Trigonometric functional equations of rectangular type
The rectangular functional equation \[ \begin{aligned} f(x_1+y_1,x_2+y_2) & +f(x_1+y_1,x_2-y_2)+f(x_1-y_1,x_2+y_2)\\ & +f(x_1-y_1,x_2-y_2)=4f(x_1,x_2)\tag{1}\end{aligned} \] for \(x_1,y_1,x_2,y_2\in\mathbb{R}\) where \(f:\mathbb{R}^2\to\mathbb{C}\) was treated by \textit{J. Aczél}, \textit{H. Haruki}, \textit{M. A. McKiernan} and \textit{G. N. Sakovič} [Aequationes Math. 1, 37-53 (1968; Zbl 0157.46102)]. Two generalizations \[ \text{Left side of (1)=4}\bigl[f(x_1,x_2)F(y_1,y_2)+F(x_1,x_2)f(y_1,y_2)\bigr],\tag{2} \] \[ \text{Left side of (1)=4}\bigl[f(x_1,x_2)f(y_1,y_2)+F(x_1,x_2)F(y_1,y_2)\bigr],\tag{3} \] for \(x_1,y_1\in G_1\) (an abelian group), \(x_2,y_2\in G_2\) (an abelian group) where \(f,F:G_1\times G_2\to\mathbb{C}\) were studied by \textit{J. K. Chung}, \textit{Pl. Kannappan} and \textit{P. K. Sahoo} [Aequationes Math. 53, No. 1-2, 36-53 (1997; Zbl 0901.39007). Corrigendum: ibid. 56, No. 1-2, 199-200 (1998; Zbl 0912.39015)]. \textit{J. K. Chung}, \textit{Pl. Kannappan} and \textit{C. T. Ng} [Math. Rep., Toyama Univ. 11, 153-165 (1988; Zbl 0659.39001)] obtained the solutions to \[ f(x+y)+f(x-y)=2f(x)F(y)+2F(x)f(y)\tag{4} \] \[ f(x+y)+f(x-y)=2f(x)f(y)+2F(x)F(y) \] where \(f,F:G\to\mathbb{C}\), \(G\) (an abelian group). In this paper the author obtains the solutions of \[ {1\over 2^n}\sum^1_{i_1,\dots,i_n=0}f\bigl(x_1+\sigma_1^{i_1}(y_1),\dots,x_n+\sigma_n^{ i_n}(y_n)\bigr)=f(x)F(y)+F(x)f(y),\tag{5} \] and \[ {1\over 2^n}\sum^1_{i_1,\dots,i_n=0}f\bigl(x_1+\sigma_1^{i_1}(y_1),\dots,x_n+ \sigma_n^{i_n}(y_n)\bigr)=f(x)f(y)+F(x)F(y),\tag{6} \] which extends (2), (3), (4), where \(f:G_1\times G_2\times\cdots\times G_n\to\mathbb{C}\) and \(\sigma_i:G_i\to G_i\) is an automorphism such that \(\sigma_i\circ\sigma_i=I\).
scientific article; zbMATH DE number 1295522