300
K. A. Rybakov
M
∗
k−1,k−1 = I
∗( j k j k−1 )
h
I
∗( j k−2 )
h
. . . I
∗( j 1 )
h
,
M
∗
k1 = I
∗( j k )
h
. . . I
∗( j 2 )
h
I
∗( j 1 )
h
,
and the ordered set of indices j k . . . j r +1 . . . j q . . . j p+1 j p . . . j 1 (1 p < q r < k)
coincides with j k . . . j 2 j 1 , i.e., the summation over m is the summation over all
possible partitions of the set of indices j k . . . j 2 j 1 into l subsets with saving their
order.
For complete orthonormal systems such as the Legendre polynomials, trigonometric functions, the Walsh functions, and the Haar functions with the standard
numeration, the function 1(t) differs from the basic function q(0, t) by the numerical
coefficient, i.e., 1(t) =
√
h q(0, t), and for h = 1 they are equal. Therefore,
1 = [
√
h 0 0 . . . ]
T
,
i.e., C i 1 =
√
h for i 1 = 0 and C i 1 = 0 for i 1 > 0. This simplifies above relations for
expansion coefficients:
C i 2 i 1 = −C i 1 i 2 , C i 1 i 1 = 0, i 1 , i 2 > 0;
C i 3 i 2 i 1 = C i 1 i 2 i 3 , i 1 , i 3 > 0;
C i 4 i 3 i 2 i 1 = −C i 1 i 2 i 3 i 4 + C i 1 i 2 C i 3 i 4 , C i 4 i 3 i 1 i 1 = −C i 1 i 1 i 3 i 4 ,
C i 4 i 4 i 2 i 1 = −C i 1 i 2 i 4 i 4 , i 1 , i 4 > 0;
C i 5 i 4 i 3 i 2 i 1 = C i 1 i 2 i 3 i 4 i 5 − C i 1 i 2 i 3 C i 4 i 5 − C i 1 i 2 C i 3 i 4 i 5 ,
C i 5 i 5 i 3 i 1 i 1 = C i 1 i 1 i 3 i 5 i 5 , i 1 , i 3 , i 5 > 0,
and so on.
In the general case M lm = 0 if k/l = 1 and i 1 , i 2 , . . . , i k > 0, where · is the
floor function.
Obtained invariant relations can reduce computational costs for the calculation of
expansion coefficients and for modeling the iterated Stratonovich stochastic integrals.
In fact, if the coefficient C i 1 i 2 ...i k has been calculated, then the coefficient C i k ...i 2 i 1
has also been calculated. Similarly, if the iterated Stratonovich stochastic integral
I
∗( j k ... j 2 j 1 )
h
has been modeled, then we can model the iterated Stratonovich stochastic
integral I
∗( j 1 j 2 ... j k )
h
by I
∗( j k ... j 2 j 1 )
h
and integrals of multiplicity less then k.
20.6 Tensor Representation
Definitions and properties of the spectral transform listed in Sect. 20.3 can be
extended to functions of several variables. We will use some notations from [26,
28] for spectral characteristics of functions of several variables. Then
K. A. Rybakov
M
∗
k−1,k−1 = I
∗( j k j k−1 )
h
I
∗( j k−2 )
h
. . . I
∗( j 1 )
h
,
M
∗
k1 = I
∗( j k )
h
. . . I
∗( j 2 )
h
I
∗( j 1 )
h
,
and the ordered set of indices j k . . . j r +1 . . . j q . . . j p+1 j p . . . j 1 (1 p < q r < k)
coincides with j k . . . j 2 j 1 , i.e., the summation over m is the summation over all
possible partitions of the set of indices j k . . . j 2 j 1 into l subsets with saving their
order.
For complete orthonormal systems such as the Legendre polynomials, trigonometric functions, the Walsh functions, and the Haar functions with the standard
numeration, the function 1(t) differs from the basic function q(0, t) by the numerical
coefficient, i.e., 1(t) =
√
h q(0, t), and for h = 1 they are equal. Therefore,
1 = [
√
h 0 0 . . . ]
T
,
i.e., C i 1 =
√
h for i 1 = 0 and C i 1 = 0 for i 1 > 0. This simplifies above relations for
expansion coefficients:
C i 2 i 1 = −C i 1 i 2 , C i 1 i 1 = 0, i 1 , i 2 > 0;
C i 3 i 2 i 1 = C i 1 i 2 i 3 , i 1 , i 3 > 0;
C i 4 i 3 i 2 i 1 = −C i 1 i 2 i 3 i 4 + C i 1 i 2 C i 3 i 4 , C i 4 i 3 i 1 i 1 = −C i 1 i 1 i 3 i 4 ,
C i 4 i 4 i 2 i 1 = −C i 1 i 2 i 4 i 4 , i 1 , i 4 > 0;
C i 5 i 4 i 3 i 2 i 1 = C i 1 i 2 i 3 i 4 i 5 − C i 1 i 2 i 3 C i 4 i 5 − C i 1 i 2 C i 3 i 4 i 5 ,
C i 5 i 5 i 3 i 1 i 1 = C i 1 i 1 i 3 i 5 i 5 , i 1 , i 3 , i 5 > 0,
and so on.
In the general case M lm = 0 if k/l = 1 and i 1 , i 2 , . . . , i k > 0, where · is the
floor function.
Obtained invariant relations can reduce computational costs for the calculation of
expansion coefficients and for modeling the iterated Stratonovich stochastic integrals.
In fact, if the coefficient C i 1 i 2 ...i k has been calculated, then the coefficient C i k ...i 2 i 1
has also been calculated. Similarly, if the iterated Stratonovich stochastic integral
I
∗( j k ... j 2 j 1 )
h
has been modeled, then we can model the iterated Stratonovich stochastic
integral I
∗( j 1 j 2 ... j k )
h
by I
∗( j k ... j 2 j 1 )
h
and integrals of multiplicity less then k.
20.6 Tensor Representation
Definitions and properties of the spectral transform listed in Sect. 20.3 can be
extended to functions of several variables. We will use some notations from [26,
28] for spectral characteristics of functions of several variables. Then