It contradicts the assumption and implies
λ
1
is not supremum of the
functional. One may notice, this is true if
F
(
x, y
)
replaced by
F
N
(
x, y
) =
=
F
(
x, y
)
−
N
β
=1
λ
β
ϕ
β
ψ
β
in (21). In particular, if
F
(
x, y
) =
F
(
y, x
)
, it
generates a self-adjoint operator,
Φ = Φ
∗
, the above conclusion is also true
for this special case of the fact described above.
From the geometrical point of view, it is known that the unit ball
B
of
L
2
(Ω)
is strictly and uniformly convex [14,15]. It is believed that the
B
∩
B
n
possesses the same property. The equations (18) create a supporting
tangent hyperplane to
B
n
in
L
2
(Ω)
. A conjecture arises that the claim made
in Corollary 5 would be true for any finite dimensional underlying spaces
R
n
. But we have had direct proof only for
n
even and
n
≤
6
. So the general
question remains still open.
One may wonder what is the condition to be imposed on
F
(
x
)
for
guaranteeing the convergence of the series (13) in space
L
1
(Ω)
and in the
Banach space of continuous functions
C
(Ω)
. The question arisen is that the
assumption (1) is not enough to ensure the convergence of the infinite sum
β
λ
β
except
F
(
x
)
generates a nuclear gradient operators [16]. However,
for our cases, according to the theories developed in [17, 18, 19], we can
establish the following Theorem. We list it with the proof omitted.
Theorem 6.
For any given function
F
(
x
)
∈
L
2
(Ω)
, the series (13)
converges uniformly in
L
1
(Ω)
. If
F
(
x
)
is continuous on
Ω
and possesses
all continuous first partial derivatives in
Ω
, then the series of expansion
(13) converges uniformly to the continuous function
F
(
x
)
.
It is worth to re-emphasize, Theorem 3 and Proposition 4 have shown
that for any high-dimensional square-integrable function
F
(
x
)
there exists
an optimal orthonormal system of its own, consisting of eigenfunctions
of its gradient operator, in terms of which
F
(
x
)
can be expanded with
shortest length and rapidest convergence. Since each element of the system
is a product of
n
single-variable functions, this may be a reliable way
for reduction of dimensionality and compact expression of information
contained in
F
(
x
)
in one-dimensional spaces. The inequality (16) provides
a posteriori
error estimate, in the process of computing the remaining error
can be precisely estimated after completion of each step of calculation, this
is thus a difference from
a priori
error estimate.
We recall that
L
2
(Ω)
and
l
2
, the space of square-summable sequences of
reals, are isometrically isomorphic. Each element of
L
2
(Ω)
has its spectral
image in
l
2
according to bases chosen in each spaces. If one identifies
the square of norm of
F
(
x
)
∈
L
2
(Ω)
with the energy or information it
carries, in terminology of physics, the outcome of Theorems presented in
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
47
