Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups
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| Format: | Preprint |
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2024
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| _version_ | 1866914786884714496 |
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| author | Aserrar, Youssef Elqorachi, Elhoucien |
| author_facet | Aserrar, Youssef Elqorachi, Elhoucien |
| contents | Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $σ:S\rightarrow S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) +\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) -\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] where $μ$ is a measure that is a linear combination of Dirac measures $(δ_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03835 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups Aserrar, Youssef Elqorachi, Elhoucien Functional Analysis 39B52 (Primary), 39B32 (Secondary) Let $S$ be a semigroup, $Z(S)$ the center of $S$ and $σ:S\rightarrow S$ is an involutive automorphism. Our main results is that we describe the solutions of the Kannappan-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) +\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] and the Van Vleck-Wilson functional equation \[\displaystyle \int_{S} f(xyt)dμ(t) -\displaystyle \int_{S} f(σ(y)xt)dμ(t)= 2f(x)g(y),\ x,y\in S,\] where $μ$ is a measure that is a linear combination of Dirac measures $(δ_{z_i})_{i\in I}$, such that $z_i\in Z(S)$ for all $i\in I$. Interesting consequences of these results are presented. |
| title | Kannappan-Wilson and Van Vleck-Wilson functional equations on semigroups |
| topic | Functional Analysis 39B52 (Primary), 39B32 (Secondary) |
| url | https://arxiv.org/abs/2405.03835 |