The Cosine-Sine Functional Equation on Semigroups

• Autorzy: Bruce Ebanks
• Języki publikacji: EN

Abstrakty

EN

The primary object of study is the "cosine-sine" functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f, g, h : S → ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles. We also discuss the special case f(xy) = f(x)g(y) + g(x)f(y) - g(x)g(y) separately, since it has an independent direct solution on a general semigroup. We give the continuous solutions on topological semigroups for both equations.(original abstract)

Słowa kluczowe

EN

Mathematics; Mathematical functions

PL

Matematyka; Funkcje matematyczne

• Czasopismo: Annales Mathematicae Silesianae
• Rocznik: 2022
• Tom: 36 (1)
• Strony: 30--52

Twórcy

autor

Bruce Ebanks

• University of Louisville, Louisville, Kentucky, USA

Bibliografia

• Chung J.K., Kannappan Pl., and Ng C.T., A generalization of the cosine-sine functional equation on groups, Linear Algebra Appl. 66 (1985), 259-277.10.1016/0024-3795(85)90137-5
• Ebanks B., Some trigonometric functional equations on monoids generated by their squares, Aequationes Math. 95 (2021), 383-391.10.1007/s00010-020-00730-5
• Ebanks B., The cosine and sine addition and subtraction formulas on semigroups, Acta Math. Hungar. (2021). DOI: 10.1007/s10474-021-01167-110.1007/s10474-021-01167-1
• Ebanks B., Ng C.T., Levi-Civita functional equations and the status of spectral synthesis on semigroups, Semigroup Forum 103 (2021), 469-494.10.1007/s00233-021-10211-z
• Stetkær H., Functional Equations on Groups, World Scientific, Singapore, 2013.10.1142/8830
• Stetkær H., A Levi-Civita functional equation on semigroups, (manuscript received July 30, 2020).
• Székelyhidi L., Convolution Type Functional Equations on Topological Abelian Groups, World Scientific, Singapore, 1991.10.1142/1406

Identyfikatory

DOI

10.2478/amsil-2021-0012