Abstract
This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series.
Similar content being viewed by others
References
Apostol, T.M.: Modular Functions and Dirichlet Series in Number Theory. Springer, New York (1990)
Ash, R.B., Novinger, W.P.: Complex Variables. Academic Press, New York (2004)
Besicovitch, A.S.: Almost Periodic Functions. Dover, New York (1954)
Bohr, H.: Zür Theorie der allgemeinen Dirichletschen Reihen. Math. Ann. 79, 136–156 (1918)
Bohr, H.: Contribution to the theory of almost periodic functions, Det Kgl. danske Videnskabernes Selskab. Matematisk-fisiske meddelelser. Bd. XX. Nr. 18, Copenhague (1943)
Bohr, H.: Almost Periodic Functions. Chelsea, New York (1951)
Corduneanu, C.: Almost Periodic Functions. Interscience publishers, New York (1968)
Jessen, B.: Some aspects of the theory of almost periodic functions. In: Proceedings of International Congress Mathematicians Amsterdam. vol. 1, pp. North-Holland, pp. 304–351 (1954)
Righetti, M., Sepulcre, J.M., Vidal, T.: The equivalence principle for almost periodic functions, available online: arXiv:1901.07917
Righetti, M.: On Bohr’s equivalence theorem, J. Math. Anal. Appl. 445 (1) (2017), 650–654. corrigendum, ibid. 449 (2017), 939–940
Sepulcre, J.M., Vidal, T.: Almost periodic functions in terms of Bohr’s equivalence relation, Ramanujan J., 46 (1) (2018), 245–267; Corrigendum, ibid, 48 (3), 685–690 (2019)
Sepulcre, J.M., Vidal, T.: Bohr’s equivalence relation in the space of Besicovitch almost periodic functions. Ramanujan J. 49(3), 625–639 (2019)
Sepulcre, J.M., Vidal, T.: A generalization of Bohr’s equivalence theorem. Complex Anal. Oper. Theory 13(4), 1975–1988 (2019)
Spira, R.: Sets of values of general Dirichlet series. Duke Math. J. 35(1), 79–82 (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
J.M. Sepulcre research was partially supported by MICIU of Spain under Project Number PGC2018-097960-B-C22.
Rights and permissions
About this article
Cite this article
Sepulcre, J.M., Vidal, T. Sets of values of equivalent almost periodic functions. Ramanujan J 56, 87–102 (2021). https://doi.org/10.1007/s11139-020-00344-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-020-00344-0
Keywords
- Almost periodic functions
- Exponential sums
- Bohr equivalence theorem
- Dirichlet series
- Bohr-equivalence relation