Interpolation with estimates in $\mathbb C^n$ and its applications

The problem of interpolation in spaces of entire functions of several complex variables of finite order and type is studied. The extension is performed from the zero set of an entire function $f$ of finite order and type. Sufficient conditions for interpolation are obtained in terms of lower bounds for the first derivatives of $f$ that are non-zero at the points in its zero set. The result is applied to the problem of the validity of the fundamental principle in spaces of solutions of homogeneous convolution equations.