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Distributional analog of a functional equation. (English) Zbl 1049.39028
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 $$\align Q_{+}[\phi](x) &= \int_{R} \phi(x-y, y)\,dy = \int_{I} \phi(x-y, y)\,dy,\\ Q_{-}[\phi](x) &= \int_{R} \phi(x+y, y)\,dy = \int_{I} \phi(x+y, y)\,dy\\ \intertext{and} R[\phi](x) &= \int_{I} \phi(x \cdot y, y) \frac{1}{y}\, dy \endalign$$ 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 $$Q_{+}^{*}[T_{1}] + Q_{-}^{*}[T_{2}] + R^{*}[T_{3}] = 0,\tag1$$ 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$.

39B52Functional equations for functions with more general domains and/or ranges
46F10Operations with distributions (generalized functions)
Full Text: DOI
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