Let \(G_ m\) denote the topological product of a sequence of discrete cyclic groups \(Z_{m_ k}\) \((k\geq 0,m_ k\geq 2)\), with the direct product measure \(\mu\) given by the pointwise measures \(\mu_ k\) for which \(\mu_ k(j)=1/m_ k\) \((j\in Z_{m_ k})\). Starting with a certain particular complete and orthonormal system of characters \(\psi_ 0,\psi_ 1,\psi_ 2,\ldots\) on \(G_ m\), the author constructs a further explicit complete and orthonormal system of functions \(\chi_ 0,\chi_ 1,\chi_ 2,\ldots\) in \(L(G_ m)\). These functions \(\chi_ n\) are not characters, but are sufficiently similar to characters to allow the author the development of a Fourier-type theory using techniques of ``Vilenkin system theory'' [compare e.g. \textit{J. Gosselin}, Trans. Am. Math. Soc. 185, 345-370 (1974; Zbl 0281.43007)]. Finally, an application of the present theory is given to Besicovitch-type almost-even arithmetical functions.