  The article is devoted to the expansions of iterated Ito stochastic integrals
based on generalized multiple Fourier series converging in the sense of norm in
the space $L_2([t, T]^k),$ $k\in\mathbb{N}.$ The method of generalized multiple
Fourier series for expansion and mean-square approximation of iterated Ito
stochastic integrals of arbitrary multiplicity $k$ ($k\in\mathbb{N}$) with
respect to components of the multidimensional Wiener process is proposed and
developed. The obtained expansions contain only one operation of the limit
transition in contrast to its existing analogues. In the article it is also
obtained the generalization of the proposed method for an arbitrary complete
orthonormal systems of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$
as well as for complete orthonormal with weight $r(t_1)\ldots r(t_k)$ systems
of functions in the space $L_2([t, T]^k),$ $k\in\mathbb{N}$. The comparison of
the considered method with the well-known expansions of iterated Ito stochastic
integrals based on the Ito formula and Hermite polynomials is given. The
convergence in the mean of degree $2n$ $(n \in \mathbb{N})$ and with
probability 1 of the proposed method is proved.
