20 Using Spectral Form of Mathematical Description …
291
f (t) defined with respect to the orthonormal basis {q(i, t)}
∞
i=0 , S is the spectral
transform and S
−1 is the spectral inversion.
Further, we will use some properties of the spectral transform [25]. Let
x(t), y(t), z(t) ∈ L 2 ([0, h]) and X = S[x(t)], Y = S[y(t)], Z = S[z(t)], then
we have:
(1) If y(t) = ˙
x(t), x(0) = 0, then Y = P X, where P is the infinite matrix with
entries
P i 1 i 2 =
h
0
q(i 1 , t) ˙
q(i 2 , t)dt + q(i 1 , 0)q(i 2 , 0), i 1 , i 2 = 0, 1, 2, . . .
(2) If y(t) =
t
0
x(τ )dτ (t h), then Y = P
−1 X , where P
−1 is the infinite matrix
with entries
P
−1
i 1 i 2
=
h
0
q(i 1 , t)
t
0
q(i 2 , τ )dτ dt, i 1 , i 2 = 0, 1, 2, . . .
(3) If z(t) = x(t)y(t), then Z = (V X)Y , where V is the three-dimensional infinite
matrix with entries
V i 1 i 2 i 3 =
h
0
q(i 1 , t)q(i 2 , t)q(i 3 , t)dt, i 1 , i 2 , i 3 = 0, 1, 2, . . .
The matrices P, P
−1 , and V are called the spectral characteristic of the differentiation operator taking into account the initial condition, the spectral characteristic of
the integration operator, and the spectral characteristic of the multiplier, respectively.
It should be emphasized that P P
−1
= P
−1 P = E, where E is the infinite identity
matrix, and V is the symmetric three-dimensional infinite matrix with respect to
any pair of indices from the triple (i 1 , i 2 , i 3 ), i.e., any section of V is the infinite
symmetric matrix.
It is important to note that the condition x(t), y(t), z(t) ∈ L 2 ([0, h]) is only
sufficient to determine spectral characteristics of these functions and to fulfill above
properties. In some cases, this condition can be weakened, e.g., we can indicate the
relationship between the spectral characteristic 1 of the unit step function 1(t) and
the infinite column matrix , where entries of are values of the orthonormal basis
{q(i, t)}
∞
i=0 at t = 0. The column matrix is called the spectral characteristic of
291
f (t) defined with respect to the orthonormal basis {q(i, t)}
∞
i=0 , S is the spectral
transform and S
−1 is the spectral inversion.
Further, we will use some properties of the spectral transform [25]. Let
x(t), y(t), z(t) ∈ L 2 ([0, h]) and X = S[x(t)], Y = S[y(t)], Z = S[z(t)], then
we have:
(1) If y(t) = ˙
x(t), x(0) = 0, then Y = P X, where P is the infinite matrix with
entries
P i 1 i 2 =
h
0
q(i 1 , t) ˙
q(i 2 , t)dt + q(i 1 , 0)q(i 2 , 0), i 1 , i 2 = 0, 1, 2, . . .
(2) If y(t) =
t
0
x(τ )dτ (t h), then Y = P
−1 X , where P
−1 is the infinite matrix
with entries
P
−1
i 1 i 2
=
h
0
q(i 1 , t)
t
0
q(i 2 , τ )dτ dt, i 1 , i 2 = 0, 1, 2, . . .
(3) If z(t) = x(t)y(t), then Z = (V X)Y , where V is the three-dimensional infinite
matrix with entries
V i 1 i 2 i 3 =
h
0
q(i 1 , t)q(i 2 , t)q(i 3 , t)dt, i 1 , i 2 , i 3 = 0, 1, 2, . . .
The matrices P, P
−1 , and V are called the spectral characteristic of the differentiation operator taking into account the initial condition, the spectral characteristic of
the integration operator, and the spectral characteristic of the multiplier, respectively.
It should be emphasized that P P
−1
= P
−1 P = E, where E is the infinite identity
matrix, and V is the symmetric three-dimensional infinite matrix with respect to
any pair of indices from the triple (i 1 , i 2 , i 3 ), i.e., any section of V is the infinite
symmetric matrix.
It is important to note that the condition x(t), y(t), z(t) ∈ L 2 ([0, h]) is only
sufficient to determine spectral characteristics of these functions and to fulfill above
properties. In some cases, this condition can be weakened, e.g., we can indicate the
relationship between the spectral characteristic 1 of the unit step function 1(t) and
the infinite column matrix , where entries of are values of the orthonormal basis
{q(i, t)}
∞
i=0 at t = 0. The column matrix is called the spectral characteristic of