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Distributional analog of a functional equation. (English) Zbl 1049.39028

Let $I:=\left(0,1\right)$, and let $D\left(I\right)$ and $D\left({I}^{2}\right)$ denote the space of all infinitely differentiable functions with compact support on $I$ and on ${I}^{2}$, respectively. The symbol ${D}^{\text{'}}\left(I\right)$ denotes the dual space of $D\left(I\right)$.

Formulas

$\begin{array}{cc}\hfill {Q}_{+}\left[\varphi \right]\left(x\right)& ={\int }_{R}\varphi \left(x-y,y\right)\phantom{\rule{0.166667em}{0ex}}dy={\int }_{I}\varphi \left(x-y,y\right)\phantom{\rule{0.166667em}{0ex}}dy,\hfill \\ \hfill {Q}_{-}\left[\varphi \right]\left(x\right)& ={\int }_{R}\varphi \left(x+y,y\right)\phantom{\rule{0.166667em}{0ex}}dy={\int }_{I}\varphi \left(x+y,y\right)\phantom{\rule{0.166667em}{0ex}}dy\hfill \\ \multicolumn{2}{c}{\text{and}}\\ \hfill R\left[\varphi \right]\left(x\right)& ={\int }_{I}\varphi \left(x·y,y\right)\frac{1}{y}\phantom{\rule{0.166667em}{0ex}}dy\hfill \end{array}$

define linear operators ${Q}_{+},{Q}_{-}$ and $R$ from $D\left({I}^{2}\right)$ into $D\left(I\right)$, 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

${Q}_{+}^{*}\left[{T}_{1}\right]+{Q}_{-}^{*}\left[{T}_{2}\right]+{R}^{*}\left[{T}_{3}\right]=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

then

${f}_{1}\left(x+y\right)+{f}_{2}\left(x-y\right)+{f}_{3}\left(xy\right)=0$

almost everywhere on ${I}^{2}$.

If ${T}_{1},{T}_{2},{T}_{3}\in {D}^{\text{'}}\left(I\right)$ 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 ${f}_{1}\left(x\right)=-\gamma {x}^{2}+{\alpha }_{2},$ ${f}_{2}\left(x\right)=\gamma {x}^{2}+{\beta }_{2}$, ${f}_{3}=4\gamma x+a$ for some real $a,\gamma ,{\alpha }_{2}$ and ${\beta }_{2}$ such that ${\alpha }_{2}+{\beta }_{2}+a=0$.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 46F10 Operations with distributions (generalized functions)
##### Keywords:
functional equations; distributions; linear operators