\section{Introduction}

\vspace{5mm}

Let $(\Omega,$ ${\rm F},$ ${\sf P})$ be a complete probability space, let 
$\{{\rm F}_t, t\in[0,T]\}$ be a nondecreasing right-continous 
family of $\sigma$-subfields of ${\rm F},$
and let ${\bf f}_t$ be a standard $m$-dimensional Wiener stochastic 
process, which is
${\rm F}_t$-measurable for any $t\in[0, T].$ We assume that the components
${\bf f}_{t}^{(i)}$ $(i=1,\ldots,m)$ of this process are independent. Consider
an Ito stochastic differential equation (SDE) 
in the integral form

\vspace{-1mm}
\begin{equation}
\label{1.5.2}
{\bf x}_t={\bf x}_0+\int\limits_0^t {\bf a}({\bf x}_{\tau},\tau)d\tau+
\int\limits_0^t B({\bf x}_{\tau},\tau)d{\bf f}_{\tau},\ \ \
{\bf x}_0={\bf x}(0,\omega).
\end{equation}

\vspace{2mm}
\noindent
Here ${\bf x}_t$ is some $n$-dimensional stochastic process 
satisfying to equation (\ref{1.5.2}). 
The nonrandom functions ${\bf a}: \Re^n\times[0, T]\to\Re^n$,
$B: \Re^n\times[0, T]\to\Re^{n\times m}$
guarantee the existence and uniqueness up to stochastic equivalence 
of a solution
of Ito SDE (\ref{1.5.2}) \cite{1}. The second integral on 
the right-hand side of (\ref{1.5.2}) is 
interpreted as an Ito stochastic integral.
Let ${\bf x}_0$ be an $n$-dimensional random variable, which is 
${\rm F}_0$-measurable and 
${\sf M}\{\left|{\bf x}_0\right|^2\}<\infty$ 
(${\sf M}$ denotes a mathematical expectation).
We assume that
${\bf x}_0$ and ${\bf f}_t-{\bf f}_0$ are independent when $t>0.$


One of the effective approaches 
to numerical integration of 
Ito SDEs is an approach based on Taylor--Ito and 
Taylor--Stratonovich expansions
\cite{Mi2}-\cite{miltret}. The most important feature of such 
expansions is a presence in them of the so-called iterated
Ito and Stratonovich stochastic integrals, which play the key 
role for solving the 
problem of numerical integration of Ito SDEs and has the 
following form

\vspace{-1mm}
\begin{equation}
\label{sodom20}
J[\psi^{(k)}]_{T,t}=\int\limits_t^T\psi_k(t_k) \ldots \int\limits_t^{t_{2}}
\psi_1(t_1) d{\bf w}_{t_1}^{(i_1)}\ldots
d{\bf w}_{t_k}^{(i_k)},
\end{equation}

\vspace{2mm}

\begin{equation}
\label{605}
J^{*}[\psi^{(k)}]_{T,t}=
\int\limits_t^{*T}\psi_k(t_k) \ldots \int\limits_t^{*t_{2}}
\psi_1(t_1) d{\bf w}_{t_1}^{(i_1)}\ldots
d{\bf w}_{t_k}^{(i_k)},
\end{equation}

\vspace{2mm}
\noindent
where every $\psi_l(\tau)$ $(l=1,\ldots,k)$ is a continuous 
non-random function 
on $[t,T],$ ${\bf w}_{\tau}^{(i)}={\bf f}_{\tau}^{(i)}$
for $i=1,\ldots,m$ and
${\bf w}_{\tau}^{(0)}=\tau,$
and 
$$
\int\limits\ \hbox{and}\ \int\limits^{*}
$$ 
denote Ito and 
Stratonovich stochastic integrals,
respectively; $i_1,\ldots,i_k = 0, 1,\ldots,m.$


Note that usually in applications 
$\psi_l(\tau)\equiv 1$ $(l=1,\ldots,k)$ and
$i_1,\ldots,i_k = 0, 1,\ldots,m$ in  
\cite{Mi2}-\cite{KlPl1} while
$\psi_l(\tau)\equiv (t-\tau)^{q_l}$ ($l=1,\ldots,k$; 
$q_1,\ldots,q_k=0, 1, 2,\ldots $) and $i_1,\ldots,i_k = 1,\ldots,m$ in
\cite{3}-\cite{32}.
 

Effective solution 
of the problem of
combined mean-square approximation for collections 
of iterated Stratonovich stochastic integrals 
(\ref{605}) of second multiplicity
composes the subject of this article.


The mean-square approximation of iterated stochastic
integrals (\ref{sodom20}), (\ref{605}) using the 
piecewise constant basis (in other
words, integrals (\ref{sodom20}), (\ref{605}) is approximated 
by integral sums) requires considerable 
computational effort \cite{7} because
it requires the division of the integration interval $[t, T]$ of 
stochastic
integrals (\ref{sodom20}), (\ref{605})
into parts ($[t, T]$ is already small because
it plays the role of the integration step in numerical methods for 
solving the Ito SDEs).

More efficient approximation methods of 
iterated stochastic
integrals (\ref{sodom20}), (\ref{605})
use Fourier series, and 
they do not require the interval $[t, T]$
to be subdivided into smaller parts. One such method was 
proposed in \cite{Mi2} and elaborated in \cite{KlPl2}, \cite{KPW}. 
This method, which received widespread use, is based on 
the Karhunen-Loeve expansion of the
Brownian bridge process \cite{Mi2} in the eigenfunctions of its 
covariance, which form a complete orthonormal
trigonometric basis in the space $L_2([t, T]).$ 

Note that in \cite{7} a more general method (called the method of 
generalized multiple Fourier series) for
the mean-square approximation of iterated Ito stochastic
integrals of form (\ref{sodom20}) was proposed. 
It is based on the generalized
multiple Fourier series that converge in mean in the space $L_2([t, T]^k)$, 
where $[t, T]^k$ is a hypercube and $k$ is
the multiplicity of iterated Ito stochastic integral. 
The method of generalized multiple 
Fourier series was developed in \cite{8}-\cite{31}.

An important feature of this method is that various complete 
orthonormal bases in the space $L_2(t, T])$ can
be used (the method proposed in \cite{Mi2} 
admits only the trigonometric basis).
Hence, we can state the problem of comparing the efficiency 
of using different complete orthonormal
bases in the space $L_2(t, T])$ in the numerical solution of Ito SDEs.
This problem has been solved in \cite{29}, \cite{30}.
In particular, in \cite{29}, \cite{30} 
it was shown that the optimal system of basis functions 
in the framework of 
the numerical solution of Ito SDEs
is the system of Legendre 
polynomials.
That is why this article is devoted to expansions
of iterated Stratonovich stochastic integrals, based
on the Fourier--Legendre series.


Usage of Fourier series according to the system of Legendre polynomials 
for approximation of iterated stochastic integrals took place for 
the first time in \cite{3}-\cite{5} (see also
\cite{6}-\cite{31}).


The results of \cite{3}
(see also \cite{4}-\cite{31}) convincingly testify
that there is a doubtless relation between 
multiplier factor $1/2$, 
which is typical for Stratonovich stochastic integral and included 
into the sum, connecting Stratonovich and Ito stochastic integrals, 
and the fact that in the point of finite discontinuity of sectionally 
smooth function $f(x)$ its Fourier series  
converges to the value $(f(x+0)+f(x-0))/2.$ 
In addition, in \cite{3}, \cite{4},
\cite{11}-\cite{16}, \cite{19}-\cite{21},
\cite{23}, \cite{25}, \cite{27}
several theorems on expansion of iterated 
Stratonovich stochastic integrals 
were formulated and proved.
As shown in these papers, the final formulas for 
expansions of iterated Stratonovich stochastic integrals 
are more compact than their analogues for iterated Ito stochastic 
integrals. 

This paper continues the study of the relationship between
generalized multiple Fourier series and iterated stochastic integrals.
We use the double Fourier--Legendre series,
summarized by Prinsheim method,
for proof of theorem 5.3 (\cite{20}, page A.294)
or theorem 3 (\cite{16}, page 59). At that the 
conditions of these theorems can be weakened.



\vspace{5mm}



\section{Method of Generalized Multiple Fourier Series}



\vspace{5mm}


Let us consider the approach to expansion of iterated
stochastic integrals \cite{7}-\cite{31} (the so-called
method of generalized
multiple Fourier series). 



Suppose that every $\psi_l(\tau)$ $(l=1,\ldots,k)$ is a continuous 
non-random function on $[t, T]$.


Define the following function on a hypercube $[t, T]^k$

\vspace{-1mm}
\begin{equation}
\label{ppp}
K(t_1,\ldots,t_k)=
\begin{cases}
\psi_1(t_1)\ldots \psi_k(t_k)\ \ \hbox{for}\ \ t_1<\ldots<t_k\\
~\\
0\ \ \hbox{otherwise}
\end{cases},\ \ \ \ \ t_1,\ldots,t_k\in[t, T],\ \ k\ge 2,
\end{equation}

\vspace{2mm}
\noindent
and 
$K(t_1)\equiv\psi_1(t_1),\ t_1\in[t, T].$  


Suppose that $\{\phi_j(x)\}_{j=0}^{\infty}$
is a complete orthonormal system of functions in the space
$L_2([t, T])$. 
 
 
The function $K(t_1,\ldots,t_k)$ is sectionally continuous in the 
hypercube $[t, T]^k.$
At this situation it is well known that the generalized 
multiple Fourier series 
of $K(t_1,\ldots,t_k)\in L_2([t, T]^k)$ is converging 
to $K(t_1,\ldots,t_k)$ in the hypercube $[t, T]^k$ in 
the mean-square sense, i.e.

\vspace{-1mm}
$$
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm lim}\cr
$\stackrel{}{{}_{p_1,\ldots,p_k\to \infty}}$\cr
}} }\biggl\Vert
K(t_1,\ldots,t_k)-
\sum_{j_1=0}^{p_1}\ldots \sum_{j_k=0}^{p_k}
C_{j_k\ldots j_1}\prod_{l=1}^{k} \phi_{j_l}(t_l)\biggr\Vert=0,
$$

\vspace{2mm}
\noindent
where
\begin{equation}
\label{ppppa}
C_{j_k\ldots j_1}=\int\limits_{[t,T]^k}
K(t_1,\ldots,t_k)\prod_{l=1}^{k}\phi_{j_l}(t_l)dt_1\ldots dt_k,
\end{equation}

$$
\left\Vert f\right\Vert=\left(\int\limits_{[t,T]^k}
f^2(t_1,\ldots,t_k)dt_1\ldots dt_k\right)^{1/2}.
$$

\vspace{3mm}

Consider the partition $\{\tau_j\}_{j=0}^N$ of $[t,T]$ such that

\begin{equation}
\label{1111}
t=\tau_0<\ldots <\tau_N=T,\ \ \
\Delta_N=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm max}\cr
$\stackrel{}{{}_{0\le j\le N-1}}$\cr
}} }\Delta\tau_j\to 0\ \ \hbox{if}\ \ N\to \infty,\ \ \
\Delta\tau_j=\tau_{j+1}-\tau_j.
\end{equation}

\vspace{2mm}

{\bf Theorem 1} \cite{7}-\cite{31}. {\it Suppose that
every $\psi_l(\tau)$ $(l=1,\ldots, k)$ is a continuous non-random function on 
$[t, T]$ and
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system  
of continuous functions in $L_2([t,T]).$ Then

\vspace{2mm}
$$
J[\psi^{(k)}]_{T,t}\  =\ 
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_k\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\ldots\sum_{j_k=0}^{p_k}
C_{j_k\ldots j_1}\Biggl(
\prod_{l=1}^k\zeta_{j_l}^{(i_l)}\ -
\Biggr.
$$

\vspace{2mm}
\begin{equation}
\label{tyyy}
-\ \Biggl.
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{N\to \infty}}$\cr
}} }\sum_{(l_1,\ldots,l_k)\in {\rm G}_k}
\phi_{j_{1}}(\tau_{l_1})
\Delta{\bf w}_{\tau_{l_1}}^{(i_1)}\ldots
\phi_{j_{k}}(\tau_{l_k})
\Delta{\bf w}_{\tau_{l_k}}^{(i_k)}\Biggr),
\end{equation}

\vspace{6mm}
\noindent
where $J[\psi^{(k)}]_{T,t}$ is defined by {\rm (\ref{sodom20}),}

\vspace{-1mm}
$$
{\rm G}_k={\rm H}_k\backslash{\rm L}_k;\
{\rm H}_k=\{(l_1,\ldots,l_k):\ l_1,\ldots,l_k=0,\ 1,\ldots,N-1\},
$$

\vspace{-1mm}
$$
{\rm L}_k=\{(l_1,\ldots,l_k):\ l_1,\ldots,l_k=0,\ 1,\ldots,N-1;\
l_g\ne l_r\ (g\ne r);\ g, r=1,\ldots,k\},
$$

\vspace{2mm}
\noindent
${\rm l.i.m.}$ is a limit in the mean-square sense$,$
$i_1,\ldots,i_k=0,1,\ldots,m,$ 
every

\vspace{-1mm}
\begin{equation}
\label{rr23}
\zeta_{j}^{(i)}=
\int\limits_t^T \phi_{j}(s) d{\bf w}_s^{(i)}
\end{equation} 

\vspace{2mm}
\noindent
is a standard Gaussian random variable
for various
$i$ or $j$ {\rm(}if $i\ne 0${\rm),}
$C_{j_k\ldots j_1}$ is the Fourier coefficient {\rm(\ref{ppppa}),}
$\Delta{\bf w}_{\tau_{j}}^{(i)}=
{\bf w}_{\tau_{j+1}}^{(i)}-{\bf w}_{\tau_{j}}^{(i)}$
$(i=0, 1,\ldots,m),$
$\left\{\tau_{j}\right\}_{j=0}^{N}$ is a partition of
the interval $[t, T],$ which satisfies the condition {\rm (\ref{1111})}.
}

\vspace{2mm}

Note that the condition of continuity of $\phi_j(x)$ can be 
weakened (see \cite{7}-\cite{16}, \cite{19}, \cite{20}).


In order to evaluate of significance of the theorem 1 for practice we will
demonstrate its transformed particular cases for 
$k=1,\ldots,4$ \cite{7}-\cite{31}

$$
J[\psi^{(1)}]_{T,t}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}
C_{j_1}\zeta_{j_1}^{(i_1)},
$$

\vspace{2mm}
\begin{equation}
\label{leto5001}
J[\psi^{(2)}]_{T,t}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\biggl(\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}
-{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}\biggr),
\end{equation}

\vspace{4mm}
$$
J[\psi^{(3)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_3\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}\sum_{j_3=0}^{p_3}
C_{j_3j_2j_1}\biggl(
\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}
-\biggr.
$$
\begin{equation}
\label{leto5002}
-{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
\zeta_{j_3}^{(i_3)}
-{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
\zeta_{j_1}^{(i_1)}-
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
\zeta_{j_2}^{(i_2)}\biggr),
\end{equation}


\vspace{6mm}
$$
J[\psi^{(4)}]_{T,t}
=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,\ldots,p_4\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\ldots\sum_{j_4=0}^{p_4}
C_{j_4\ldots j_1}\biggl(
\prod_{l=1}^4\zeta_{j_l}^{(i_l)}
\biggr.
-
$$
$$
-
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
\zeta_{j_3}^{(i_3)}
\zeta_{j_4}^{(i_4)}
-
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_4}^{(i_4)}-
$$
$$
-
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
\zeta_{j_2}^{(i_2)}
\zeta_{j_3}^{(i_3)}
-
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_4}^{(i_4)}-
$$
$$
-
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_3}^{(i_3)}
-
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}
\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}+
$$
$$
+
{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}
{\bf 1}_{\{i_3=i_4\ne 0\}}
{\bf 1}_{\{j_3=j_4\}}
+
{\bf 1}_{\{i_1=i_3\ne 0\}}
{\bf 1}_{\{j_1=j_3\}}
{\bf 1}_{\{i_2=i_4\ne 0\}}
{\bf 1}_{\{j_2=j_4\}}+
$$
\begin{equation}
\label{leto5003}
+\biggl.
{\bf 1}_{\{i_1=i_4\ne 0\}}
{\bf 1}_{\{j_1=j_4\}}
{\bf 1}_{\{i_2=i_3\ne 0\}}
{\bf 1}_{\{j_2=j_3\}}\biggr),
\end{equation}

\vspace{5mm}
\noindent
where ${\bf 1}_A$ is the indicator of the set $A$.
 

\vspace{5mm}


\section{Theorem on Expansion of Iterated Stochastic
Stratonovich Integrals}


\vspace{5mm}


In a number of works of the author \cite{12}-\cite{16}, 
\cite{19}-\cite{21}, \cite{25}, \cite{27}
the theorem 1 has been adapted for iterated Stratonovich stochastic integrals
(\ref{605}) of multiplicities 2 to 5.
For example, we can formulate the following theorem.

\vspace{2mm}

{\bf Theorem 2} \cite{12}-\cite{16}, 
\cite{19}-\cite{21}, \cite{27}.
{\it Suppose that 
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of 
Legendre polynomials or trigonometric functions in the space $L_2([t, T]).$
At the same time $\psi_2(s)$ is a continuously differentiable 
function on $[t, T]$ and $\psi_1(s), \psi_3(s)$ are two times 
continuously differentiable functions on $[t, T]$. Then 

\vspace{-1mm}
\begin{equation}
\label{a}
J^{*}[\psi^{(2)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\ \ (i_1,i_2=1,\ldots,m),
\end{equation}

\begin{equation}
\label{feto19000ab}
J^{*}[\psi^{(3)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2,p_3\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}\sum_{j_3=0}^{p_3}
C_{j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\ \
(i_1,i_2,i_3=0, 1,\ldots,m),
\end{equation}

\begin{equation}
\label{feto19000a}
J^{*}[\psi^{(3)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p\to \infty}}$\cr
}} }
\sum\limits_{j_1,j_2,j_3=0}^{p}
C_{j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\ \
(i_1,i_2,i_3=1,\ldots,m),
\end{equation}

\begin{equation}
\label{uu}
J^{*}[\psi^{(4)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p\to \infty}}$\cr
}} }
\sum\limits_{j_1, \ldots, j_4=0}^{p}
C_{j_4 j_3 j_2 j_1}\zeta_{j_1}^{(i_1)}
\zeta_{j_2}^{(i_2)}\zeta_{j_3}^{(i_3)}\zeta_{j_4}^{(i_4)}\ \
(i_1,\ldots,i_4=0, 1,\ldots,m),
\end{equation}


\vspace{6mm}
\noindent
where $J^{*}[\psi^{(k)}]_{T,t}$ is defined by {\rm (\ref{605})}, and
$\psi_l(s)\equiv 1$ $(l=1,\ldots,4)$ in {\rm (\ref{feto19000ab})}, 
{\rm (\ref{uu});} another notations see in the theorem {\rm 1.}
}

\vspace{2mm}


Note that the proof of the theorem 2 for the case $k=2$ based
on the proof of the following equality

\vspace{-1mm}
\begin{equation}
\label{5t11}
\frac{1}{2}
\int\limits_t^T\psi_1(t_1)\psi_2(t_1)dt_1
=\sum_{j_1=0}^{\infty}
C_{j_1j_1},
\end{equation}

\vspace{2mm}
\noindent
where $C_{j_1j_1}$ is defined by (\ref{ppppa})
for $k=2,$ $j_1=j_2,$ 
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of 
Legendre polynomials or trigonometric functions in the space $L_2([t, T]).$


According to the standard relation between 
Ito and Stratonovich stochastic integrals we can write down w. p. 1

\vspace{-3mm}
\begin{equation}
\label{oop51}
J^{*}[\psi^{(2)}]_{T,t}=
J[\psi^{(2)}]_{T,t}+
\frac{1}{2}{\bf 1}_{\{i_1=i_2\ne 0\}}
\int\limits_t^T\psi_1(t_1)\psi_2(t_1)dt_1.
\end{equation}

\vspace{2mm}

From the other side according to (\ref{leto5001})


\vspace{1mm}
$$
J[\psi^{(2)}]_{T,t}=
\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\left(\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}
-{\bf 1}_{\{i_1=i_2\ne 0\}}
{\bf 1}_{\{j_1=j_2\}}\right)=
$$

\vspace{1mm}
\begin{equation}
\label{yes2001}
=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}
-{\bf 1}_{\{i_1=i_2\ne 0\}}\sum_{j_1=0}^{\infty}
C_{j_1j_1}.
\end{equation}

\vspace{2mm}

From (\ref{5t11})--(\ref{yes2001}) we obtain (\ref{a}).


The proof of the theorem 2 for the case $k=2$ (see (\ref{a}))
based on the multiple (double) Fourier--Legendre
series and analogous trigonometric Fourier series (see the theorem 1).
This uses double integration by parts, which leads to 
the requirement of double continuous differentiability of the function
$\psi_1(\tau)$ 
at the interval $[t, T]$.


In this article we formulate and prove the analogue
of the theorem 2 for the case $k=2$
but under the weakened conditions:
the functions
$\psi_1(\tau),$  $\psi_2(\tau)$ only one time
continuously differentiable 
at the interval $[t, T]$.
At that we will use 
the double Fourier--Legendre series, summarized by Prinsheim method.




\vspace{5mm}


\section{Sufficient Conditions of Convergence of Double Fourier--Legendre 
Series, Summable by Prinsheim Method}


\vspace{5mm}


Let $P_j(x)$ $(j=0, 1, 2,\ldots )$ be the Legendre polynomial.
Consider the function $f(x,y)$, defined for $(x,y)\in [-1,1]^2.$

Consider the double Fourier--Legendre series, summable by
Prinsheim method, corresponding 
to the function 
$f(x,y)$

\vspace{-1mm}
$$
\lim_{n,m\to\infty}
\sum_{j=0}^n\sum_{i=0}^m \frac{1}{2}\sqrt{(2j+1)(2i+1)}{C}_{ij}^{*}
P_i(x)P_j(y)
\stackrel{\sf{def}}{=}
$$

\begin{equation}
\label{555}
\stackrel{\sf{def}}{=}
\sum_{i,j=0}^{\infty}\frac{1}{2}\sqrt{(2j+1)(2i+1)}{C}_{ij}^{*}
P_i(x)P_j(y),
\end{equation}

\vspace{2mm}
\noindent
where

\vspace{-1mm}
\begin{equation}
\label{777}
{C}_{ij}^{*}=\frac{1}{2}\sqrt{(2j+1)(2i+1)}\int\limits_{[-1,1]^2}
f(x,y)P_i(x)P_j(y)dxdy.
\end{equation}

\vspace{2mm}

{\bf Theorem 3}\ \cite{star}.\ {\it Let $f(x,y)\in L_2([-1,1]^2)$
and for some point $(x,y)\in (-1,1)^2$ the following condition
is satisfied 

\vspace{-1mm}
\begin{equation}
\label{444}
\int\limits_{[-1,1]^2}
\left(\frac{f(x,y) - f(u,v)}{(x-u)(y-v)}\right)^2 dudv<\infty.
\end{equation}

\vspace{2mm}

Then, in the point $(x,y)$ the double Fourier--Legendre series
of form {\rm (\ref{555})} converges to the function
$f(x,y)$.}

\vspace{2mm}

Note that the theorem 3 is the generalization for the case
of functions of two variables 
of the well-known theorem about sufficient
conditions of convergence of the Fourier--Legendre series
\cite{suet}.

Consider the generalization for the case of two variables
\cite{star}
the theorem about equiconvergence for the Fourier--Legendre
series
\cite{suet}.

\vspace{2mm}

{\bf Theorem 4}\ \cite{star}.\ {\it Let $f(x,y)\in L_2([-1,1]^2)$ and 
the function

\vspace{-1mm}
$$
f(x,y)(1-x^2)^{-1/4}(1-y^2)^{-1/4}
$$

\vspace{2mm}
\noindent
is integrable on
$[-1,1]^2.$  Moreover, let

\vspace{-1mm}
$$
|f(x,y)-f(u,v)|\le G(y)|x-u|+H(x)|y-v|,
$$

\vspace{2mm}
\noindent
where $G(y), H(x)$ are bounded functions on $[-1,1]^2$.

Then for all $(x,y)\in(-1,1)^2$ the following equality
is satisfied 

\vspace{2mm}
$$
\lim\limits_{n,m\to\infty}
\biggl(\sum_{j=0}^n\sum_{i=0}^m \frac{1}{2}\sqrt{(2j+1)(2i+1)}{C}_{ij}^{*}
P_i(x)P_j(y)-\biggr.
$$

\begin{equation}
\label{888}
-\biggl.
(1-x^2)^{-1/4}(1-y^2)^{-1/4}S_{nm}({\rm arccos}x,{\rm arccos}y,F)
\biggr)=0.
\end{equation}

\vspace{4mm}

At that, the convergence is uniform on 

\vspace{-1mm}
$$
[-1+\varepsilon, 1-\varepsilon]\times[-1+\delta,1-\delta]\ \
\forall\ \varepsilon, \delta>0,
$$ 

\vspace{3mm}
\noindent
and $S_{nm}(\theta,\varphi,F)$ is a partial sum of
the double trigonometric Fourier series of the
auxiliary function

\vspace{1mm}
$$
F(\theta,\varphi)=\sqrt{|{\rm sin}\theta|}\sqrt{|{\rm sin}\varphi|}
f({\rm cos}\theta,{\rm cos}\varphi),\ \
\theta,\varphi\in[0, \pi],
$$

\vspace{4mm}
\noindent
the Fourier coefficient ${C}_{ij}^{*}$ has form {\rm (\ref{777}).}
}

\vspace{2mm}

From the theorem 4, for example, follows that  
for all $(x,y)\in(-1,1)^2$ the following equality is
fulfilled 

\begin{equation}
\label{8888}
\lim\limits_{n,m\to\infty}
\biggl(\sum_{j=0}^n\sum_{i=0}^m \frac{1}{2}\sqrt{(2j+1)(2i+1)}{C}_{ij}^{*}
P_i(x)P_j(y)-f(x,y)\biggr)=0.
\end{equation}

\vspace{4mm}

At that the convergence is uniform on 

\vspace{-1mm}
$$
[-1+\varepsilon, 1-\varepsilon]\times[-1+\delta,1-\delta]\ \
\forall\ \varepsilon, \delta>0,
$$ 

\vspace{2mm}
\noindent
if the corresponding conditions of convergence 
of the double trigonometric Fourier series of the auxiliary 
function 

\vspace{-2mm}
\begin{equation}
\label{999}
g(x,y)=f(x,y)(1-x^2)^{1/4}(1-y^2)^{1/4}
\end{equation}

\vspace{2mm}
\noindent
are satisfied.

Note also that the theorem 4 does not imply any
conclusions on the behavior of the double Fourier--Legendre series 
on the boundary of
the square 
$[-1,1]^2.$

\vspace{2mm}

{\it For each $\delta>0$ let us call the exact upper edge 
of difference $|f({\bf t}')-f({\bf t}'')|$ 
in the set
of all points ${\bf t}'$, ${\bf t}''$ which 
belong 
to the domain $D$, 
as the module of 
continuity of function
$f({\bf t})$ $({\bf t}=(t_1,\ldots,t_k))$ in the 
$k$-dimentional domain
$D$ $(k\ge 1)$, 
moreover distance
satisfies the formula
$\rho({\bf t}',{\bf t}'')<\delta.$}

\vspace{2mm}

{\it We will declare that the function 
of $k$ $(k\ge 1)$ variables  
$f({\bf t})$ $({\bf t}=(t_1,\ldots,t_k))$
belongs 
to the Holder class with 
the parameter {\rm 1} $(f({\bf t})\in C^{1}(D))$ 
in the domain $D$ 
if the module of 
continuity of function
$f({\bf t})$ $({\bf t}=(t_1,\ldots,t_k))$ 
in the domain $D$ has the order $O(\delta).$}

\vspace{2mm}

In 1967, Zhizhiashvili L.V.
proved
that the rectangular sums of multiple trigonometric Fourier series 
in the hypercube $[t,T]^k$ of the function of $k$ variables 
converge uniformly in the hypercube 
to this function, if it 
belongs
to $C^{\alpha}([t,T]^k),$ $\alpha>0$ (definition
of the Holder class with the parameter $\alpha>0$ may be found in 
the well-known mathematical analysis manuals).


For example, the following theorem is true.

\vspace{2mm}

{\bf Theorem 5}\ \cite{IP}.\ {\it If the function
$f(x_1,\ldots,x_n)$ is periodic with period $2\pi$ with respect to each
variable and belongs in $\mathbb{R}^n$ to the Holder class $C^{\alpha}$
for any $\alpha>0,$ then the rectangular partial
sums of the multiple trigonometric Fourier series of a function
$f(x_1,\ldots,x_n)$ converge to this function uniformly in 
$\mathbb{R}^n$.}

\vspace{2mm}

{\bf Lemma 1}.\ {\it 
Let the fiunction $f(x,y)$ satisfies
to the following condition 

\vspace{-2mm}
$$
|f(x,y)-f(x_1,y_1)|\le C_1|x-x_1|+C_2|y-y_1|,
$$

\vspace{2mm}
\noindent
where $C_1, C_2<\infty$, and $(x,y),$ $(x_1,y_1)\in [-1,1]^2.$

Then, the following inquality is fulfilled 

\vspace{-2mm}
\begin{equation}
\label{2222}
|g(x,y)-g(x_1,y_1)|\le K\rho^{1/4},
\end{equation}

\vspace{2mm}
\noindent
where $g(x,y)$ 
has form
{\rm(\ref{999})}{\rm,}\ 

\vspace{-3mm}
$$
\rho=\sqrt{(x-x_1)^2+(y-y_1)^2},
$$ 

\vspace{3mm}
\noindent
$(x,y)$ and $(x_1,y_1)\in [-1,1]^2,$ $K<\infty.$}
                                  
\vspace{2mm}

{\bf Proof.}\ First, we assume that $x\ne x_1,$ $y\ne y_1.$ In this case
we have

\vspace{3mm}
$$
|g(x,y)-g(x_1,y_1)|=
|(1-x^2)^{1/4}(1-y^2)^{1/4}(f(x,y)-f(x_1,y_1))+
$$

$$
+
f(x_1,y_1)((1-x^2)^{1/4}(1-y^2)^{1/4}
-(1-x_1^2)^{1/4}(1-y_1^2)^{1/4})|\le C_1|x-x_1|+C_2|y-y_1|+
$$

\begin{equation}
\label{6666}
+C_3|(1-x^2)^{1/4}(1-y^2)^{1/4}
-(1-x_1^2)^{1/4}(1-y_1^2)^{1/4}|,\ \ \ C_3<\infty.
\end{equation}

\vspace{5mm}

Moreover

\vspace{-1mm}
$$
|(1-x^2)^{1/4}(1-y^2)^{1/4}
-(1-x_1^2)^{1/4}(1-y_1^2)^{1/4}|=
$$

$$
=|(1-x^2)^{1/4}((1-y^2)^{1/4}-(1-y_1^2)^{1/4})
+(1-y_1^2)^{1/4}((1-x^2)^{1/4}-(1-x_1^2)^{1/4})|\le
$$

\begin{equation}
\label{5555}
\le |(1-y^2)^{1/4}-(1-y_1^2)^{1/4}|+
|(1-x^2)^{1/4}-(1-x_1^2)^{1/4}|,
\end{equation}

\vspace{4mm}

$$
|(1-x^2)^{1/4}-(1-x_1^2)^{1/4}|=
$$

$$
=|((1-x)^{1/4}-(1-x_1)^{1/4})(1+x)^{1/4}+
(1-x_1)^{1/4}((1+x)^{1/4}-(1+x_1)^{1/4})|\le
$$

\begin{equation}
\label{4444}
\le K_1(|(1-x)^{1/4}-(1-x_1)^{1/4}|+
|(1+x)^{1/4}-(1+x_1)^{1/4}|),\ \ \ K_1<\infty.
\end{equation}

\vspace{7mm}

Not difficult to see that

\vspace{1mm}
$$
|(1 \pm x)^{1/4}-(1 \pm x_1)^{1/4}|=
\frac{|(1\pm x) - (1\pm x_1)|}
{((1\pm x)^{1/2}+(1 \pm x_1)^{1/2})((1\pm x)^{1/4}+(1 \pm x_1)^{1/4})}=
$$

\begin{equation}
\label{3333}
=|x_1-x|^{1/4} \frac{|x_1-x|^{1/2}}{(1\pm x)^{1/2}+(1 \pm x_1)^{1/2}}
\cdot \frac{|x_1-x|^{1/4}}{(1\pm x)^{1/4}+(1 \pm x_1)^{1/4}}
\le |x_1-x|^{1/4}.
\end{equation}

\vspace{4mm}

The last inequality follows from the obvious inequalities


$$
\frac{|x_1-x|^{1/2}}{(1\pm x)^{1/2}+(1 \pm x_1)^{1/2}}\le 1,
$$

$$
\frac{|x_1-x|^{1/4}}{(1\pm x)^{1/4}+(1 \pm x_1)^{1/4}}\le 1.
$$

\vspace{4mm}


From (\ref{6666})--(\ref{3333}) we obtain 

\vspace{-1mm}
$$
|g(x,y)-g(x_1,y_1)|\le
C_1|x-x_1|+C_2|y-y_1|+C_4(|x_1-x|^{1/4}+|y_1-y|^{1/4})\le
$$

$$
\le 
C_5 \rho + C_6 \rho^{1/4} \le K \rho^{1/4},
$$

\vspace{3mm}
\noindent
where $C_5, C_6, K <\infty.$ 


The cases $x=x_1, y\ne y_1$ and $x\ne x_1, y=y_1$ 
can be considered analogously to the case
$x\ne x_1, y\ne y_1$. At that, the consideration 
begins from the inequalities 

\vspace{-1mm}
$$
|g(x,y)-g(x_1,y_1)|\le
K_2 |(1-y^2)^{1/4}f(x,y)-
(1-y_1^2)^{1/4}f(x_1,y_1)|
$$

\vspace{2mm}
\noindent
($x=x_1,\ y\ne y_1$) and

\vspace{-1mm}
$$
|g(x,y)-g(x_1,y_1)|\le  
K_2 |(1-x^2)^{1/4}f(x,y)-
(1-x_1^2)^{1/4}f(x_1,y_1)|
$$

\vspace{2mm}
\noindent
($x\ne x_1,\ y=y_1$),
where $K_2<\infty$. Lemma 1 is proven. 

\vspace{2mm}

Lemma 1 and theorem 5 imply,
that rectangular sums of double trigonometric
Fourier series of the function $g (x,y)$ converge uniformly in 
the square $[-1,1]^2$
to the function $g (x,y)$. This, in turn, means that
the equality (\ref{8888}) holds.


Not difficult to see that under the conditions of 
lemma 1 the condition 
(\ref{444}) is fulfilled in each point $(x,y)\in (-1,1)^2.$


\vspace{5mm}

\section{Main Theorem}

\vspace{5mm}


{\bf Theorem 6}\ \cite{30a}. {\it 
Suppose that 
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of 
Legendre polynomials in the space $L_2([t, T]).$
Moreover, $\psi_1(s), \psi_2(s)$ are 
continuously differentiable functions on $[t, T]$. Then 
for the iterated Stratonovich stochastic integral

\vspace{-1mm}
$$
J^{*}[\psi^{(2)}]_{T,t}={\int\limits_t^{*}}^T\psi_2(t_2)
{\int\limits_t^{*}}^{t_2}\psi_1(t_1)d{\bf f}_{t_1}^{(i_1)}
d{\bf f}_{t_2}^{(i_2)}\ \ (i_1, i_2=1,\ldots,m)
$$

\vspace{2mm}
\noindent
the following converging in the mean-square
sence expansion  

\vspace{-1mm}
\begin{equation}
\label{jes}
J^{*}[\psi^{(2)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{
\hfil#\hfil\cr
{\rm l.i.m.}\cr
$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr
}} }\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}
C_{j_2j_1}\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}
\end{equation}

\vspace{2mm}
\noindent
is valid, where the notations see in the theorem {\rm 2.}
}

\vspace{2mm}

{\bf Proof.}\ Let us prove the equality

\vspace{-1mm}
$$
\frac{1}{2}
\int\limits_t^T\psi_1(t_1)\psi_2(t_1)dt_1
=\sum_{j_1=0}^{\infty}
C_{j_1j_1},
$$

\vspace{2mm}
\noindent
where $C_{j_1j_1}$ is defined by formula (\ref{ppppa})
for $k=2$ and $j_1=j_2.$ At that
$\{\phi_j(x)\}_{j=0}^{\infty}$ is a complete orthonormal system of 
Legendre polynomials in the space $L_2([t, T]).$

Consider the auxiliary function

\vspace{-1mm}
$$
K'(t_1,t_2)=\left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1\ge t_2\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1\le t_2
\end{matrix}
\right.,\ \ \ t_1,t_2\in[t,T]
$$

\vspace{2mm}
\noindent
and prove that 

\vspace{-1mm}
\begin{equation}
\label{9090}
|K'(t_1,t_2)-K'(t_1^{*},t_2^{*})|\le L(|t_1-t_1^{*}|+|t_2-t_2^{*}|),
\end{equation}

\vspace{3mm}
\noindent
where $L<\infty,$ and $(t_1,t_2)$, $(t_1,t_2)\in [t, T]^2.$


By Lagrange formula for
the functions $\psi_1(t_1^{*}),$ $\psi_2(t_1^{*})$ on the interval
$[{\rm min}\{t_1, t_1^{*}\}, {\rm max}\{t_1, t_1^{*}\}]$ 
and for the functions 
$\psi_1(t_2^{*}),$ $\psi_2(t_2^{*})$ on the interval
$[{\rm min}\{t_2, t_2^{*}\}, {\rm max}\{t_2, t_2^{*}\}]$ 
we obtain

\vspace{2mm}
$$
|K'(t_1,t_2)-K'(t_1^{*},t_2^{*})|\le 
\left|\ \left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1\ge t_2\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1\le t_2
\end{matrix}\right.
-\left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ t_1^{*}\ge t_2^{*}\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1^{*}\le t_2^{*}
\end{matrix}
\right.\ \right|+
$$

\vspace{2mm}
\begin{equation}
\label{uuuu1}
+
L_1|t_1-t_1^{*}|+L_2|t_2-t_2^{*}|,\ \ \ L_1, L_2<\infty.
\end{equation}

\vspace{6mm}

We have

$$
\left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1\ge t_2\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1\le t_2
\end{matrix}
\right.
-\left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1^{*}\ge t_2^{*}\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1^{*}\le t_2^{*}
\end{matrix}\right.
=
$$

\vspace{3mm}
\begin{equation}
\label{uuuu2}
=\left\{
\begin{matrix}
0,\ \ \ t_1\ge t_2,\  t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \ 
t_1\le t_2,\ t_1^{*}\le t_2^{*}\cr\cr
\psi_2(t_1)\psi_1(t_2)-\psi_1(t_1)\psi_2(t_2),\ \ \
t_1\ge t_2,\ t_1^{*}\le t_2^{*}\cr\cr
\psi_1(t_1)\psi_2(t_2)-\psi_2(t_1)\psi_1(t_2),\ \ \
t_1\le t_2,\ t_1^{*}\ge t_2^{*}
\end{matrix}.\right.
\end{equation}

\vspace{6mm}

By Lagrange formula for the functions
$\psi_1(t_2),$ $\psi_2(t_2)$ at the interval
$[{\rm min}\{t_1, t_2\}, {\rm max}\{t_1, t_2\}]$ 
we obtain the estimate

\vspace{1mm}
$$
\left|\ \left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1\ge t_2\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1\le t_2
\end{matrix}\right.
-\left\{
\begin{matrix}\psi_2(t_1)\psi_1(t_2),\ \ t_1^{*}\ge t_2^{*}\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1^{*}\le t_2^{*}
\end{matrix}
\right.\ \right|\le
$$

\begin{equation}
\label{ppp1}
\le L_3 |t_2-t_1|
\left\{
\begin{matrix}
0,\ \ \ t_1\ge t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \
t_1\le t_2,\ t_1^{*}\le t_2^{*}\cr\cr
1,\ \ \ t_1\le t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \
t_1\ge t_2,\ t_1^{*}\le t_2^{*}
\end{matrix},\right.\ \ \ L_3<\infty. 
\end{equation}

\vspace{5mm}

Let us show that if $t_1\le t_2,\ t_1^{*}\ge t_2^{*}$ or
$t_1\ge t_2,\ t_1^{*}\le t_2^{*}$, then the following 
inequality is satisfied  

\vspace{-1mm}
\begin{equation}
\label{r5r5}
|t_2-t_1|\le |t_1^{*}-t_1|+|t_2^{*}-t_2|.
\end{equation}

\vspace{3mm}

First, consider the case $t_1\ge t_2,\ t_1^{*}\le t_2^{*}$.
For this case

\vspace{-1mm}
$$
t_2+(t_1^{*}-t_2^{*})\le t_2\le t_1.
$$

\vspace{2mm}

Then 
$$
(t_1^{*}-t_1)-(t_2^{*}-t_2)\le t_2-t_1\le 0
$$ 

\vspace{2mm}
\noindent
and (\ref{r5r5}) is satisfied.

For the case $t_1\le t_2,\ t_1^{*}\ge t_2^{*}$ we obtain

\vspace{-1mm}
$$
t_1+(t_2^{*}-t_1^{*})\le t_1\le t_2.
$$

\vspace{2mm}


Then 
$$
(t_1-t_1^{*})-(t_2-t_2^{*})\le t_1-t_2\le 0
$$ 

\vspace{2mm}
\noindent
and also (\ref{r5r5}) is satisfied.

From (\ref{ppp1}) and (\ref{r5r5}) we have

\vspace{3mm}
$$
\left|\ \left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1\ge t_2\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1\le t_2
\end{matrix}\right.
-\left\{
\begin{matrix}
\psi_2(t_1)\psi_1(t_2),\ \ t_1^{*}\ge t_2^{*}\cr\cr
\psi_1(t_1)\psi_2(t_2),\ \ t_1^{*}\le t_2^{*}
\end{matrix}
\right.\ \right|\le
$$

\vspace{5mm}
$$
\le L_3 (|t_1^{*}-t_1|+|t_2^{*}-t_2|)
\left\{
\begin{matrix}
0,\ \ \ t_1\ge t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \ 
t_1\le t_2,\ t_1^{*}\le t_2^{*}\cr\cr
1,\ \ \ t_1\le t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \ 
t_1\ge t_2,\ t_1^{*}\le t_2^{*}
\end{matrix}\ \right. \le
$$

\vspace{5mm}
$$
\le
L_3 (|t_1^{*}-t_1|+|t_2^{*}-t_2|)
\left\{
\begin{matrix}
1,\ \ \ t_1\ge t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \ 
t_1\le t_2,\ t_1^{*}\le t_2^{*}\cr\cr
1,\ \ \ t_1\le t_2,\ t_1^{*}\ge t_2^{*}\ \ \ \hbox{or}\ \ \ 
t_1\ge t_2,\ t_1^{*}\le t_2^{*}
\end{matrix}=\right. 
$$

\vspace{3mm}
\begin{equation}
\label{t2t2}
=L_3 (|t_1^{*}-t_1|+|t_2^{*}-t_2|).
\end{equation}

\vspace{7mm}

From (\ref{uuuu1}), (\ref{t2t2}) we obtain (\ref{9090}).
Let 


$$
t_1=\frac{T-t}{2}x+\frac{T+t}{2},\ \ \
t_2=\frac{T-t}{2}y+\frac{T+t}{2},
$$

\vspace{3mm}
\noindent
where $x, y \in [-1,1].$ 
Then

\vspace{-1mm}
$$
K'(t_1,t_2)\equiv K^{*}(x,y)
=\left\{
\begin{matrix}
\psi_2\left(h(x)\right)
\psi_1\left(h(y)\right),\ \ x\ge y\cr\cr
\psi_1\left(h(x)\right)
\psi_2\left(h(y)\right),\ \ x\le y
\end{matrix}\right.,
$$

\vspace{2mm}
\noindent
where $x, y\in[-1,1]$ and

\vspace{-2mm}
$$
h(x)=\frac{T-t}{2}x+\frac{T+t}{2}.
$$

\vspace{3mm}

Inequality (\ref{9090}) can be rewritten in the form 

\vspace{-1mm}
\begin{equation}
\label{4343}
|K^{*}(x,y)- K^{*}(x^{*},y^{*})|\le L^{*}(|x-x^{*}|+|y-y^{*}|),
\end{equation}

\vspace{3mm}
\noindent
where $L^{*}<\infty$ and $(x,y)$, $(x^{*},y^{*})\in [-1, 1]^2.$


Thus, the function $K^{*} (x,y)$ satisfies the conditions of the
lemma 1 and hence for the function $K^{*}(x,y)(1-x^2)^{1/4}(1-y^2)^{1/4}$ 
the inequality (\ref{2222}) is true.


Due to continuous differentiability of functions 
$\psi_1\left(h(x)\right)$ and
$\psi_2\left(h(x)\right)$ on the interval
$[-1,1]$ we have
$K^{*}(x,y)\in L_2([-1,1]^2)$. In addition



$$
\int\limits_{[-1,1]^2}
\frac{K^{*}(x,y)dxdy}{(1-x^2)^{1/4}(1-y^2)^{1/4}}\le
C \Biggl(~\int\limits_{-1}^{1}\frac{1}{(1-x^2)^{1/4}}
\int\limits_{-1}^{x}\frac{1}{(1-y^2)^{1/4}}dydx+\Biggr.
$$

$$
\Biggl.+
\int\limits_{-1}^{1}\frac{1}{(1-x^2)^{1/4}}
\int\limits_{x}^{1}\frac{1}{(1-y^2)^{1/4}}dydx\Biggr)<\infty,\ \ \ C<\infty.
$$

\vspace{4mm}


Thus, the conditions of the theorem 4 are met for the function $K^{*} (x,y).$ 
Furthermore, due to inequality (\ref{4343})
the function $K^{*}(x, y)$ satisfies the condition 
(\ref{444}) at each point $(x,y)\in (-1,1)^2.$


Note that all mentioned properties of the function
$K^{*}(x,y),$ $x,y \in [-1,1]$ also true 
for the function $K'(t_1,t_2),$ $t_1, t_2\in [t, T].$


Let us expand the function $K'(t_1,t_2)$ into a multiple 
(double) Fourier--Legendre series in the square $[t, T]^2$, summable 
by the method of rectangular sums, i.e.


$$
K'(t_1,t_2)=
\lim_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1}\sum_{j_2=0}^{n_2}
\int\limits_t^T\int\limits_t^TK'(t_1,t_2)
\phi_{j_1}(t_1)\phi_{j_2}(t_2)dt_1 dt_2\cdot
\phi_{j_1}(t_1)\phi_{j_2}(t_2)=
$$

\vspace{1mm}
$$
=\lim_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1}\sum_{j_2=0}^{n_2}\Biggl(~
\int\limits_t^T\psi_2(t_2)\phi_{j_2}(t_2)
\int\limits_t^{t_2}\psi_1(t_1)\phi_{j_1}(t_1)dt_1dt_2+\Biggr.
$$

$$
\Biggl. +
\int\limits_t^T\psi_1(t_2)\phi_{j_2}(t_2)
\int\limits_{t_2}^{T}\psi_2(t_1)\phi_{j_1}(t_1)dt_1\Biggr)dt_2
\phi_{j_1}(t_1)\phi_{j_2}(t_2)
=
$$


\begin{equation}
\label{334.ye}
=\lim_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1}\sum_{j_2=0}^{n_2}\left(C_{j_2j_1}+
C_{j_1j_2}\right)
\phi_{j_1}(t_1)\phi_{j_2}(t_2),\ \ \ (t_1, t_2)\in (t, T)^2,
\end{equation}


\vspace{5mm}
\noindent
moreover, the convergence of the series (\ref{334.ye}) is uniform on any
rectangle

\vspace{-1mm}
$$
[t+\varepsilon, T-\varepsilon]\times[t+\delta,T-\delta]\ \
\forall\ \varepsilon, \delta>0\ \hbox{(in particular, we can 
take}\ \varepsilon=\delta).
$$

\vspace{2mm}

In addition, the series (\ref{334.ye}) converges to 
$K'(t_1,t_2)$ at any inner point of the square $[t, T]^2.$


Note that theorems 3 and 4 do not answer the question of convergence
of the series
(\ref{334.ye}) on a border of the square
$[t, T]^2.$


In obtaining (\ref{334.ye}) we replaced the order of
integration in the second repeated integral.


Let us put $t_1=t_2$ in (\ref{334.ye}).
After that, let us rewrite the limit on 
the right-hand side of (\ref{334.ye}) as two limits.
Let us replace $j_1$ with $j_2$, $j_2$ with $j_1$, 
$n_1$ with $n_2$ and $n_2$ with $n_1$ in the second limit. Thus, we get


\vspace{-3mm}
\begin{equation}
\label{5656}
\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1} \sum_{j_2=0}^{n_2} C_{j_2j_1}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)=\frac{1}{2}\psi_1(t_1)\psi_2(t_1),\ \ \ t_1\in (t, T).
\end{equation}

\vspace{4mm}


According to the above reasoning, equality
(\ref{5656}) holds uniformly on every interval
$[t+\varepsilon, T-\varepsilon]$\ $\forall$ $\varepsilon>0.$
Additionally, (\ref{5656}) holds at each interior point
of the interval $[t, T].$

\vspace{2mm}

{\bf Lemma 2}.\ {\it 
Under the conditions of theorem {\rm 6} the series
{\rm (\ref{5656})} for $n_1=n_2$ converges at $t_1=t,\ T.$
At that, the series 

\vspace{-1mm}
$$
\lim\limits_{n\to\infty}
\sum_{j_1=0}^{n} \sum_{j_2=0}^{n} C_{j_2j_1}\phi_{j_1}(t)
\phi_{j_2}(t),\ \ \ 
\lim\limits_{n\to\infty}
\sum_{j_1=0}^{n} \sum_{j_2=0}^{n} C_{j_2j_1}\phi_{j_1}(T)
\phi_{j_2}(T)
$$

\vspace{4mm}
\noindent
may not converge to $\psi_1 (t)\psi_2 (t) / 2$ 
and $\psi_1 (T)\psi_2(T)/2$ respectively.}


\vspace{2mm}


The proof of the lemma 2 is given in the section 7.


Let us integrate equality (\ref{5656}) at the interval $[t+\varepsilon,
T-\varepsilon]$ $\forall \varepsilon>0$. 
Due to uniform convergence of the series (\ref{5656}) we can replace the 
places of the series and the integral 

\vspace{-1mm}
\begin{equation}
\label{5657}
\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1} \sum_{j_2=0}^{n_2} C_{j_2j_1}
\int\limits_{t+\varepsilon}^{T-\varepsilon}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)dt_1=\int\limits_{t+\varepsilon}^{T-\varepsilon}
\frac{1}{2}\psi_1(t_1)\psi_2(t_1)dt_1.
\end{equation}

\vspace{3mm}

Consider the left-hand side of (\ref{5657})

\vspace{-1mm}
$$
\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1} \sum_{j_2=0}^{n_2} C_{j_2j_1}
\int\limits_{t+\varepsilon}^{T-\varepsilon}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)dt_1=
$$

$$
=\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1} \sum_{j_2=0}^{n_2} C_{j_2j_1}\biggl(
\int\limits_{t}^{T}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)dt_1-\int\limits_{t}^{t+\varepsilon}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)dt_1-
\int\limits_{T-\varepsilon}^{T}\phi_{j_1}(t_1)
\phi_{j_2}(t_1)dt_1\biggr)=
$$

$$
=\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{n_1} \sum_{j_2=0}^{n_2} C_{j_2j_1}\biggl(
{\bf 1}_{\{j_1=j_2\}}-
\biggl(\phi_{j_1}(\theta)
\phi_{j_2}(\theta)+\phi_{j_1}(\lambda)\phi_{j_2}(\lambda)\biggr)
\varepsilon\biggr)=
$$

$$
=\lim\limits_{n_1,n_2\to\infty}
\sum_{j_1=0}^{\min\{n_1,n_2\}}C_{j_1j_1}-
\varepsilon\biggl(\sum_{j_1,j_2=0}^{\infty}
C_{j_2j_1}\phi_{j_1}(\theta)
\phi_{j_2}(\theta)-
\sum_{j_1,j_2=0}^{\infty}
C_{j_2j_1}\phi_{j_1}(\lambda)
\phi_{j_2}(\lambda)\biggr)=
$$

\begin{equation}
\label{566}
=\sum_{j_1=0}^{\infty}C_{j_1j_1}-
\varepsilon\biggl(\sum_{j_1,j_2=0}^{\infty}
C_{j_2j_1}\phi_{j_1}(\theta)
\phi_{j_2}(\theta)-
\sum_{j_1,j_2=0}^{\infty}
C_{j_2j_1}\phi_{j_1}(\lambda)
\phi_{j_2}(\lambda)\biggr),
\end{equation}

\vspace{7mm}
\noindent
where $\theta\in [t,t+\varepsilon],$ $\lambda\in [T-\varepsilon,T]$.


In obtaining (\ref{566}) we used the theorem
on the mean value for the integral and orthonormality of functions
$\phi_{j}(x)$ for $j=0, 1, 2\ldots$


Substitute (\ref{566}) into (\ref{5657})
and implement in the resulting equality
the passage to the limit $\lim\limits_{\varepsilon\to+0}$. As a result
we come to the equality

\vspace{-2mm}
$$
\frac{1}{2}\int\limits_t^T\psi_1(t_1)\psi_2(t_1)dt_1
=\sum_{j_1=0}^{\infty}
C_{j_1j_1}.
$$

\vspace{3mm}

In obtaining the last equality we used the relation

\vspace{-1mm}
$$
\sum_{j_1,j_2=0}^{\infty}
C_{j_2j_1}\phi_{j_1}(s)
\phi_{j_2}(s)<\infty,\ \ s\in[t,t+\varepsilon],\
[T-\varepsilon,T].
$$

\vspace{3mm}


The theorem is proven. 

\vspace{2mm}

{\bf Remark 1.} {\it On the basis of {\rm(\ref{9090})} it can be 
argued that the 
function $K'(t_1, t_2)$ belongs to the Holder class 
with parameter $1$ in $[t, T]^2.$ Hence by theorem $5$ this 
function can be expanded into uniformly convergent double 
trigonometric Fourier series on the square $[t, T]^2,$ 
which summarized by Prinsheim method.
This means that theorem $6$ will remain valid if in 
this theorem instead of a double Fourier--Legendre 
series we consider a double trigonometric Fourier series. 
However, the expansions of iterated stochastic integrals 
obtained by using the system of
Legendre polynomials are essentially easier than their 
analogues obtained by using the trigonometric system of functions
{\rm (}see section {\rm 8)}.}


\vspace{5mm}


\section{About Some Properties of Legendre Polynomials}


\vspace{5mm}


For further consideration, we will need some well-known properties 
of Legendre polynomials, as well as some modifications 
of these properties \cite{suet}, \cite{Gob}.


The complete orthonormal system
of Legendre polynomials in the space $L_2([t, T])$
looks as follows

\vspace{-1mm}
\begin{equation}
\label{4009}
\phi_j(x)=\sqrt{\frac{2j+1}{T-t}}P_j\biggl(\biggl(
x-\frac{T+t}{2}\biggr)\frac{2}{T-t}\biggr),\ \ \ j=0, 1, 2,\ldots,
\end{equation}

\vspace{2mm}
\noindent
where $P_j(x)$ is the Legendre polynomial. 

It is known that the Legendre polynomial $P_j (x)$ is represented, 
for example, as 

\vspace{-1mm}
$$
P_j(x)=\frac{1}{2^j j!} \frac{d^j}{dx^j}\left(x^2-1\right)^j.
$$

\vspace{3mm}

At the edges of the interval of orthogonality the Legendre 
polynomials satisfy the following relations

\vspace{-1mm}
$$
P_j(1)=1,\ \ \ P_j(-1)=(-1)^j, 
$$

$$
P_{j+1}(1)-P_j(1)=0,\ \ \ P_{j+1}(-1)+P_j(-1)=0,
$$

\vspace{2mm}
\noindent
where $j=0, 1, 2, \ldots$


Relation of the Legendre polynomial $P_j(x)$ with derivatives 
of the Legendre polynomials $P_{j+1}(x)$ and $P_{j-1}(x)$ is expressed 
by the following equality

\vspace{-1mm}
$$
P_j(x)=\frac{1}{2j+1}\left(P_{j+1}^{'}(x)-
P_{j-1}^{'}(x)\right),\ \ \
j=1, 2,\ldots
$$

\vspace{2mm}


The recurrent relation has the form

\vspace{-1mm}
$$
xP_{j}(x)=\frac{(j+1)P_{j+1}(x)+jP_{j-1}(x)}{2j+1},\ \ \ j=1, 2,\ldots
$$

\vspace{2mm}


Orthogonality of Legendre polynomial $P_j(x)$ to any polynomial
$Q_k (x)$ of lesser degree we write in the following form

\vspace{-1mm}
$$
\int\limits_{-1}^1 Q_k(x) P_j(x)dx=0,\ \ \ k=0, 1, 2,\ldots,j-1.
$$

\vspace{3mm}

From the property

\vspace{-1mm}
$$
\int\limits_{-1}^1 P_k(x)P_j(x)dx=\left\{
\begin{matrix}
0\ \ &\hbox{if}\ \
k\ne j\cr\cr
2/(2j+1)\ \ &\hbox{if}\ \ k=j
\end{matrix}
\right.
$$

\vspace{4mm}
\noindent
it follows that the orthonormal on the interval $[-1,1]$ Legendre polynomials 
determined by the relation


$$
P_j^{*}(x)=\sqrt{\frac{2j+1}{2}}P_j(x),\ \ \ j=0, 1, 2,\ldots
$$

\vspace{3mm}

It is well known that there is an estimate


\begin{equation}
\label{otit987}
|P_j(y)| <\frac{K}{\sqrt{j+1}(1-y^2)^{1/4}},\ \ \ 
y\in (-1, 1),\ \ \ j=1, 2,\ldots,
\end{equation}

\vspace{3mm}
\noindent
where constant $K$ does not depends on $y$ and $j$.

Moreover

\vspace{-3mm}
\begin{equation}
\label{q1}
|P_j(x)|\le 1,\ \ \ x\in[-1,1],\ \ \ j=0,\ 1,\ldots
\end{equation}

\vspace{3mm}

The Christoffel-Darboux formula has the form


\vspace{-1mm}
\begin{equation}
\label{ty}
\sum\limits_{j=0}^n (2j+1)P_j(x)P_j(y)=
(n+1)\frac{P_n(x)P_{n+1}(y)-P_{n+1}(x)P_{n}(y)}{y-x}.
\end{equation}

\vspace{2mm}

In particular, from (\ref{ty}) for $x=\pm 1$ we obtain

\vspace{-1mm}
\begin{equation}
\label{yy3}
\sum\limits_{j=0}^n (2j+1)P_j(y)=
(n+1)\frac{P_{n+1}(y)-P_{n}(y)}{y-1},
\end{equation}

\begin{equation}
\label{yy4}
\sum\limits_{j=0}^n (2j+1)(-1)^{j}P_j(y)=
(n+1)(-1)^n\frac{P_{n+1}(y)+P_{n}(y)}{y+1}.
\end{equation}

\vspace{3mm}

From the other side

\vspace{-2mm}
$$
\sum\limits_{j=0}^n (2j+1)P_j(y)=1+\sum\limits_{j=1}^n (2j+1)P_j(y)=
$$

$$
=1+\sum\limits_{j=1}^n (P_{j+1}^{'}(y)-P_{j-1}^{'}(y))=
1+\biggl(\sum\limits_{j=1}^n (P_{j+1}(y)-P_{j-1}(y))\biggr)'=
$$

\begin{equation}
\label{yy2}
=1+(P_{n+1}(x)+P_n(x)-x-1)'=(P_n(x)+P_{n+1}(x))'
\end{equation}

\vspace{2mm}
\noindent
and

\vspace{-2mm}
$$
\sum\limits_{j=0}^n (2j+1)(-1)^j P_j(y)=1+\sum\limits_{j=1}^n (-1)^j
(2j+1)P_j(y)=
$$

$$
=1+\sum\limits_{j=1}^n (-1)^j (P_{j+1}^{'}(y)-P_{j-1}^{'}(y))=
1+\biggl(\sum\limits_{j=1}^n (-1)^j(P_{j+1}(y)-P_{j-1}(y))\biggr)'=
$$

\begin{equation}
\label{yy1}
=1+((-1)^n(P_{n+1}(x)-P_n(x))-x+1)'=(-1)^n(P_{n+1}(x)-P_{n}(x))'.
\end{equation}

\vspace{8mm}


Comparing (\ref{yy3}) with (\ref{yy2}) and (\ref{yy4}) with (\ref{yy1}) 
we obtain

\vspace{1mm}
\begin{equation}
\label{uu1}
(n+1)\frac{P_{n+1}(y)-P_{n}(y)}{y-1}=(P_n(x)+P_{n+1}(x))',
\end{equation}

\begin{equation}
\label{uu2}
(n+1)\frac{P_{n+1}(y)+P_{n}(y)}{y+1}
=(P_{n+1}(x)-P_{n}(x))'.
\end{equation}


\vspace{5mm}


\section{Proof of the Lemma 2}


\vspace{5mm}


So, it is required to prove that in the conditions of theorem 6 
the following values 
are finite

\vspace{-1mm}
\begin{equation}
\label{rr1}
\lim\limits_{n\to\infty}
\sum\limits_{j_2=0}^n
\sum\limits_{j_1=0}^n
C_{j_2j_1}\phi_{j_2}(T)\phi_{j_1}(T),\ \ \
\lim\limits_{n\to\infty}
\sum\limits_{j_2=0}^n
\sum\limits_{j_1=0}^n
C_{j_2j_1}\phi_{j_2}(t)\phi_{j_1}(t).
\end{equation}

\vspace{3mm}

We have

\vspace{-2mm}
$$
\sum\limits_{j_2=0}^n
\sum\limits_{j_1=0}^n
C_{j_2j_1}\phi_{j_2}(T)\phi_{j_1}(T)=
$$

$$
=\frac{1}{4}
\sum\limits_{j_2=0}^n
\sum\limits_{j_1=0}^n
(2j_2+1)(2j_1+1)\int\limits_{-1}^{1}
\psi_2(h(y))P_{j_2}(y)
\int\limits_{-1}^{y}
\psi_1((h(y_1))P_{j_1}(y_1)dy_1 dy=
$$

$$
=\frac{1}{4}
\int\limits_{-1}^{1}
\psi_2(h(y))
\sum\limits_{j_2=0}^n
(2j_2+1)P_{j_2}(y)
\int\limits_{-1}^{y}
\psi_1((h(y_1))\sum\limits_{j_1=0}^n
(2j_1+1)P_{j_1}(y_1)dy_1 dy=
$$

$$
=\frac{1}{4}
\int\limits_{-1}^{1}
\psi_2(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)+P_{n}(y_1))\biggr)d(P_{n+1}(y)+P_{n}(y))=
$$

$$
=\frac{1}{4}
\int\limits_{-1}^{1}
\psi_1(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)+P_{n}(y_1))\biggr)d(P_{n+1}(y)+P_{n}(y))+
$$

$$
+\frac{1}{4}
\int\limits_{-1}^{1}
\Delta(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)+P_{n}(y_1))\biggr)d(P_{n+1}(y)+P_{n}(y))=
$$

$$
=\frac{1}{4}I_1+\frac{1}{4}I_2,
$$

\vspace{4mm}
\noindent
where 

\vspace{-3mm}
\begin{equation}
\label{191}
\Delta(h(y))=\psi_2((h(y))-\psi_1((h(y)),\ \ \ 
h(y)=\frac{T-t}{2}y+\frac{T+t}{2}.
\end{equation}

\vspace{3mm}

Moreover

\vspace{-3mm}
$$
I_1=\frac{1}{2}
\biggl(\int\limits_{-1}^{1}
\psi_1((h(y))
d(P_{n+1}(y)+P_{n}(y))\biggr)^2=
$$

$$
=\frac{1}{2}\biggl(2\psi_1(T)-\int\limits_{-1}^{1}
(P_{n+1}(y)+P_{n}(y))\psi_1'((h(y))\frac{T-t}{2}dy\biggr)^2<C_1<\infty,
$$

\vspace{5mm}
\noindent
where constant $C_1$ does not depends on $n$ and $\psi_1'$ is
a derivative of
$\psi_1$ with respect to the variable $y.$


By Lagrange formula we obtain

\vspace{-1mm}
$$
\Delta(h(y))=\psi_2\biggl(\frac{1}{2}(T-t)(y-1)+T\biggr)-
\psi_1\biggl(\frac{1}{2}(T-t)(y-1)+T\biggr)=
$$

\begin{equation}
\label{hu}
=\psi_2(T)-\psi_1(T)+(y-1)\biggl(\psi_2'(\xi_y)-\psi_1'(\theta_y)\biggr)
\frac{1}{2}(T-t)=C_1+\alpha_y(y-1),
\end{equation}

\vspace{5mm}
\noindent
where $|\alpha_y|<\infty$ and $C_1=\psi_2(T)-\psi_1(T).$


Consider the integral $I_2$ and substitute (\ref{hu}) into the 
integral $I_2$

\vspace{-1mm}
$$
I_2=I_3+I_4,
$$

\vspace{1mm}
\noindent

$$
I_3=\int\limits_{-1}^{1}
\alpha_y(y-1)
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)+P_{n}(y_1))\biggr)d(P_{n+1}(y)+P_{n}(y)),
$$

$$
I_4=
C_1\int\limits_{-1}^{1}
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)+P_{n}(y_1))\biggr)d(P_{n+1}(y)+P_{n}(y)).
$$

\vspace{4mm}

Integrating by parts and using (\ref{uu1}), we obtain

\vspace{-1mm}
$$
I_3=
\int\limits_{-1}^{1}
\frac{\alpha_y(y-1)(n+1)(P_{n+1}(y)-P_{n}(y))}{y-1}
\biggl(
\psi_1((h(y))(P_{n+1}(y)+P_{n}(y))-\biggr.
$$

$$
\biggl.-
\int\limits_{-1}^{y}
(P_{n+1}(y_1)+P_{n}(y_1))\psi_1'((h(y_1))\frac{1}{2}(T-t)dy_1\biggr)dy.
$$

\vspace{4mm}

Applying the estimate (\ref{otit987}) and taking into account the 
boundedness of $\alpha_y$ and $\psi_1'((h(y_1))$, we will have
$|I_3|<\infty.$

Change the integration order in the integral $I_4$

\vspace{-1mm}
$$
I_4=
C_1\int\limits_{-1}^{1}\psi_1((h(y_1))
\biggl(\int\limits_{y_1}^{1}
d(P_{n+1}(y)+P_{n}(y))\biggr)d(P_{n+1}(y_1)+P_{n}(y_1))=
$$

$$
=C_1\int\limits_{-1}^{1}\psi_1((h(y_1))d(P_{n+1}(y_1)+P_{n}(y_1))
\int\limits_{-1}^{1}
d(P_{n+1}(y)+P_{n}(y))-
$$

$$
-C_1\int\limits_{-1}^{1}\psi_1((h(y_1))
\biggl(\int\limits_{-1}^{y_1}
d(P_{n+1}(y)+P_{n}(y))\biggr)d(P_{n+1}(y_1)+P_{n}(y_1))=
$$

$$
=I_5-I_6.
$$

\vspace{2mm}

Consider the integral $I_5$

\vspace{-1mm}
$$
I_5=2C_1\int\limits_{-1}^{1}\psi_1((h(y_1))d(P_{n+1}(y_1)+P_{n}(y_1))=
$$

$$
=2C_1\biggl(2\psi_1(T)-
\int\limits_{-1}^{1}
(P_{n+1}(y_1)+P_{n}(y_1))
\psi_1'((h(y_1))\frac{1}{2}(T-t)dy_1\biggr).
$$

\vspace{2mm}

Applying the estimate (\ref{q1}) and taking into account 
the boundedness of $\psi_1'((h (y_1))$, we will
have
$|I_5|<\infty.$


Since (see (\ref{hu}))

\vspace{-3mm}
$$
\psi_1(h(y))=
\psi_1\biggl(\frac{1}{2}(T-t)(y-1)+T\biggr)=
$$

$$
=\psi_1(T)+(y-1)\psi_1'(\theta_y)
\frac{1}{2}(T-t)=C_2+\beta_y(y-1),
$$

\vspace{5mm}
\noindent
where $|\beta_y|<\infty$ and $C_2=\psi_1(T),$ then


$$
I_6=
C_3\int\limits_{-1}^{1}
\biggl(\int\limits_{-1}^{y_1}
d(P_{n+1}(y)+P_{n}(y))\biggr)d(P_{n+1}(y_1)+P_{n}(y_1))+
$$

$$
+C_1\int\limits_{-1}^{1}\beta_{y_1}(y_1-1)
\biggl(\int\limits_{-1}^{y_1}
d(P_{n+1}(y)+P_{n}(y))\biggr)d(P_{n+1}(y_1)+P_{n}(y_1))=
$$

$$
=\frac{C_3}{2}\biggl(\int\limits_{-1}^{1}
d(P_{n+1}(y)+P_{n}(y))\biggr)^2+
$$

$$
+C_1\int\limits_{-1}^{1}\frac{\beta_{y_1}(y_1-1)
(n+1)(P_{n+1}(y_1)-P_{n}(y_1))}{y_1-1}
\biggl(\int\limits_{-1}^{y_1}
d(P_{n+1}(y)+P_{n}(y))\biggr)dy_1=
$$

$$
=2C_3+
C_1\int\limits_{-1}^{1}\beta_{y_1}
(n+1)(P_{n+1}(y_1)-P_{n}(y_1))
(P_{n+1}(y_1)+P_{n}(y_1))dy_1.
$$

\vspace{3mm}

Applying the estimate (\ref{otit987}) and taking into account 
the boundedness of $\beta_{y_1}$, we will have $|I_6|<\infty.$

Thus the boundedness of the first series in (\ref{rr1}) is proved.


Let us prove the boundedness of the second series in (\ref{rr1}). 
We have

\vspace{1mm}
$$
\sum\limits_{j_2=0}^n
\sum\limits_{j_1=0}^n
C_{j_2j_1}\phi_{j_2}(t)\phi_{j_1}(t)=
$$

$$
=\frac{1}{4}
\sum\limits_{j_1,j_2=0}^n
(2j_2+1)(2j_1+1)(-1)^{j_1+j_2}\int\limits_{-1}^{1}
\psi_2(h(y))P_{j_2}(y)
\int\limits_{-1}^{y}
\psi_1((h(y_1))P_{j_1}(y_1)dy_1 dy
$$

$$
=\frac{1}{4}
\int\limits_{-1}^{1}
\psi_2(h(y))
\sum\limits_{j_2=0}^n
(2j_2+1)P_{j_2}(y)(-1)^{j_2}
\int\limits_{-1}^{y}
\psi_1((h(y_1))\sum\limits_{j_1=0}^n
(2j_1+1)P_{j_1}(y_1)(-1)^{j_1}dy_1 dy=
$$

$$
=\frac{(-1)^{2n}}{4}
\int\limits_{-1}^{1}
\psi_2(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y))
$$

$$
=\frac{1}{4}
\int\limits_{-1}^{1}
\psi_1(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y))+
$$

$$
+\frac{1}{4}
\int\limits_{-1}^{1}
\Delta(h(y))
\biggl(\int\limits_{-1}^{y}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y))=
$$

$$
=\frac{1}{4}J_1+\frac{1}{4}J_2,
$$

\vspace{7mm}
\noindent
where $\Delta(h(y)),$
$h(y)$ are defined by relations (\ref{191}).

Moreover

\vspace{-2mm}
$$
J_1=\frac{1}{2}
\biggl(\int\limits_{-1}^{1}
\psi_1((h(y))
d(P_{n+1}(y)-P_{n}(y))\biggr)^2=
$$

\begin{equation}
\label{hh1}
=\frac{1}{2}\biggl(2(-1)^{n}\psi_1(t)-\int\limits_{-1}^{1}
(P_{n+1}(y)-P_{n}(y))\psi_1'((h(y))\frac{T-t}{2}dy\biggr)^2<K_1<\infty,
\end{equation}

\vspace{2mm}
\noindent
where constant $K_1$ does not depends on $n$ and $\psi_1'$ is
a derivative of
$\psi_1$ with respect to the variable $y.$


By Lagrange formula we obtain

\vspace{-1mm}
$$
\Delta(h(y))=\psi_2\biggl(\frac{1}{2}(T-t)(y+1)+t\biggr)-
\psi_1\biggl(\frac{1}{2}(T-t)(y+1)+t\biggr)=
$$

\begin{equation}
\label{hu11}
=\psi_2(t)-\psi_1(t)+(y+1)\biggl(\psi_2'(\mu_y)-\psi_1'(\rho_y)\biggr)
\frac{1}{2}(T-t)=K_2+\gamma_y(y+1),
\end{equation}

\vspace{3mm}
\noindent
where $|\gamma_y|<\infty$ and $K_2=\psi_2(t)-\psi_1(t).$

Consider the integral $J_2$

\vspace{-1mm}
$$
J_2=
\int\limits_{-1}^{1}
\Delta(h(y))
d(P_{n+1}(y)-P_{n}(y))
\int\limits_{-1}^{1}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))-
$$

$$
-
\int\limits_{-1}^{1}
\Delta(h(y))
\biggl(\int\limits_{y}^{1}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y))=
$$

$$
=J_3J_4-J_5.
$$

\vspace{3mm}

The integral $J_4$ appeared when considering the integral $J_1$ 
(see (\ref{hh1})),
where it was shown that $|J_4|<\infty$. Similarly, $|J_3|<\infty.$

Consider the integral $J_5$ and substitute (\ref{hu11})
into the integral $J_5$

\vspace{-2mm}
$$
J_5=J_6+J_7,
$$

$$
J_6=\int\limits_{-1}^{1}
\gamma_y(y+1)
\biggl(\int\limits_{y}^{1}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y)),
$$

$$
J_7=
K_2\int\limits_{-1}^{1}
\biggl(\int\limits_{y}^{1}
\psi_1((h(y_1))
d(P_{n+1}(y_1)-P_{n}(y_1))\biggr)d(P_{n+1}(y)-P_{n}(y)).
$$

\vspace{4mm}

Integrating by parts and using (\ref{uu2}), we obtain

\vspace{1mm}
$$
J_6=
\int\limits_{-1}^{1}
\frac{\gamma_y(y+1)(n+1)(P_{n+1}(y)+P_{n}(y))}{y+1}
\biggl(
-\psi_1((h(y))(P_{n+1}(y)-P_{n}(y))-\biggr.
$$

$$
\biggl.-
\int\limits_{y}^{1}
(P_{n+1}(y_1)-P_{n}(y_1))\psi_1'((h(y_1))\frac{1}{2}(T-t)dy_1\biggr)dy.
$$

\vspace{2mm}


Applying the estimate (\ref{otit987}) and taking into account the  
boundedness of $\gamma_y$ and $\psi_1'((h(y_1))$, we have $|J_6|<\infty.$


Change the integration order in the integral $J_7$ 

\vspace{-1mm}
$$
J_7=
K_2\int\limits_{-1}^{1}
\psi_1((h(y_1))
\biggl(\int\limits_{-1}^{y_1}
d(P_{n+1}(y)-P_{n}(y))\biggr)d(P_{n+1}(y_1)-P_{n}(y_1))=
$$

$$
=K_2\int\limits_{-1}^{1}
\psi_1((h(y_1))d(P_{n+1}(y_1)-P_{n}(y_1))
\int\limits_{-1}^{1}
d(P_{n+1}(y)-P_{n}(y))-K_2J_8=
$$

$$
=K_2 J_4 2(-1)^{n} - K_2J_8,
$$

\vspace{1mm}
\noindent
where

\vspace{-2mm}
$$
J_8=\int\limits_{-1}^{1}
\psi_1((h(y_1))
\biggl(\int\limits_{y_1}^{1}
d(P_{n+1}(y)-P_{n}(y))\biggr)d(P_{n+1}(y_1)-P_{n}(y_1)).
$$

\vspace{4mm}

Since (see (\ref{hu11}))

\vspace{-2mm}
$$
\psi_1(h(y))=\psi_1\biggl(\frac{1}{2}(T-t)(y+1)+t\biggr)=
$$

\begin{equation}
\label{ooo1}
=\psi_1(t)+(y+1)\psi_1'(\rho_y)
\frac{1}{2}(T-t)=K_3+\varepsilon_y(y+1),
\end{equation}

\vspace{4mm}
\noindent
where $|\varepsilon_y|<\infty$ and $K_3=\psi_1(t),$ then

\vspace{1mm}
$$
J_8=K_3\int\limits_{-1}^{1}
\biggl(\int\limits_{y_1}^{1}
d(P_{n+1}(y)-P_{n}(y))\biggr)d(P_{n+1}(y_1)-P_{n}(y_1))+
$$

$$
+\int\limits_{-1}^{1}
\varepsilon_y(y+1)
\biggl(\int\limits_{y_1}^{1}
d(P_{n+1}(y)-P_{n}(y))\biggr)d(P_{n+1}(y_1)-P_{n}(y_1))=
$$

$$
=\frac{K_3}{2}\biggl(\int\limits_{-1}^{1}
d(P_{n+1}(y)-P_{n}(y))\biggr)^2+
$$

$$
+\int\limits_{-1}^{1}
\frac{\varepsilon_{y_1}(y_1+1)(n+1)(P_{n+1}(y_1)+P_{n}(y_1))}{y_1+1}
(P_{n}(y_1)-P_{n+1}(y_1))dy=
$$

\begin{equation}
\label{qqq1}
=2K_3+\int\limits_{-1}^{1}
\varepsilon_{y_1}(n+1)(P_{n+1}(y_1)+P_{n}(y_1))
(P_{n}(y_1)-P_{n+1}(y_1))dy.
\end{equation}

\vspace{4mm}

For obtaining (\ref{qqq1}) we used (\ref{uu2}).
Applying the estimate (\ref{otit987}) and taking into account 
the boundedness of $\varepsilon_{y_1}$,
we obtain $|J_8|<\infty.$

Thus, the boundedness of the second series in (\ref{rr1}) 
is proved. The lemma 2 is proved.


\vspace{5mm}

\section{Expansions of Iterated Stratonovich Stochastic Integrals
of First and Second Multiplicity, Based on the Fourier--Legendre
Series}


\vspace{5mm}


We will use the following notations for iterated 
Stratonovich stochastic integrals of 1 and 2 multiplicities

\vspace{-1mm}
\begin{equation}
\label{ll1}
I_{(l_1)T,t}^{*(i_1)}
={\int\limits_t^{*}}^T(t-t_1)^{l_1} d{\bf f}_{t_1}^{(i_1)},
\end{equation}

\begin{equation}
\label{ll2}
I_{(l_1 l_2)T,t}^{*(i_1i_2)}
={\int\limits_t^{*}}^T(t-t_2)^{l_2}{\int\limits_t^{*}}^{t_{2}}
(t-t_1)^{l_1} d{\bf f}_{t_1}^{(i_1)}d{\bf f}_{t_2}^{(i_2)},
\end{equation}

\vspace{2mm}
\noindent
where $l_1, l_2=0, 1,\ldots;$ $i_1,\ldots,i_k=1,\ldots,m.$

Note that together with the iterated Stratonovich stochastic integrals
of higher multiplicities than the second, the stochastic integrals
(\ref{ll1}) and (\ref{ll2}) are included in the so-called
unified Taylor--Stratonovich expansion \cite{32}. 
This expansion can be used to construction of
strong (definition 
of strong numerical method see, for example, in \cite{KlPl2})
numerical methods for Ito SDEs.

                                                
Consider the expansions of some stochastic integrals 
(\ref{ll1}) and (\ref{ll2}),
obtained by using the theorem 6 and formula (\ref{jes})


$$
I_{(0)T,t}^{*(i_1)}=\sqrt{T-t}\zeta_0^{(i_1)},
$$

\vspace{1mm}

\begin{equation}
\label{4002}
I_{(1)T,t}^{*(i_1)}=-\frac{(T-t)^{3/2}}{2}\biggl(\zeta_0^{(i_1)}+
\frac{1}{\sqrt{3}}\zeta_1^{(i_1)}\biggr),
\end{equation}

\vspace{2mm}


\begin{equation}
\label{4003}
I_{(2)T,t}^{*(i_1)}=\frac{(T-t)^{5/2}}{3}\biggl(\zeta_0^{(i_1)}+
\frac{\sqrt{3}}{2}\zeta_1^{(i_1)}+
\frac{1}{2\sqrt{5}}\zeta_2^{(i_1)}\biggr),
\end{equation}

\vspace{2mm}

\begin{equation}
\label{4004}
I_{(00)T,t}^{*(i_1 i_2)}=
\frac{T-t}{2}\biggl(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{i=1}^{\infty}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right)\biggr),
\end{equation}

\vspace{6mm}


$$
I_{(01)T,t}^{*(i_1 i_2)}=-\frac{T-t}{2}I_{(00)T,t}^{*(i_1 i_2)}
-\frac{(T-t)^2}{4}\biggl(
\frac{1}{\sqrt{3}}\zeta_0^{(i_1)}\zeta_1^{(i_2)}+\biggr.
$$

\vspace{2mm}
$$
+\biggl.\sum_{i=0}^{\infty}\biggl(
\frac{(i+2)\zeta_i^{(i_1)}\zeta_{i+2}^{(i_2)}
-(i+1)\zeta_{i+2}^{(i_1)}\zeta_{i}^{(i_2)}}
{\sqrt{(2i+1)(2i+5)}(2i+3)}-
\frac{\zeta_i^{(i_1)}\zeta_{i}^{(i_2)}}{(2i-1)(2i+3)}\biggr)\biggr),
$$

\vspace{8mm}

$$
I_{(10)T,t}^{*(i_1 i_2)}=-\frac{T-t}{2}I_{(00)T,t}^{*(i_1 i_2)}
-\frac{(T-t)^2}{4}\biggl(
\frac{1}{\sqrt{3}}\zeta_0^{(i_2)}\zeta_1^{(i_1)}+\biggr.
$$

\vspace{2mm}
\begin{equation}
\label{4006}
+\biggl.\sum_{i=0}^{\infty}\biggl(
\frac{(i+1)\zeta_{i+2}^{(i_2)}\zeta_{i}^{(i_1)}
-(i+2)\zeta_{i}^{(i_2)}\zeta_{i+2}^{(i_1)}}
{\sqrt{(2i+1)(2i+5)}(2i+3)}+
\frac{\zeta_i^{(i_1)}\zeta_{i}^{(i_2)}}{(2i-1)(2i+3)}\biggr)\biggr),
\end{equation}


\vspace{8mm}


$$
I_{(02)T,t}^{*(i_1 i_2)}=-\frac{(T-t)^2}{4}I_{(00)T,t}^{*(i_1 i_2)}
-(T-t)I_{(01)T,t}^{*(i_1 i_2)}+
\frac{(T-t)^3}{8}\biggl(
\frac{2}{3\sqrt{5}}\zeta_2^{(i_2)}\zeta_0^{(i_1)}+\biggr.
$$

\vspace{2mm}
$$
+\frac{1}{3}\zeta_0^{(i_1)}\zeta_0^{(i_2)}+
\sum_{i=0}^{\infty}\biggl(
\frac{(i+2)(i+3)\zeta_{i+3}^{(i_2)}\zeta_{i}^{(i_1)}
-(i+1)(i+2)\zeta_{i}^{(i_2)}\zeta_{i+3}^{(i_1)}}
{\sqrt{(2i+1)(2i+7)}(2i+3)(2i+5)}+
\biggr.
$$

\vspace{2mm}
$$
\biggl.\biggl.+
\frac{(i^2+i-3)\zeta_{i+1}^{(i_2)}\zeta_{i}^{(i_1)}
-(i^2+3i-1)\zeta_{i}^{(i_2)}\zeta_{i+1}^{(i_1)}}
{\sqrt{(2i+1)(2i+3)}(2i-1)(2i+5)}\biggr)\biggr),
$$

\vspace{8mm}


$$
I_{(20)T,t}^{*(i_1 i_2)}=-\frac{(T-t)^2}{4}I_{(00)T,t}^{*(i_1 i_2)}
-(T-t)I_{(10)T,t}^{*(i_1 i_2)}+
\frac{(T-t)^3}{8}\biggl(
\frac{2}{3\sqrt{5}}\zeta_0^{(i_2)}\zeta_2^{(i_1)}+\biggr.
$$

\vspace{2mm}
$$
+\frac{1}{3}\zeta_0^{(i_1)}\zeta_0^{(i_2)}+
\sum_{i=0}^{\infty}\biggl(
\frac{(i+1)(i+2)\zeta_{i+3}^{(i_2)}\zeta_{i}^{(i_1)}
-(i+2)(i+3)\zeta_{i}^{(i_2)}\zeta_{i+3}^{(i_1)}}
{\sqrt{(2i+1)(2i+7)}(2i+3)(2i+5)}+
\biggr.
$$

\vspace{2mm}
$$
\biggl.\biggl.+
\frac{(i^2+3i-1)\zeta_{i+1}^{(i_2)}\zeta_{i}^{(i_1)}
-(i^2+i-3)\zeta_{i}^{(i_2)}\zeta_{i+1}^{(i_1)}}
{\sqrt{(2i+1)(2i+3)}(2i-1)(2i+5)}\biggr)\biggr),
$$

\vspace{8mm}



$$
I_{(11)T,t}^{*(i_1 i_2)}=-\frac{(T-t)^2}{4}I_{(00)T,t}^{*(i_1 i_2)}
-\frac{(T-t)}{2}\left(I_{(10)T,t}^{*(i_1 i_2)}+
I_{(01)T,t}^{*(i_1 i_2)}\right)+
$$

\vspace{2mm}
$$
+
\frac{(T-t)^3}{8}\biggl(
\frac{1}{3}\zeta_1^{(i_1)}\zeta_1^{(i_2)}
+
\sum_{i=0}^{\infty}\biggl(
\frac{(i+1)(i+3)\left(\zeta_{i+3}^{(i_2)}\zeta_{i}^{(i_1)}
-\zeta_{i}^{(i_2)}\zeta_{i+3}^{(i_1)}\right)}
{\sqrt{(2i+1)(2i+7)}(2i+3)(2i+5)}+
\biggr.
$$

\vspace{2mm}
$$
\biggl.\biggl.+
\frac{(i+1)^2\left(\zeta_{i+1}^{(i_2)}\zeta_{i}^{(i_1)}
-\zeta_{i}^{(i_2)}\zeta_{i+1}^{(i_1)}\right)}
{\sqrt{(2i+1)(2i+3)}(2i-1)(2i+5)}\biggr)\biggr),
$$

\vspace{7mm}

$$
I_{(3)T,t}^{*(i_1)}=-\frac{(T-t)^{7/2}}{4}\biggl(\zeta_0^{(i_1)}+
\frac{3\sqrt{3}}{5}\zeta_1^{(i_1)}+
\frac{1}{\sqrt{5}}\zeta_2^{(i_1)}+
\frac{1}{5\sqrt{7}}\zeta_3^{(i_1)}\biggr),
$$

\vspace{6mm}
\noindent
where every
\begin{equation}
\label{zq1}
\zeta_{j}^{(i)}=
\int\limits_t^T \phi_{j}(s) d{\bf f}_s^{(i)}\ \ (i=1,\ldots,m)
\end{equation} 

\vspace{2mm}
\noindent
is a standard Gaussian random variable
for 
different
$i$ or $j$. 


Note the simplicity of the formulas (\ref{4002}), (\ref{4003}).
For comparison, we present analogs of the formulas 
(\ref{4002}), (\ref{4003}), 
obtained in \cite{KPW} (see also \cite{KlPl2}) using the method, proposed
in \cite{Mi2}


\vspace{-1mm}
\begin{equation}
\label{42}
I_{(1)T,t}^{(i_1)q}=-\frac{{(T-t)}^{3/2}}{2}
\biggl(\zeta_0^{(i_1)}-\frac{\sqrt{2}}{\pi}\biggl(\sum_{r=1}^{q}
\frac{1}{r}
\zeta_{2r-1}^{(i_1)}+\sqrt{\alpha_q}\xi_q^{(i_1)}\biggr)
\biggr),
\end{equation}

\vspace{1mm}
\begin{equation}
\label{46}
I_{(2)T,t}^{(i_1)q}=
(T-t)^{5/2}\biggl(
\frac{1}{3}\zeta_0^{(i_1)}+\frac{1}{\sqrt{2}\pi^2}
\biggl(\sum_{r=1}^{q}\frac{1}{r^2}\zeta_{2r}^{(i_1)}+
\sqrt{\beta_q}
\mu_q^{(i_1)}\biggr)-\biggr.
$$
$$
\biggl.-\frac{1}{\sqrt{2}\pi}\biggl(\sum_{r=1}^q
\frac{1}{r}\zeta_{2r-1}^{(i_1)}+
\sqrt{\alpha_q}\xi_q^{(i_1)}\biggr)\biggr),
\end{equation}

\vspace{2mm}
\noindent
where $\zeta_j^{(i)}$ is defined by formula (\ref{zq1}),
$\phi_j(s)$ is a complete orthonormal system of trigonometric
functions in the space
$L_2([t, T]),$ and
$\zeta_0^{(i)},$ $\zeta_{2r}^{(i)},$
$\zeta_{2r-1}^{(i)},$ $\xi_q^{(i)},$ $\mu_q^{(i)}$ $(r=1,\ldots,q;$
$i=1,\ldots,m)$ are independent 
standard Gaussian random variables, $i_1=1,\ldots,m,$ 

\vspace{-1mm}
$$
\xi_q^{(i)}=\frac{1}{\sqrt{\alpha_q}}\sum_{r=q+1}^{\infty}
\frac{1}{r}\zeta_{2r-1}^{(i)},\ \ \
\alpha_q=\frac{\pi^2}{6}-\sum_{r=1}^q\frac{1}{r^2},
$$

$$
\mu_q^{(i)}=\frac{1}{\sqrt{\beta_q}}\sum_{r=q+1}^{\infty}
\frac{1}{r^2}~\zeta_{2r}^{(i)},\ \ \ 
\beta_q=\frac{\pi^4}{90}-\sum_{r=1}^q\frac{1}{r^4}.
$$

\vspace{3mm}

Another example of obvious advantage of the Legendre polynomials 
over the trigonometric functions (in the framework of the considered 
problem) is the truncated expansion 
of the iterated Stratonovich stochastic integral
$I_{(10)T, t}^{*(i_1 i_2)}$, obtained by theorem 6, in which
instead of the double Fourier--Legendre series is taken 
the double trigonometric Fourier series (which is acceptable 
according to the remark 1)

\vspace{-2mm}
$$
I_{(10)T,t}^{*(i_1 i_2)q}=-(T-t)^{2}\Biggl(\frac{1}{6}
\zeta_{0}^{(i_1)}\zeta_{0}^{(i_2)}-\frac{1}{2\sqrt{2}\pi}
\sqrt{\alpha_q}\xi_q^{(i_2)}\zeta_0^{(i_1)}+\Biggr.
$$

$$
+\frac{1}{2\sqrt{2}\pi^2}\sqrt{\beta_q}\Biggl(
\mu_q^{(i_2)}\zeta_0^{(i_1)}-2\mu_q^{(i_1)}\zeta_0^{(i_2)}\Biggr)+
$$

$$
+\frac{1}{2\sqrt{2}}\sum_{r=1}^{q}
\Biggl(-\frac{1}{\pi r}
\zeta_{2r-1}^{(i_2)}
\zeta_{0}^{(i_1)}+
\frac{1}{\pi^2 r^2}\left(
\zeta_{2r}^{(i_2)}
\zeta_{0}^{(i_1)}-
2\zeta_{2r}^{(i_1)}
\zeta_{0}^{(i_2)}\right)\Biggr)-
$$

$$
-
\frac{1}{2\pi^2}\sum_{\stackrel{r,l=1}{{}_{r\ne l}}}^{q}
\frac{1}{r^2-l^2}\Biggl(
\zeta_{2r}^{(i_1)}
\zeta_{2l}^{(i_2)}+
\frac{l}{r}
\zeta_{2r-1}^{(i_1)}
\zeta_{2l-1}^{(i_2)}
\Biggr)+
$$

$$
+
\sum_{r=1}^{q}
\Biggl(\frac{1}{4\pi r}\left(
\zeta_{2r}^{(i_1)}
\zeta_{2r-1}^{(i_2)}-
\zeta_{2r-1}^{(i_1)}
\zeta_{2r}^{(i_2)}\right)
$$

\begin{equation}
\label{944}
+
\Biggl.\Biggl.
\frac{1}{8\pi^2 r^2}\left(
3\zeta_{2r-1}^{(i_1)}
\zeta_{2r-1}^{(i_2)}+
\zeta_{2r}^{(i_2)}
\zeta_{2r}^{(i_1)}\right)\Biggr)\Biggr),
\end{equation}

\vspace{5mm}
\noindent
where the meaning of the notations included 
in (\ref{42}), (\ref{46}) is preserved.



An analogue of the formula (\ref{944}) (for the case of Legendre polynomials)
is (according to (\ref{4004}) and (\ref{4006})) the following representation

$$
I_{(10)T,t}^{*(i_1 i_2)q}=-\frac{T-t}{2}I_{(00)T,t}^{*(i_1 i_2)q}
-\frac{(T-t)^2}{4}\biggl(
\frac{1}{\sqrt{3}}\zeta_0^{(i_2)}\zeta_1^{(i_1)}+\biggr.
$$

\vspace{1mm}
\begin{equation}
\label{5500}
+\biggl.\sum_{i=0}^{q}\biggl(
\frac{(i+1)\zeta_{i+2}^{(i_2)}\zeta_{i}^{(i_1)}
-(i+2)\zeta_{i}^{(i_2)}\zeta_{i+2}^{(i_1)}}
{\sqrt{(2i+1)(2i+5)}(2i+3)}+
\frac{\zeta_i^{(i_1)}\zeta_{i}^{(i_2)}}{(2i-1)(2i+3)}\biggr)\biggr),
\end{equation}

\vspace{4mm}
\noindent
where

\vspace{-2mm}
$$
I_{(00)T,t}^{*(i_1 i_2)q}=
\frac{T-t}{2}\biggl(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{i=1}^{q}
\frac{1}{\sqrt{4i^2-1}}\left(
\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-
\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right)\biggr),
$$

\vspace{4mm}
\noindent
which is obviously substantially simpler than (\ref{944}).

Here it is necessary to pay special attention on the fact
that the representation (\ref{5500}) includes a single sum with 
an upper summation limit $q,$
while the representation (\ref{944}) includes a double sum 
with the same summation limit. In numerical simulation, 
obviously, the formula (\ref{5500}) is more economical 
in terms of computational cost, than its analogue (\ref{944}).


There is another feature that should be noted
in connection with the formula (\ref{944}). This formula was first
obtained in \cite{KPW} by the method from \cite{Mi2}.
However, due to its peculiarities (see the section 1)
the method from \cite{Mi2} leads to the iterated application of the operation
of the limit transition (in contrast to the theorems 1, 2, and 6, in 
which limit transition is performed only once)
in the implementation of expansions
of the iterated stochastic integrals.
This means that from the method \cite{Mi2}, generally speaking,
does not follow the mean-square convergence of 
$I_{(10)T,T}^{*(i_1 i_2)q}$
of the form (\ref{944}) to the $I_{(10)T,T}^{*(i_1 i_2)}$ when $q\to\infty.$
The same applies to some others 
approximations of iterated Stratonovich stochastic integrals,
obtained in \cite{KPW} by the method from \cite{Mi2}.


The validity of the formula

\vspace{-1mm}
$$
\lim\limits_{q\to\infty}
{\sf M}\left\{\left(I_{(10)T,t}^{*(i_1 i_2)}-I_{(10)T,t}^{*(i_1 i_2)q}
\right)^2\right\}=0,
$$

\vspace{2mm}
\noindent
where $I_{(10)T,T}^{*(i_1 i_2)q}$ is defined by the formula (\ref{944}),
follows from the theorems 2, 6.



\vspace{5mm}


