Functional equations generating space-densifying curves
The first part of the paper offers the result that there are infinitely many linearly independent continuous solutions of the functional equation (1) \( f(x)=-f(2x)\) for \(x\) in \([0,1]\) or in \(\mathbb{R}\) or in \(\mathbb{C}\), with values in \(\mathbb{R}\) or in \(\mathbb{C}\) or in a subset of a metric space, the graphs of which are \(\alpha\)-dense there. In Example 1 it is verified that a class of functions satisfies (1) and the conclusion is drawn that ``therefore, this class is equivalent to the set of solutions to functional equation (1)''. The general continuous solution to this functional equation can clearly be chosen arbitrarily on \(]1/2,1]\) or \([-1,-1/2[ \cup ]1/2,1]\) or on \(\{z |1/2<|z|\leq 1\}\), respectively, as long as it is continuous there and \(\lim_{x\to 1/2} f(x)= - f(1)\) for the first domain, with the additional condition \(\lim_{x\to -1/2} f(x)=-f(-1)\) for the second and with \(\lim_{r\to 1/2} f(r e^{it})=-f(e^{it})\) for the third domain, further \(f(0)=0\), and then \(f\) can be extended to its whole domain by equation (1) itself. Similar remarks apply to the functional equation \(f(x)=-f(cx)\), also mentioned in the paper. The second part of the paper contains similar statements about the functional equation \(f(x)+f(2x)+f(3x)=0\), to which also noncontinuous solutions are offered.
scientific article; zbMATH DE number 1471218