Theorem 3.
Any given
F
(
x
)
∈
L
2
(Ω)
,
F
=
0, can be expressed
in the form of series (13) with all positive eignevalues and associated
eignefunctions generated by the sequence of gradient operators
{
Φ
β
}
. The
series (13) converges exponentially to
F
(
x
)
in the norm of
L
2
(Ω)
.
Proof.
If the process of construction described in (10) terminates at a
finite step
N
, the validity of the first part of Theorem is apparent. Now
suppose that (10) becomes an infinite series (13) when
N
→ ∞
. It is
obvious from (10), however large is
N
, one always has
F
N
2
≥
0
, and
N
β
=1
λ
2
β
F
0
2
.
(14)
The necessary condition of convergence for the series on the left side is
λ
N
→
0 as
N
→ ∞
, and (6) implies that the following relationship holds
uniformly on
B
n
,
λ
N
+1
= sup
ψ
α
∈
B
α
Ω
F
N
(
x
)
α
ψ
α
(
x
α
)
d
Ω =
F
N
(
x
)
,
α
ϕ
α
N
+1
L
2
(Ω)
≥
≥
F
N
(
x
)
,
α
ψ
α
L
2
(Ω)
,
∀
ψ
α
∈
B
α
,
here
α
ϕ
α
N
+1
is a solution of (6) and (8) with
F
0
(
x
)
replaced by
F
N
(
x
)
.
Therefore, when
N
→ ∞
,
lim
N
→∞
F
N
(
x
)
,
α
ψ
α
L
2
(Ω)
→
0
,
∀
ψ
α
∈
B
α
, α
= 1
,
2
,
· · ·
, n.
It is evident that
B
n
is a fundamental set, i.e., it spans
L
n
2
(0
,
1)
, is a
dense subset of
L
2
(Ω)
. The above condition suffices for
F
N
to converge
weakly-star to some element
F
∞
which is equivalent to 0 in the weak
topology [4],
lim
N
→∞
sup
B
n
Ω
F
N
(
x
)
α
ψ
α
(
x
α
)
d
Ω = sup
B
n
F
∞
,
α
ψ
α
L
2
(Ω)
= 0
.
Recall the fact that the set
E
=
α
e
γ
α
α
, γ
α
∈
N
consisting of
combinations of orthonormal bases of
L
(
α
)
2
(0
,
1)
is a complete orthonormal
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
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