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A class of functional equations associated with almost periodic functions

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Abstract

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.

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References

  1. Almira, J.M., Abu-Helaiel, Kh.F.: On solutions of \(f(x)+f(a_1x)+\ldots +f(a_Nx)=0\) and related equations. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity 9, 3–17 (2011)

  2. Ash, R.B.: Complex Variables. Academic Press, London (1971)

    MATH  Google Scholar 

  3. Besicovitch, A.S.: Almost Periodic Functions. Dover, New York (1954)

    Google Scholar 

  4. Bohr, H.: Zur Theorie der fastperiodischen Funktionen. (German) III. Dirichletentwicklung analytischer Funktionen. Acta Math. 47(3), 237–281 (1926)

    Article  MathSciNet  Google Scholar 

  5. Bohr, H.: Almost Periodic Functions. Chelsea, New York (1951)

    MATH  Google Scholar 

  6. Corduneanu, C.: Almost Periodic Functions. Interscience Publishers, New York (1968)

    MATH  Google Scholar 

  7. Favorov, SYu.: Zeros of holomorphic almost-periodic functions, zeros of holomorphic almost periodic functions. J. Anal. Math. 84, 51–66 (2001)

    Article  MathSciNet  Google Scholar 

  8. Jessen, B.: Some aspects of the theory of almost periodic functions. In: Proceedings of International Congress Mathematicians Amsterdam, vol. 1, pp. 304–351. North-Holland (1954)

  9. Mas, A., Sepulcre, J.M.: The projections of the zeros of exponential polynomials with complex frequencies. Colloq. Math. 158(1), 91–102 (2019)

    Article  MathSciNet  Google Scholar 

  10. Mora, G.: A note on the functional equation \(F(z)+F(2z)+\cdots +F(nz)=0\). J. Math. Anal. Appl. 340, 466–475 (2008)

    Article  MathSciNet  Google Scholar 

  11. Mora, G., Sepulcre, J.M.: The zeros of Riemann zeta partial sums yield solutions to \(f(x)+f(2x)+ \cdots +f(nx)=0\). Mediterr. J. Math. 10(3), 1221–1232 (2013)

    Article  MathSciNet  Google Scholar 

  12. Sepulcre, J.M., Vidal, T.: On the analytic solutions of the functional equations \(w_1f(a_1z)+w_2f(a_2z)+\cdots +w_nf(a_nz)=0\). Mediterr. J. Math. 12, 667–678 (2015)

    Article  MathSciNet  Google Scholar 

  13. Yang, S.J., Wu, H.Z., Zhang, Q.B.: Generalization of Vandermonde determinants. Linear Algebra Appl. 336, 201–204 (2001)

    Article  MathSciNet  Google Scholar 

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Correspondence to J. M. Sepulcre.

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J. M. Sepulcre was supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE).

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Sepulcre, J.M., Vidal, T. A class of functional equations associated with almost periodic functions. Aequat. Math. 95, 91–105 (2021). https://doi.org/10.1007/s00010-020-00732-3

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  • DOI: https://doi.org/10.1007/s00010-020-00732-3

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