Timeline for Rearrangements of a power series at the boundary of convergence
Current License: CC BY-SA 3.0
5 events
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| Feb 14, 2014 at 19:44 | comment | added | Andrés E. Caicedo | (Reposting comment lost on migration.) The question seems well defined to me; but it is a pointwise problem related to a known open problem, so sadly I doubt there is much current literature on it. Rearranging the series for f gives us a new series that converges on some points of the boundary. The domain of this rearrangement (on the boundary) may be different from the domain of the original series. Currently, I do not see clearly how to even describe the collection of sets obtainable as domains of a given f. The problem being asked is harder. | |
| Feb 14, 2014 at 9:54 | history | migrated | from math.stackexchange.com (revisions) | ||
| Jan 20, 2014 at 12:42 | comment | added | echinodermata | Let $E_{\sigma}$ be the set of points $z$ where the series $f_{\sigma}(z)$ converges; each $f_{\sigma}$ is a function with domain $E_{\sigma}$. If it's too painful to consider a set of functions that don't even have the same domain, you could give up some generality and use some sort of "aggregate": Say, a function $F:\mathbb{C} \rightarrow \mathcal{P}(\mathbb{C})$ which takes each value of $z$ and returns the (possibly empty) set of convergent values of the series at $z$ over all rearrangements. In any case, you don't have to take "space" too seriously in the vein of a typical function space. | |
| Jan 19, 2014 at 6:36 | comment | added | user68061 | The question is not very well defined, because I don't quite understand what is understood by the resulting ``function'' in the case when, let's say, series is nowhere convergent? Your link is very interesting, maybe you should add it at least as a comment to the question? The problem for all "partially defined" functions seem to monstrous)) | |
| Jan 19, 2014 at 6:24 | history | answered | user68061 | CC BY-SA 3.0 |