Real function to entire functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:07:28Z http://mathoverflow.net/feeds/question/105918 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105918/real-function-to-entire-functions Real function to entire functions unknown (google) 2012-08-30T09:23:58Z 2012-08-30T22:21:58Z <p>Let $g:[0,\infty) \rightarrow \mathbb{R}$ be an increasing function. Is there a way to construct an entire function $f(z)$ such that $f(x)=g(|x|)$ for all real $x$?</p> http://mathoverflow.net/questions/105918/real-function-to-entire-functions/105925#105925 Answer by jbc for Real function to entire functions jbc 2012-08-30T10:54:18Z 2012-08-30T10:54:18Z <p>You certainly need some smoothness condition for this to be possible. In fact, the condition is that $f$ be extendable to an even, real-analytic function on the line which kind of makes the result a tautology since then $f$ has the form $\sum_{n=1}^\infty a_{2n} z^{2n}$.</p> http://mathoverflow.net/questions/105918/real-function-to-entire-functions/105936#105936 Answer by Bazin for Real function to entire functions Bazin 2012-08-30T12:35:08Z 2012-08-30T22:21:58Z <p>You should keep in mind analytic continuation. Let $f:\mathbb R\rightarrow\mathbb R$ be a $C^\infty$ function ; if there exists an entire function extending $f$, then</p> <p>(1) It is unique by analytic continuation,</p> <p>(2) The function $f$ must be real-analytic.</p> <p>Of course (2) is not sufficient: think about the real-analytic $x\mapsto\frac{1}{1+x^2}$, which does NOT have an entire extension, since by analytic continuation, that extension should coincide with $\mathbb C\ni z\mapsto \frac{1}{1+z^2}$, which has poles at $\pm i$. For a real-analytic $f$, you can formulate a criterion dealing with the size of derivatives. Such a function has an entire extension iff $$\forall R>0,\exists C_R,\forall n\in\mathbb N,\quad \vert f^{(n)}(0)\vert\le C_R\frac{n!}{R^n}.$$ On the other hand, real-analyticity of a $C^\infty$ $f$ on the real line is equivalent to $$\forall x\in \mathbb R,\exists r>0,\exists C>0, \exists R>0,\forall n\in\mathbb N,\quad \Vert f^{(n)}\Vert_{L^\infty(B(x,r))}\le C\frac{n!}{R^n}.$$</p>