Wikipedia:Reference desk/Archives/Mathematics/2014 October 29

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October 29

Real Analytic Extension from Closed Nowhere Dense Set and Coefficient Control

As far as I am aware, given a real valued function f on a CND set of reals, there is an analytic function on the reals that extends it; and given any finite sequence of reals and real a, there is an analytic function that with power series around a so an initial segment of coefficients is the finite sequence. Are both of these true at the same time? That is can we always extend f to an analytic function with any finite sequence it's coeffs. around a point a? More interestingly, and the question I am more curious about: given a real e > 0, natural n, and f on a CND; if I want an analytic F extending f so the first n terms in the expansion around a given point are within e of f(x) for all x in f's domain, is there always such an F? And what constraints are placed on the coefficients of that expansion? The point the series is expanded about can be arbitrary, but it can be assumed in f's domain. If anyone knows an direct results, that would be wonderful; but I'd be more than happy with links to papers considering the matter, or something related to it - I'm not an analyst, not at all, but need something along these lines for what I am working on, just a nice starting point would be wonderful, I can narrow the results to my needs from there (I'm at a loss for how to get started and need to control approximations coming from taylor series in a specific fashion). Thank you for any and all help:-)Phoenixia1177 (talk) 17:18, 29 October 2014 (UTC)

Assume that both the domain of f and f are unbounded; you can even go further and assume that f is increasing and bounded below by x times a constant, if it makes things simpler.Phoenixia1177 (talk) 17:40, 29 October 2014 (UTC)

This looks like far too much to hope for. Consider the Principle of Permanence; if your CND set contains an accumulation point, the analytic extension is uniquely determined, so you certainly can't control the power series.--80.109.80.31 (talk) 19:37, 29 October 2014 (UTC)

I just realized that your first claim is surely false. Define $f(1/n)=f(0)=0$ for integer $n$, and $f(2)=1$. This is a definition on a closed nowhere dense set of reals, but it cannot be extended to an analytic function.--80.109.80.31 (talk) 20:30, 29 October 2014 (UTC)

You're absolutely right (thanks for pointing that out :-) ). What if we assume that f has no accumulation points?Phoenixia1177 (talk) 20:32, 29 October 2014 (UTC)