuser/Springer, Basel, 2013), p. x+592

    Book  Google Scholar 

  19. J.L. Walsh, A mean value theorem for polynomials and harmonic polynomials. Bull. Am. Math. Soc. 42, 923–930 (1936)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The study was carried out within the Fundamental Research Programme funded by the Ministry of Education and Science of Ukraine, Project No. 0118U003138. The authors want to thank the referee. We highly appreciate his/her comments and suggestions, which significantly contributed to improving the quality of this publication.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, USA

    Anastasiia Minenkova

  2. Department of Mathematics, Vasyl’ Stus Donetsk National University, Vinnytsia, Ukraine

    Olga Trofimenko

Authors
  1. Anastasiia Minenkova
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  2. Olga Trofimenko
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Corresponding author

Correspondence to Anastasiia Minenkova .

Editor information

Editors and Affiliations

  1. Georgia Southern University, Statesboro, GA, USA

    Martha Abell

  2. Georgia Southern University, Statesboro, GA, USA

    Emil Iacob

  3. Georgia Southern University, Statesboro, GA, USA

    Alex Stokolos

  4. Georgia Southern University, Statesboro, GA, USA

    Sharon Taylor

  5. ICREA, Passeig de Lluís Companys, Barcelona, Spain ; Centre de Recerca Matemàtica, Barcelona, Spain

    Sergey Tikhonov

  6. Georgia Southern University, Statesboro, GA, USA

    Jiehua Zhu

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Minenkova, A., Trofimenko, O. (2019). Integral Identities for Polyanalytic Functions. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_17

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