A study of various results for a class of entire Dirichlet series with complex frequencies

Niraj Kumar; Garima Manocha

Mathematica Bohemica (2018)

  • Volume: 143, Issue: 1, page 1-9
  • ISSN: 0862-7959

Abstract

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Let F be a class of entire functions represented by Dirichlet series with complex frequencies ∑ a k e 〈 λ k , z 〉 for which ( | λ k | / e ) | λ k | k ! | a k | is bounded. Then F is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. F is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to F have also been established.

How to cite

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Kumar, Niraj, and Manocha, Garima. "A study of various results for a class of entire Dirichlet series with complex frequencies." Mathematica Bohemica 143.1 (2018): 1-9. <http://eudml.org/doc/294877>.

@article{Kumar2018,
abstract = {Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k \{\rm e\}^\{\langle \lambda ^k, z\rangle \}$ for which $(|\lambda ^k|/\{\rm e\})^\{|\lambda ^k|\} k!|a_k|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.},
author = {Kumar, Niraj, Manocha, Garima},
journal = {Mathematica Bohemica},
keywords = {Dirichlet series; Banach algebra; topological zero divisor; division algebra; continuous linear functional; total set},
language = {eng},
number = {1},
pages = {1-9},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A study of various results for a class of entire Dirichlet series with complex frequencies},
url = {http://eudml.org/doc/294877},
volume = {143},
year = {2018},
}

TY - JOUR
AU - Kumar, Niraj
AU - Manocha, Garima
TI - A study of various results for a class of entire Dirichlet series with complex frequencies
JO - Mathematica Bohemica
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 143
IS - 1
SP - 1
EP - 9
AB - Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_k {\rm e}^{\langle \lambda ^k, z\rangle }$ for which $(|\lambda ^k|/{\rm e})^{|\lambda ^k|} k!|a_k|$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.
LA - eng
KW - Dirichlet series; Banach algebra; topological zero divisor; division algebra; continuous linear functional; total set
UR - http://eudml.org/doc/294877
ER -

References

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  1. Khoi, L. H., Coefficient multipliers for some classes of Dirichlet series in several complex variables, Acta Math. Vietnam. 24 (1999), 169-182. (1999) Zbl0942.32001MR1710776
  2. Kumar, N., Manocha, G., 10.1016/S0252-9602(13)60105-8, Acta Math. Sci., Ser. B, Engl. Ed. 33 (2013), 1571-1578. (2013) Zbl1313.30007MR3116603DOI10.1016/S0252-9602(13)60105-8
  3. Kumar, N., Manocha, G., 10.1016/j.joems.2012.10.008, J. Egypt. Math. Soc. 21 (2013), 21-24. (2013) Zbl1277.30004MR3040754DOI10.1016/j.joems.2012.10.008
  4. Kumar, N., Manocha, G., Certain results on a class of entire functions represented by Dirichlet series having complex frequencies, Acta Univ. M. Belii, Ser. Math. 23 (2015), 95-100. (2015) Zbl1336.30004MR3373834
  5. Larsen, R., Banach Algebras---An Introduction, Pure and Applied Mathematics 24. Marcel Dekker, New York (1973). (1973) Zbl0264.46042MR0487369
  6. Larsen, R., Functional analysis---An Introduction, Pure and Applied Mathematics 15. Marcel Dekker, New York (1973). (1973) Zbl0261.46001MR0461069
  7. Srivastava, R. K., Some growth properties of a class of entire Dirichlet series, Proc. Natl. Acad. Sci. India, Sect. A 61 (1991), 507-517. (1991) Zbl0885.30004MR1169262
  8. Srivastava, R. K., On a paper of Bhattacharya and Manna, Internal Report (1993), IC/93/417. (1993) 

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