  Let f_n be a sequence of analytic functions in a domain U with a common
attracting fixed point z_0. Suppose that f_n converges to f_0 uniformly on each
compact subset of U and that z_0 is a Siegel point of f_0. We establish a
sufficient condition for the immediate basins of attraction $\mathcal
A^*(z_0,f_n,U)$ to form a sequence that converges to the Siegel disk of f_0 as
to the kernel with respect to z_0. The same condition is shown to imply
convergence of the Koenings functions associated with f_n to that of f_0. Our
method allows us also to obtain a kind of quantitative result for analytic
one-parametric families.
