Remarks on \(L^1\)-solutions of dilation equations (Q2490650)

The problem of the existence of non-trivial \(L^{1}\)-solutions \(f:\mathbb{R} \rightarrow \mathbb{C}\) of the dilation equation \[ f(x)=\sum_{n=0}^{N} c_{n} f(\alpha x - \beta_{n}) \] is studied. Throughout the paper it is assumed that \(N\) is a positive integer, \(\alpha >1\) and \(\beta_{0} < \ldots < \beta_{N}\) are real constants, and \(c_{0}, \ldots , c_{N}\) are complex constants such that \[ \sum_{n=0}^{N} c_{n} =\alpha. \] First the author gives the necessary condition for the existence of a non-trivial \(L^{1}\)-solution of this equation under the additional assumption that the constants \(c_{0}, \ldots , c_{N}\) are positive real numbers. The next two theorems say that if there exists a non-trivial \(L^{1}\)-solution of the dilation equation for some constants \(\alpha\), \(\beta_{0}, \ldots , \beta_{N}\), and \(c_{0}, \ldots , c_{N}\), then there exists a non-trivial \(L^{1}\)-solution of the dilation equation for other constants. In the first of these theorems the orders of the two dilation equations are the same, whereas in the second one the constant \(\alpha\) is the same for both equations. In the last part of the paper the author shows applications of these theorems.