The cosine and sine addition and subtraction formulas on semigroups
| Autor: | B. Ebanks |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Acta Mathematica Hungarica. 165:337-354 |
| ISSN: | 1588-2632 0236-5294 |
| DOI: | 10.1007/s10474-021-01167-1 |
| Popis: | The cosine addition formula on a semigroup S is the functional equation $$g(xy) = g(x)g(y) - f(x)f(y)$$ for all $$x,y \in S$$ . We find its general solution for $$g,f \colon S \to \mathbb{C}$$ , using the recently found general solution of the sine addition formula $$f(xy) = f(x)g(y) + g(x)f(y)$$ on semigroups. A simpler proof of this latter result is also included, with some details added to the solution. We also solve the cosine subtraction formula $$g(x\sigma(y)) = g(x)g(y) + f(x)f(y)$$ on monoids, where $$\sigma$$ is an automorphic involution. The solutions of these functional equations are described mostly in terms of additive and multiplicative functions, but for some semigroups there exist points where f and/or g can take arbitrary values. The continuous solutions on topological semigroups are also found. |
| Databáze: | OpenAIRE |
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