formal power series converges if and only if it converges along every line

Suppose $T(x)$ denotes the formal power series $\sum_{\alpha}a_{\alpha}x^{\alpha},$ using the multi-index notation, where $x=(x_{1},\ldots,x_{N})$ and $a_{\alpha}\in\mathbb{C}.$ Fixing $v\in{\mathbb{R}}^{N}$ and we can also talk of the formal power series in $t\in\mathbb{R}$

\begin{split}\displaystyle T(tv)&\displaystyle=\sum_{\alpha}a_{\alpha}(tv)^{% \alpha}\\ &\displaystyle=\sum_{\alpha}a_{\alpha}v^{\alpha}t^{\lvert\alpha\rvert}\\ &\displaystyle=\sum_{k=0}^{\infty}\left(\sum_{\lvert\alpha\rvert=k}a_{\alpha}v% ^{\alpha}\right)t^{k}.\end{split}

Theorem.

Suppose $T(x)$ is a formal power series in $x\in{\mathbb{R}}^{N}$ . Suppose $T(tv)$ is a convergent power series in $t\in\mathbb{R}$ for all $v\in{\mathbb{R}}^{N}$ . Then $T$ is convergent.

The other direction, if $T(x)$ converges then $t\mapsto T(tv)$ converges, is obvious.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	formal power series converges if and only if it converges along every line
Canonical name	FormalPowerSeriesConvergesIfAndOnlyIfItConvergesAlongEveryLine
Date of creation	2013-03-22 17:42:11
Last modified on	2013-03-22 17:42:11
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	5
Author	jirka (4157)
Entry type	Theorem
Classification	msc 13H05
Classification	msc 13B35
Classification	msc 13J05
Classification	msc 13F25