Distributional analog of a functional equation
Let \(I:= (0,1)\), and let \(D(I)\) and \(D(I^{2})\) denote the space of all infinitely differentiable functions with compact support on \(I\) and on \(I^{2}\), respectively. The symbol \(D'(I)\) denotes the dual space of \(D(I)\). Formulas \[ \begin{aligned} Q_{+}[\phi](x) &= \int_{R} \phi(x-y, y)\,dy = \int_{I} \phi(x-y, y)\,dy,\\ Q_{-}[\phi](x) &= \int_{R} \phi(x+y, y)\,dy = \int_{I} \phi(x+y, y)\,dy\\ \text{and} R[\phi](x) &= \int_{I} \phi(x \cdot y, y) \frac{1}{y}\, dy \end{aligned} \] define linear operators \(Q_{+},Q_{-}\) and \(R\) from \(D(I^{2})\) into \(D(I)\), whereas \(Q_{+}^{*}, Q_{-}^{*}\) and \( R^{*}\) denote their adjoint operators. If \(f_{1}\),\(f_{2}\) and \(f_{3}\) are locally integrable functions and every \(T_{i}\) is the regular distribution corresponding to \(f_{i}\) (i = 1, 2, 3) (this is written as \(T_{i} = \lambda_{f_{i}}\)) and \[ Q_{+}^{*}[T_{1}] + Q_{-}^{*}[T_{2}] + R^{*}[T_{3}] = 0,\tag{1} \] then \[ f_{1}(x+y) + f_{2}(x-y) + f_{3}(xy) = 0 \] almost everywhere on \(I^{2}\). If \(T_{1}, T_{2}, T_{3} \in D'(I)\) satisfy equation (1), then they are of the form: \(T_{1} = \lambda_{f_{1}}\), \(T_{2} = \lambda_{f_{2}}\) and \(T_{3} = \lambda_{f_{3}}\), where \(f_{1}(x) = -\gamma x^{2} + \alpha_{2},\) \(f_{2}(x) = \gamma x^{2} + \beta_{2}\), \( f_{3} = 4 \gamma x + a\) for some real \(a, \gamma, \alpha_{2}\) and \( \beta_{2}\) such that \( \alpha_{2} + \beta_{2} + a = 0\).
scientific article; zbMATH DE number 2073724