292
K. A. Rybakov
Dirac delta function [25]. In fact, the spectral characteristic definition is formally
extended to functions, for which the expansion coefficients can be calculated using
Eq. 20.5. It defines the linear functional on the set of spectral characteristics of
functions and P
−1
= 1, P1 = (if an additional condition for the pointwise
convergence is fulfilled).
Moreover, it is possible to apply the spectral form of mathematical description not
only for deterministic functions, but also for random processes. If f (t) is a random
process satisfying the condition E
h
0
f
2
(t)dt < ∞, where E is the expectation, then
f (t) is determined by random expansion coefficients, which are also represented
as the infinite column matrix, and this matrix is called the spectral characteristic of
the random process f (t) defined with respect to the orthonormal basis {q(i, t)}
∞
i=0 .
The spectral characteristic definition can be extended to a class of random processes,
for which Eq. 20.5 is applicable. Thus, the spectral characteristic V of standard
Gaussian white noise v(t) is an infinite column matrix, whose entries are independent
random variables having a standard normal distribution. It is related to the spectral
characteristic W of the standard Wiener random process w(t) as P
−1
V = W, PW =
V [30, 31]. Note that the spectral characteristic V defines the random linear functional
on the set of spectral characteristics of functions.
In addition, we should indicate one more property of the spectral transform:
(4) The spectral transform preserves the norm and the inner product, i.e.,
h
0
x
2
(t)dt = X
T X,
h
0
x(t)y(t)dt = X
T Y.
As an example of using the spectral form of mathematical description, we will
represent the spectral characteristic of the function
x k (t) =
t
0
g k (τ k ) . . .
τ 3
0
g 2 (τ 2 )
τ 2
0
g 1 (τ 1 )dτ 1 dτ 2 . . . dτ k
(20.7)
by spectral characteristics of functions g l (t), l = 1, 2, . . . , k.
Consider a System of Ordinary Differential Equations (SODE):
˙
x 1 (t) = g 1 (t), ˙
x 2 (t) = g 2 (t)x 1 (t), . . . , ˙
x k (t) = g k (t)x k−1 (t),
x 1 (0) = x 2 (0) = · · · = x k (0) = 0.
(20.8)
The spectral form of mathematical description was proposed for the dynamical
systems analysis. In this context, SODE (Eq. 20.8) can be considered as a mathematical model of the dynamical system, for which functions g l (t) and x l (t) are input
and output signals (l = 1, 2, . . . , k), respectively.
K. A. Rybakov
Dirac delta function [25]. In fact, the spectral characteristic definition is formally
extended to functions, for which the expansion coefficients can be calculated using
Eq. 20.5. It defines the linear functional on the set of spectral characteristics of
functions and P
−1
= 1, P1 = (if an additional condition for the pointwise
convergence is fulfilled).
Moreover, it is possible to apply the spectral form of mathematical description not
only for deterministic functions, but also for random processes. If f (t) is a random
process satisfying the condition E
h
0
f
2
(t)dt < ∞, where E is the expectation, then
f (t) is determined by random expansion coefficients, which are also represented
as the infinite column matrix, and this matrix is called the spectral characteristic of
the random process f (t) defined with respect to the orthonormal basis {q(i, t)}
∞
i=0 .
The spectral characteristic definition can be extended to a class of random processes,
for which Eq. 20.5 is applicable. Thus, the spectral characteristic V of standard
Gaussian white noise v(t) is an infinite column matrix, whose entries are independent
random variables having a standard normal distribution. It is related to the spectral
characteristic W of the standard Wiener random process w(t) as P
−1
V = W, PW =
V [30, 31]. Note that the spectral characteristic V defines the random linear functional
on the set of spectral characteristics of functions.
In addition, we should indicate one more property of the spectral transform:
(4) The spectral transform preserves the norm and the inner product, i.e.,
h
0
x
2
(t)dt = X
T X,
h
0
x(t)y(t)dt = X
T Y.
As an example of using the spectral form of mathematical description, we will
represent the spectral characteristic of the function
x k (t) =
t
0
g k (τ k ) . . .
τ 3
0
g 2 (τ 2 )
τ 2
0
g 1 (τ 1 )dτ 1 dτ 2 . . . dτ k
(20.7)
by spectral characteristics of functions g l (t), l = 1, 2, . . . , k.
Consider a System of Ordinary Differential Equations (SODE):
˙
x 1 (t) = g 1 (t), ˙
x 2 (t) = g 2 (t)x 1 (t), . . . , ˙
x k (t) = g k (t)x k−1 (t),
x 1 (0) = x 2 (0) = · · · = x k (0) = 0.
(20.8)
The spectral form of mathematical description was proposed for the dynamical
systems analysis. In this context, SODE (Eq. 20.8) can be considered as a mathematical model of the dynamical system, for which functions g l (t) and x l (t) are input
and output signals (l = 1, 2, . . . , k), respectively.