Theorem concerning analytic continuation

A. M. Lukatskii

Matematičeskie zametki, Tome 10 (1971) no. 1, pp. 57-62.

Résumé

A. I. Markushevich obtained the following representation of a function in its holomorphicity star with a sequence $\{m_\nu\}$, for which $m_{\nu+1}/m_\nu\to\infty$: $$f(z)=\lim\limits_{\nu\to\infty}\left\{\sum_0^{m_{2\nu}}\theta_k\frac{f^{(k)}(z_0)}{k!}(z-z_0)^k+\sum_0^{m_{2\nu-1}}(1-\theta_k)\frac{f^{(k)}(z_0)}{k!}(z-z_0)^k\right\}$$. Here it is proved that this condition is necessary; more precisely, $\overline{\lim\limits_{\nu\to\infty}}\frac{m_{\nu+1}}{m_\nu}=\infty$ . This result is derived from certain properties of over-convergent power series.