Construction of Weights on the Semigroup $({\mathbb N}, \, +)$ Using some Standard Functions

Shreema S. Bhatt1 and H. V. Dedania1

1Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Gujarat, India.

Abstract: A \emph{weight} on the semigroup $(\mathbb N, +)$ of natural numbers is a function $\omega : {\mathbb N} \longrightarrow (0, \infty)$ satisfying the submultiplicativity $\omega(m + n) \leq \omega(m)\omega(n)$ for all $m, n \in {\mathbb N}$. In this simple paper, we exhibit that some standard functions such as $c\cosh(n)$, $c\sinh(n)$, $ n^{k}+c$, $(n + c)^{k}$, $e^{n^c}$, $e^{-n^c}$, $\log(n^k) + c$, $[\log(n) + c]^k$, and much more are weights on $\mathbb N$ under certain conditions on the constant $c$.
Keywords: Semigroup, Weight, Hyperbolic functions, Exponential function, and Logarithmic function.

Cite this article as: Shreema S. Bhatt and H. V. Dedania, Construction of Weights on the Semigroup $({\mathbb N}, \, +)$ Using some Standard Functions, Int. J. Math. And Appl., vol. 9, no. 3, 2021, pp. 41-47.

  1. Tom M. Apostol, Introduction to Analytic Number Theory, Springer International Student Edition, Narosa Publishing House, New Delhi, (1989).
  2. Shreema S. Bhatt, Weights on Groups and Semigroups, and their relevance in Banach Algebras, M.Phil. Dissertation, Sardar Patel University, (2019).
  3. T. D. Blackmore, Weak Amenability of Discrete Semigroup Algebras, Semigroup Forum, 55(1997), 196-205.
  4. H. G. Dales, Banach Algebras and Automatic Continuity, London Math. Soc. Monographs, Vol 24, Clarendon Press, Oxford, (2000).
  5. H. G. Dales and H. V. Dedania, Weighted Convolution Algebras on Subsemigroups of the Real Line, Dissertationes Mathematicae (Rosprawy Matematicze), 459(2009), 1-60.
  6. H. R. Ebrahimi Vishki, B. Khodsiani and A. Rejali, Arens regularity of certain weighted semigroup algebras and countability, Semigroup Forum, 92(2016), 304–310.
  7. N. Gr${\o}$nb${\ae}$k, Amenability of Weighted Discrete Semigroup Algebras on Cancellative Semigroups, Proc. Roy. Soc. Edinburgh Series-A, 110(1988), 351-360.
  8. N. K. Nikolskii, Selected Problems of Weighted Approximations and Spectral Analysis, Proc. Steklov Inst. Math., 120(1974), 1-278.
  9. J. H. Shah, Vector-valued Weighted Discrete Semigroup Algebras, Ph.D. Thesis, Sardar Patel University, (2008).
  10. M. P. Thomas, A Non-standard Ideal of a Radical Banach Algebra of Power Series, Acta Math., 152(1984), 199-217.