this paper is to assert that for any given
F
there exists an optimal basis
in
L
2
(Ω)
which furnishes the element with an image sharply concentrated
on a few of spectrum-lines in
l
2
, if the latter is equipped with canonical
basis. This may be in marked contrast with a flat spread of spectral lines
with respect to a casually chosen basis for spectral analysis as it happens
in many cases of practices.
The results presented in this paper may find wide applications in
computational mathematics and engineering sciences, particularly in the
field of control theory and automation [20]. Take a typical example, if a
hypersurface or mainfold in
R
n
,
x
n
=
F
(
x
1
, x
2
, . . . , x
n
−
1
)
, is needed to
be stored, the amount of data is measured as
N
n
−
1
+
N, N
is the mean
number of discrete samplings for each variable. If
l
terms are taken in
(13) to represent
F
, the amount of data to be stored or processed will be
reduced to
nlN
, a
1
/N
n
−
2
times less than previously needed. Engineering
practice had shown, sometimes to take two to three terms of (13) would
be precise enough to represent a given higher dimensional function by the
sum of products of one-variable functions [20].
The problems we investigated in this paper are related to a topic posed
and studied by Liapunov A.M. at the beginning of 20
th
century, he called it
power series integral equations and imposed severe restriction on the given
function. He required
F
(
x
1
, x
2
, . . . , x
n
)
to be totally symmetric, that means
the exchange of any two among
n
variables retains
F
unchanged [21,22].
The general properties of
n
-multilinear forms have been elucidated in [5,
6, 7]. Krasnoselsky M.A. proved that if
F
is strictly positive and totally
symmetric,
F
∈
L
2
(Ω)
,
0
< m
≤
F
≤
M <
∞
, the following integral
equation
Φ
n
i
=1
ϕ
(
t
i
) =
=
Ω
F
(
s, t
1
, t
2
,
· · ·
t
n
ϕ
(
t
1
)
ϕ
(
t
2
)
· · ·
ϕ
(
t
n
)
dt
1
dt
2
· · ·
dt
n
=
λϕ
(
s
)
possesses at least one positive eigenvalue [22]. Wainberg M.M. had
shown that for a totally symmetric
F
∈
L
2
(Ω)
all components of the
gradient operator
Φ
generated by
F
are compact, and the functional
Φ
n
i
=1
ϕ
(
t
i
)
, ϕ
(
s
)
is weak continuous respect to
ϕ
, it achieves its
supremum value on the unit ball [21]. It is obvious, the results we obtained
in this paper cover most cases studied by these earlier investigators.
The author expresses his gratitude to professors Lin Qun, Guo Lei,
Qin Hua-Shu, Guo Bao-Zhu and Cheng Dai-Zhan of the Institute of
48
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
