Operators and generalized functions, decomposable with respect to systems of coordinates (Q1178167)

Let \((\ell_ 1,\ell_ 2)\) be a pair of lines in \(\mathbf R^ 2\), \(\ell_ 1\neq\ell_ 2\). A linear operator \(T\) on \(L_ 2(\mathbf R^ 2)\) is called decomposable with respect to \((\ell_ 1,\ell_ 2)\) if it can be represented as \(T=T_ 1T_ 2\), where \(T_ 1\) acts only along \(\ell_ 1\) and \(T_ 2\) acts only along \(\ell_ 2\). The following problem is treated: given two pairs of lines \(\ell_ 1\), \(\ell_ 2\), \(\tilde\ell_ 1\), \(\tilde\ell_ 2\), defined by the equations \(x=0\), \(y=0\), \(ax+by=h\), \(cx+dy=r\), respectively, find all linear operators (unitary operators, generalized functions) decomposable with respect to both pairs of lines. All the three problems are particular cases of finding all generalized functions such that \[ f(x,y)=g_ 1(x)g_ 2(y)=g_ 3(ax+by)g_ 4(cx+dy), \] where \(x,y\) are either from \(\mathbf R\) or from \(\mathbf R^ 2\). Putting \(\theta=ad(bc)^ {-1}\) the author gives a complete solution to the problem in \(\theta<0\), and also some partial results for the (much harder) case \(\theta>0\).