diverges to infinity as
N
→ ∞
[10], and the right side of (16) tends to
zero exponentially as claimed in the Theorem. The unconditionality of
convergence of (13) will be provided by Proposition 4.
In the above discussion we did not touch upon the properties of the
set of eigenfunctions. We will see below, it is quite similar to the case of
symmetrical integral operators, the set of all eigenfuctions generated by
(12) constitutes an orthonormal system in
L
2
(Ω)
.
Proposition 4.
For arbitrarily given
F
(
x
)
∈
L
2
(Ω)
,
F
= 0
, the set
of all eigenfunctions
α
ϕ
α
β
, β
= 1
,
2
, . . .
of the sequence of gradient
operators
Φ
β
= (Φ
β
1
, . . . ,
Φ
βn
)
, defined by (12), constitutes an orthonormal
system as an ingredient part of some complete orthonormal basis of
L
2
(Ω)
.
Proof.
By definition of
F
N
, the identities
F
β
,
α
ϕ
α
β
= 0
hold
for all
β
= 1
,
2
, . . . , N
. Each
ϕ
α
N
+1
can be decomposed uniquely
as
ϕ
α
N
+1
=
a
α
ϕ
α
N
+
b
α
¯
ϕ
α
N
+1
,
ϕ
α
N
⊥
¯
ϕ
α
N
+1
, and
aα
,
bα
be constants of
normalization. A substitution for
ϕ
α
N
+1
yields
α
ϕ
α
N
+1
=
=
C
0
α
ϕ
α
N
+
C
1
n
i
=1
¯
ϕ
i
N
+1
α
=
i
ϕ
α
N
+
· · ·
+
C
n
α
¯
ϕ
α
N
+1
=
C
0
α
ϕ
α
N
+
P
N
+1
.
Clearly,
P
N
+1
⊥
α
ϕ
α
N
. Thus, due to the identities said above,
λ
N
+1
=
F
N
,
α
ϕ
α
N
+1
=
C
0
F
N
,
α
ϕ
α
N
+
F
N
, P
N
+1
=
=
F
N
−
1
, P
N
+1
. It follows
α
ϕ
α
N
⊥
α
ϕ
α
N
+1
. Similar analysis of
ϕ
α
β
for
β
=
N
−
1
, . . . ,
1
in succession, one obtains
λ
N
+1
= sup
ψ
α
∈
B
α
Ω
F
N
(
x
)
ψ
α
d
Ω =
= sup
ψ
α
∈
B
α
ψ
α
∈
(
L
(
α
)
N
)
⊥
Ω
F
0
(
x
)
α
ψ
α
d
Ω =
Ω
F
0
(
x
)
α
ϕ
α
N
+1
d
Ω
.
(17)
This being true for all
N
∈
N
follows the set of eigenfunctions
α
ϕ
α
β
, β
= 1
,
2
, . . .
constitutes a orthonormal system in
L
2
(Ω)
. Since
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
43
