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Description
One of the methods of studying the problem of analytical continuation of power series is interpolation of the coefficients of the series.
With this approach
Le Roy and Lindelöf obtained conditions under which a series analytically extends into a sector. Note that the theorem, gave a connection between the sector and the growth of the interpolation function. More precisely, the type of interpolation function must be less than $\pi$ on the closed half-plane $Re z \geq 0$.
We weakened the condition of less than $\pi$ on the fact that the sum of the indicator (growth) on directions $\frac{\pi}{2}$ and $-\frac{\pi}{2}$ is less than $2\pi$.
Also we obtain the multivariate version of this theorem, i.e. establish a connection between the growth of the interpolating function of the coefficients on the imaginary subspace and the multivariate sectoral domain where the multiple series is analytically extends.
