Due to the limitation of space we report here the primary Theorems
with abridged proofs. The detailed proofs will be contained in a separate
paper.
Let
F
(
x
) =
F
(
x
1
,
x
2
, . . . ,
x
n
)
be an arbitrarily given function defined
on the unit cube
Ω
in
R
n
,
F
∈
L
2
(Ω)
,
Ω
F
(
x
1
, x
2
, ..., x
n
)
2
dx
1
dx
2
...dx
n
M
2
<
∞
,
(1)
where
x
= (
x
1
,
x
2
, . . . ,
x
n
)
is a point in
R
n
. It is intended to find a set of one-
variable functions
ϕ
(
x
α
)
whose product
ϕ
1
(
x
1
)
ϕ
2
(
x
2
)
. . . ϕ
n
(
x
n
)
∈
L
2
(Ω)
would best, or optimally, approximate
F
(
x
)
with the least-square deviation:
L
=
Ω
F
(
x
1
, x
2
, ..., x
n
)
−
ϕ
1
(
x
1
)
ϕ
2
(
x
2
)
. . . ϕ
n
(
x
n
)
2
d
Ω = min
,
(2)
where
d
Ω =
dx
1
dx
2
. . . dx
n
.
Suppose each
ψ
α
(
x
α
)
is taken from the unit balls
B
α
∈
L
(
α
)
2
(0
,
1)
,
B
α
=
{
ψ
α
L
2
(0
,
1)
≤
1
}
,
α
= 1
,
2
, . . . , n
, the above requirement (2) can
be rewritten as
L
= inf
ψ
α
∈
B
α
Ω
(
F
(
x
)
−
λ
n
α
=1
ψ
α
(
x
α
))
2
d
Ω
.
(3)
Opening up the brackets on the right side we have
Ω
(
F
(
x
)
−
λ
n
α
=1
ψ
α
(
x
α
))
2
d
Ω =
=
Ω
(
F
2
(
x
)
−
2
λF
(
x
)
n
α
=1
ψ
α
(
x
α
) +
λ
2
n
α
=1
(
ψ
α
(
x
α
))
2
)
d
Ω
.
It is easy to verify that (3) holds if and only if there exists a product of
n
functions
α
ϕ
α
,
ϕ
α
∈
B
α
, and a real
λ
∈
R
,
λ
= 0
, which enable the
following functional to achieve supremum on all unit balls
B
α
,
λ
= sup
ψ
α
∈
B
α
Ω
F
(
x
)
n
α
=1
ψ
α
(
x
α
)
d
Ω =
=
Ω
F
(
x
1
, x
2
, . . . , x
n
)
ϕ
1
(
x
1
)
ϕ
2
(
x
2
)
. . . ϕ
n
(
x
n
)
dx
1
dx
2
. . . dx
n
.
(4)
34
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
