Criterion for boundedness of power series
Question Give a necessary and sufficient criterion for the sequence so that the power series is bounded for all .
Consider a power series that is convergent for all , thus defining a function . Are there criteria to decide whether is bounded (which e.g. is the case for the series with for and for n odd)? Some general remarks:
- A necessary condition for to be bounded is that is the only non-zero or there are infinitely many non-zero 's which change sign infinitely many times.
- Changing a finite set of 's (except ) does leave the subspace of bounded power series.
- The subspace of bounded power series is "large" in the sense that it is both a linear subspace (closed under sums and scalar multiples) and an algebra (closed under products). It includes all functions of the form , where is any entire function . The question whether the subspace of bounded power series contains only these functions seems to be open.