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8
МАТЕМАТИКА
Ji an SONG
(Chinese Academy of Engineering, Beijing 100038)
OPTIMAL REPRESENTATION OF MULTIVARIATE FUNCTIONS OR DATA IN VISUALIZABLE LOW-DIMENSIONAL SPACES1
It is intended to find the best representation of high-dimensional functions or multivariate data in the L2 (ft) space with the fewest number ofterms, each ofthem is a combination ofone-variable function. A system of non-linear integral equations has been derived as an eigenvalue problem of gradient operator in the above-said space. It is proved that the complete set ofeigenfunctions generated by the gradient operator constitutes an orthonormal system, and any function of L2(ft) can be expanded with the fewest terms and exponential rapidity of convergence. It is also proved as a Corollary, all eigenvalues of the integral operators has multiplicity equal to 1 if the dimension of the underlying space Rn is n = 2, 4 and 6.
The analysis and processing of massive amount of multivariate data or high-dimensional functions have become a basic need in many areas of scientific exploration and engineering. To reduce the dimensionality for compact representation and visualization of high-dimensional information appear imperative in exploratory research and engineering modeling. Since D.Hilbert raised the 13th problem in 1900, the study on possibility to express high-dimensional functions via composition of lower-dimensional functions has gained considerable success [1,2]. Nonetheless, no methods of realization are ever indicated, and not even all integrable functions can be treated this way, a fortiori functions in L2 (H). The common practice is to expand high-dimensional functions into a convergent series in terms of a chosen orthonormal basis with lower dimensional ones. However, the length and rapidity of convergence of the expansion heavily depend upon the choice of basis. In this paper an attempt is made to seek an optimal basis for a given function provided with fewest terms and rapidest convergence. All elements of the optimal basis turned out to be products of single-variable functions taken from the unit balls of ingredient spaces. The proposed theorems and schemes may find wide applications in data processing, visualization, computing, engineering simulation and decoupling of nonlinear control systems. The facts established in the theorems may have their own theoretical interests.
The paper is published without any redaction.
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\x2, ... ,xn) be an arbitrarily given function defined on the unit cube Q in Rn , F e L2(Q),
f (F(x1,x2,...,xn))2 dx1 dx2...dxn < M2 < to, (1)
where x = (x1,x2 ,...,xn) is a point in Rn. It is intended to find a set of one-variable functions ^(xa) whose product (x1)^2(x2)... (xn) e L2(Q) would best, or optimally, approximate F(x) with the least-square deviation:
L = J (F(x1, x2,..., xn) - ^(x1 )^2(x2)... ^n(xn))2 dQ = min, (2)
n
where dQ = dx1 dx2 ... dxn.
Suppose each -^a(xa) is taken from the unit balls Ba e L2a)(0,1),
L2(01) < 1}, a = 1, 2,...,n, the above requirement (2) can
be rewritten as
L = mf / (F(x) - A J] ))2dQ. (3)
w a J
n a=1
Opening up the brackets on the right side we have
n
(x) — All -0a(xa^2-
/10
(F(x) - A J]>a(xa))2dQ =
n
/n n
(F2(x) - 2AF(x) J] фа(xa) + A2 Д (^a(xa))2)dQ.
a=1 a=1
It is easy to verify that (3) holds if and only if there exists a product of n functions^ , e Ba, and a real A e R, A = 0, which enable the
a
following functional to achieve supremum on all unit balls Ba,
// 0
F (x)JJ ^a(xa )dQ
n
Л ™2 Л/ 1\ 2/ 2\ ,„n/ n\ 7 1 7 2 7 n
a=1
= J F(x1, x2,..., xn)^1(x1 )^2(x2)... ^n(xn)dx1 dx2 ... dxn. (4) n
n
For the convenience of discussion in the sequel we distinguish the same spaces (0,1), a = 1,2,...,n, and construct a product space (0,1) = 4^(0,1) x L>2)(0,1) x ... x L^n)(0,1), L^(0,1) =
n
= {E , e L2a)(0,1), a = 1,2,...,n, p = 1,2,...,n} •
в
a=1
After introduction of inner product for ф, v e Щ(0,1), ф = J^J фс
n
V = П V-,
a=1
a=1
nn
<ф,^> = (П Ф- П
\a=1 a=1
with induced natural norm
Va ) =
n n „ 1
П<фа,va> = п/ ■
a=1 a=1 ^ 0
фа(ха)va(xa)dxa,
пф-
a=1
L2 (0,1)
П
a=1
lL2a)(0,1),
Ln (0,1) becomes a linear normed space. The Ln (0,1) defined above can be embedded into L2 (Q) with preserved norm and becomes a dense subset of the latter [3, 4], while L2a)(0,1) is a closed subspace of L2 (Q), since ll^a||=(a)(0 i) ll^alL2 (n) always holds on Q the other hand, for the multilinear functional f : Ln (0,1) ^ R, defined by
П ^^ = / F(x) П
V=1 ' a
x)TT ^a(xa)dn,
(5)
the following inequality holds for every element of Ln(0,1):
f П*"
<
МП
where M is the lower bound defined in (1). Hence f is bounded and, by Banach-Steinhaus theorem, is totally continuous in Ln(0,1) [5].
Let Ba be the unit ball of 4a)(0,1), Ba = e L2a)(0,1),
\\^a\\ < 1}, and Bn = Bi x B2 x •■•B„. First of all we need the following Lemma.
Lemma 1. The n-linear form (5) can achieve its supremum on Bn. Whenever F(x) = 0, the supremum A is positive,
(a)
A = sup f F(x)TT фа^ > 0. 'Фа eßaJ V"
(6)
n
n
n
a
a
Proof. Since all unit balls Ba, a = 1, 2,...,n, are weak compact, by the Banach-Alaoglu theorem, every sequence |^al enabling (4) to approach supremum contains a subsequence weakly converging to some
element in Ba [5,6]. Now we show that there exists a sequence ^
a=1
n
k = 1, 2,..., that converges weakly to an element e Ln(0, 1)
a=1
which the following functional achieves its supremum,
lim f F(x1, •••,xn)TT#dQ = in *
= [ F (x1, •••,xn)TT ^adQ = A = sup f F(x)T] (xa)dQ,
or
lim f F(x1, •••,xn)^ ^ -Д pa]dQ = 0.
J n л,
Due to the identity
Ша -П
pa =
= П # - p1 ■■■$? + p1 ■■■$? - pV2 ■■■$? +
+ p1 p2 ■ ■■p"-1 ^n -П
p
we have
/ f П « - П p
£
i=1
dQ =
F ■ (^ - pi)dxi
i-1
n о
Пр* П dQ
а=1 e=i+1
n „ i—1 n
E / F^ Ра ■ П ^fdQi,
i=1 n а=1 e=i+1
where dQi = dx1 • • • dxi • • • dxn; i means without ith coordinate, and
Fk (x1, ■■■ ,xi, ■■■ ,xn) = J F (x1, ■■■ ,xn)[^k (xi) - pi(xi)] dxi.
о
n
а
1
n
n
1
By the weak compactness of Ba, for any fixed point (x1, ••• , x', ••• , xn)
in Q,, F^x1, ••• ,x', ••• , xn) ^ 0 as (^k(x') — ^'(x')) tends weakly to 0. By the Dominated Convergence Theorem [7], ||Fk ||L2(ni) ^ 0, k= 1, 2, ..., n, hence
F (x1, ••• ,xn)
Ш* "П ^
dQ
n
n
(Hi) ^ 0
i=1
as k ^ œ. This is to be shown for the first part of the Lemma.
To verify the second statement we take an orthonormal basis \ ef (xa),
{n
Wela (xa),
a=1
7a = 1, 2, ••• >, each Ya runs over N independently. Since L>(0,1) is dense
in L2 (Q), E becomes an orthonormal basis of the latter [4, 8]. Any element F e L2 (Q) can be expressed uniquely in the form of Fourier series,
n
F (x1,... ,xn) = J] РвП , Рз = (
в=1 a=1 \ a / L2(H)
By assumption F = 0, there must be some =0. Let =(sign ) ea.
a
a
Substitute the above series into (6), and take inner product with just defined , we obtain immediately À > |Pk | > 0, what is claimed in the Lemma. Now we proceed to establish the necessary conditions which a solution of (4), II , should satisfy. Suppose the expression (4) achieves its
a
supremum at some element G Bn. According to Lagrange Principle,
a
must satisfy the following conditions with a multiplier À' [5, 9]:
DU F(x\ ••• ,xn) n ^x — X'(/n n — 1) 1=0,
\n a M a a / L2(n)
(7)
where D denotes the Gateaux directional derivative with respect to all <^a, A' is a real to be determined. According to the rules of differentiation,
n
Df = ^ Df, Df is the partial derivative with respect to Let h'
be arbitrary element in 4г)(0, 1), t e[0, 1], A = 2A'. A straightforward computation yields
DJ = lim —
i^Q t
/(П^ (P + th'
а=г
ча=1
Due to the arbitrariness of h', we have
- nun =Ыи - Api,hv,, =0
a=i / f 4°(Q.1)
ФиП - Ap' = 0, i = 1, 2,- ■ ■ n, (8)
or, after unfolding,
Apj(xj) = $/ n Pa) =
= / F ^ • (xl)' • • ^ • • P" • •
• • • dx^ • • dx", i = 1, 2,- • • n, (8')
where i means absence of ith coordinate, Qj is the (n — 1) dimensional unit cube without xj.
The operator $ = <^1, $2,...,$") generated by G-derivative and defined in (8') is called the gradient operator of functional (5). Now we proceed to prove the following, the primary theorem as a start-point for further investigation.
Theorem 2. For any given F(x) e L2(Q), F(x) = 0, its gradient operator $ or, equivalently, the system of homogeneous integral equations (8') possesses at least one positive eigenvalue. The greatest eigenvalue and its associated eigenfunctions satisfy (4), and are a solution of this supremum problem.
Proof (abridged). It is known that for any given F(x) e L2(Q), all components of its gradient operator defined by (8), are compact [6, 7], so the range (B"-1) is a compact subset in ¿2^(0, 1), here
?"— 1 _ ) T T „l,a{rva
B" 1 = < JJ r(xa), ||^a|| < 1 f. By Lemma 1, there exists a sequence in B", < JJ f, weakly convergent to JJ pa e B" as k ^ to so that
V y ,-y
4 a=i
the following holds,
Ит(фЛПФ- , Фк
vi / / l2(0,1)
Ф» П Va , vO = A, i = 1, 2,- ■ ■ ,n.
V-=i / / L (0,1) Now we construct new functions,
nk И = Афк - ф/П Фа) =
= Афк(хг) - i F(x1,x2,- ■ ■ ,хп)ф1(х1)
• • • • • ^(x^dx1- • • dx*- • • dxn, \\ < 1, i = 1, 2, • • • ,n.
An accurate calculation of the norm of nk shows that Lemma 1 implies also while —a approaches weakly to its limit nk strongly tends
a a
to zero. Namely,
lim <nk ,П^> = lim
кк
Афк - Фг(Д Фа)
а=г
= 0, i = 1, 2,
This indicates, the sequence {-^a, a = 1, 2, ■ ■ ■ , n, k = 1, 2, ■ ■ ■ } has a strong limit, denoted again by , a = 1, 2,..., n}, which satisfies (8). Taking inner product for (8') with we obtain finally,
n ^
a = 1 a = i
= /F(x1,x2,' ■ ',x'v(xV(x2)' ■ ■ ^n(x")dx1dx2' ■ ■ dx" =A, (9)
n
that is, is a solution of (8), as claimed in the Theorem.
n
Let A1 and ^ are the greatest eigenvalue and its associated
1
a=1
eigenfunctions, respectively. Construct new function
F1(x1, x2, ■ ■ ■ , xn) = F0(x1, x2, ■ ■ ■ , xn) - A1 Д (xa).
a=1
2
n
Let F1 — 0. It is easy to check that the norm of Fiin L2 (H) can be calculated
as
||Fi||2 = (Fq - A^p?,Fo - A^p
= ||Fo||2 - 2A^Fo, J] + Ai (J] pa, J] P?) = IIFo||2 - A2.
So long as F1 — 0, the Lemma 1 and Theorem 2 are applicable for F1 as well. Having F0 replaced by F1 in (8), one gets a new operator and, correspondingly, new system of equations (8').
By Theorem 2, it possesses at least one positive eigenvalue A2 and associated eigenfunctions p? of (8) with F replaced by F1. Similarly, we have a
n n n
F2 = Fi - A^ p? = Fo - A^ p?-A2 n P?,
a=1 a=1 a=1
with its norm
II F ||2 _ || F ||2 A2 _ || F ||2 \2 \2
|| F2 | — || F11| - A2 — ||Fo| - Ai - A2.
The process can be continued inductively, if FN — 0,
n N n
Fn — Fn-1 - An J} Pn — Fo - ^ A, J] P?, (10)
a=1 3=1 a=1
and
N
||FN ||2 — ||FN-1||2 - AN — ||F0|2 - ^^ A3. (11)
,3=1
Further, each FN generates its own gradient operator — ($N 1,..., $Nn),
$Ni(n Pn+J — f Fn(x)JJ pN+1 (x?— An+1 pN+1 (x4),
^a=i ' q a=i
i — 1, 2,- ■ ■ ,n. (12)
If the process continues infinitely, (10) becomes an infinite series. Now we prove that the following equality holds as N in the norm of L2(H),
x n
Fclx1, x2, ■ ■ ■ ,xn) = ^ A^ Д pa(xa). (13)
в=1 a=1
Theorem 3. Any given F(x) e L2(Q), 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 }. The series (13) converges exponentially to F(x) in the norm of L2(Q).
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 — to. It is obvious from (10), however large is N, one always has ||FN || > 0 , and
N
EA? < llFoII2 • (14)
3=1
The necessary condition of convergence for the series on the left side is An —^0 as N — to, and (6) implies that the following relationship holds uniformly on Bn,
An+1 = sup i Fn (x)TT ^a(xa)dQ=( Fn(x), TT ^N+1 ) > y V \ a / L2(H)
>
, V^a G Ba ,
a / L(П)
here +i is a solution of (6) and (8) with F0 (x) replaced by FN (x).
a
Therefore, when N ^то,
lim ( Fn(x),TT^a ) ^ 0, V^a G Ba, a = 1, 2,- ■ ■ ,n.
N^œ \ /
\ a / l2(Q)
It is evident that Bn is a fundamental set, i.e., it spans L£(0, 1), is a dense subset of L2 (Q). The above condition suffices for FN to converge weakly-star to some element Fœ which is equivalent to 0 in the weak topology [4],
lim sup i Fn (x) TT ^a(xa )dQ = sup / Fœ, TT^a ) = 0.
N^œ bn / Bn \ /
n a \ a / L2 (n)
Recall the fact that the set E = < eaa, Ya G N > consisting of
a
combinations of orthonormal bases of L>a)(0, 1) is a complete orthonormal
basis for L2 (П). NoticeE с Bn , by Lemma 1, it implies
0 = sup /F^H^M > sup/ F^^e^M > 0.
Bn \ / E
Which means all Fourier coefficients of Fœ are zero, hence Fœ= 0. Then (13) holds in the sense
oo
||Fo||2 — Y, A?. (15)
3=1
Now we proceed to justify the second statement claimed in the Theorem about the convergence rapidity of the expansion (15). By assumption all F? — 0, and due to (10) and (11), the following relations and the continuous multiplication are well defined,
H-Fi||2 _ ||F0||2 — A2 A2
— 1 —
I î? II 2 llî?ll2 II TP Il2 :
F0II IIFq II IIFQH
Ifn ||2 !fn-i||2 — AN л AN
= 1 -
1/7 II2 Iii? II2 II TP Il2 :
FN-1 II IIFN- 1II IIFN-lI
I F II2 M F M2 il F M2 и F и2 11 F и2 N / A2
IF N II IIFN II IIFN-lI IIF2I IIF1II ^ / -, A3
I TP II2 II TP II2 II TP II2 II TP II2 II TP II2 П ( Il г? 1,2
|FoII IIFN-lII IIFN—21 IIFlI IIF0II IIFe-1
2 A?
Denote n, =--—namely A , = o> ||F>-i ||. By the proved previously,
HF?-1H
0 < a? < 1, and notice that the inequality (1 - a) < e-a always holds, then we have
n - E ?2
||Fn ||2 — ||Fo||2n (1 - a?) < ||Fo||2 e ,
3=1
or
-, N
-I £ "I
2 'J-ß
|FnII < IIFoII e (16)
Let Rn+1 = IIFn II2 = ^ be the sum of residual part of (15). It is
3=N+1
easy to show the sum
N N , 2
fc=1 k=1 k
diverges to infinity as N ^ to [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 L2 (H).
Proposition 4. For arbitrarily given F(x) e L2(H), F = 0, the set
of all eigenfunctions < P = 1, 2,... > of the sequence of gradient
^ a '
operators $3 = (^31,..., $3n), defined by (12), constitutes an orthonormal system as an ingredient part of some complete orthonormal basis of L2 (H).
Proof. By definition of FN, the identities ^F^, ^ ^^ = 0 hold
for all P = 1, 2,...,N. Each ^N+1 can be decomposed uniquely as ^N+1 = + ba^N+1, ^N^N+1, and aa, 6a be constants of
normalization. A substitution for ^N+1 yields
n ^N+1 =
=c0 n ^N+1 n ^N + ■ ■ ■ +Cn n +1 =C0 JJ ^N+Pn+1.
a i=1 a=i a a
Clearly, Pn+1 ^N. Thus, due to the identities said above,
a
An+1 = ^FN, n ^N+1^ = FN, n ^ + (FN,PN+1 > = = (Fn-1, Pn+1 >. It follows J] ^N^N+1. Similar analysis of
a a
for P = N — 1,..., 1 in succession, one obtains
An+1 — sup / Fn (x) J n
i ВДП ^d^— / ВДЦ +1 dn. (17)
nn
— sup I F0(x)|| ^ F0(x)
n a ^ n
e(LNa))±
This being true for all N G N follows the set of eigenfunctions 111 ^ = 1, 2,--- f constitutes a orthonormal system in L2 (Q). Since
n
for any orthonormal set S in a Hilbert space there is a complete orthonormal basis that contains S as its subset [11]. The Proposition is thus justified.
Remark. It appears the remarkable maximum property of gradient operators expressed in (17), which is entirely analogous with compact self-adjoint operators in Hilbert spaces. The Proposition also shows there are as many different orthonormal bases as the cardinality of different elements in L2(ft).
Corollary 5. If the dimension of underlying space Rn with n = 2, 4 and 6, all eigenvalues of gradient operators defined by (11) for arbitrarily given F(x) G L2(Q), F(x) = 0, have multiplicity no more than 1.
Proof (abridged). Suppose the contrary, if there exist two different eigenfunctions p1 = pN+1,1 and p2 = pN+1,2 corresponding to the
a a
same eigenvalue ÀN+1, which enable the following functional to achieve its supremum on Bn,
Àn+1 = SUp /n (TT ^a) =
^a
= sup / Fn(x)TT = /n(TT pN+1,k), k = 1, 2. (18)
J n a a
Now construct a new element p3 = H(tpa + (1 - t)pa), t G [0, 1], and
a
put it into (18). After exposing the product, we have
/n (P3, t) = /n (n (tpa + (1 - t)pa^ =
a
= t/N ( n pa) + tn-1 (1 -1)/^ E P2 n pa)+
^ a ' M=1 J
+tn-2 (1 - t)2/J E p2 p2 n pa) +... ...+12 (1 - t)n-2 /J e p1 p1 n
+t(i - t)n-1 /n ( £ pa ) + (1 - t)n/N ( П pa ). (19)
i=1
By Proposition 4, pi and p2 are mutually orthogonal. Dropping index N for brevity, we have the necessary condition,
Df = f(E hi n pa) = E (pi, hi> A = 0, Vhi G 4i}(0,1).
\ A ~ / A
Take h = p2 in the above we see that the second term and the second from the end are zero. For computing the rest terms in the exposition (19) we invoke the sufficient conditions for f to achieve its maximum at p. If the dimension of the underlying space, Rn, n is even, then the following equations must be satisfied [9, 11],
D' f(n P?) =0' k = 1, 2,- ■ -,n - 1. (20)
^ a '
A direct calculation shows
Df (n P?) = 2f (E hn P?) - E (hi, h> A = 0
? ij 4
for arbitrary hi and h2 taken from 4^(0, 1). Let hi = h2 = p2. We get the value of the third term,
iw (E n^ =
V4=j ij ' 2
The third term from the end of (19) is in complete symmetry with the above, and, due to (20), all the rest terms are zero except the first and last ones. Then,
fN(<p3, t)= ^ + r-2(i-t)^ + t2(i-tr-2^ + (i-tr^xN+1.
It follows from the assumption, fN(p3, 0) = fN(p3, 1) = AN+i, and it reaches minimum at t = 1/2. Let t = 1/2. It yields
Since + ip%\\ = V2, iPl "t-^2 G Ba, we have
v 2
In (v>3, -^j = In (n = (2 + n) ■ 2"t Aw+1 = p{n)\N+l.
It is evident, if p(n) > 1 then (p(n))i > 1, so that Jj" ^ G Bn_
A direct computation shows this is possible only for n = 2, 4 and 6. In these cases if p= pN+iii, and P= Pn+i,2 both render the supremum
AN+i to the functional , then along the direction of the middle point p3 of segment joining p^ and pf, also provides the supremum AN+i to . By the assumption p1 = p2, there must be infinite amount of different elements in Bn at each of them attains its supremum. This contradicts the compactness of gradient operators. That is, for n = 2, 4, and 6, the multiplicity of any eigenvalue of (12) is no more than 1. This completes the proof of the corollary.
The case n = 2 may cause particular theoretical interest [12,13]. Let F(x,y) be defined on the unit rectangle B2 of the plane and be square-integrable. By Theorem 2, it generates a gradient operator and (8) is reduced to
1 1
= y F(x,y)^(y)dy = Ap, p = J F(x,y)p(x)dx = A^. (21) 0 0
Apparently, p and ^ are eigenfunctions of self-adjoint operators
p = A2 p, = A2
Corollary 5 claims for this case that all eigenvalues of $ have multiplicity no more than 1. Indeed, suppose the contrary. Let p1 (x)^1(y) and p2 (x)^2(y) provide the same supremum A1 on B2,
A1 = sup / F (x, y)p(x)-^(y)dxdy = 2 J n
= y F(x,y)pfe(x)^fe(y)dxdy, k = 1, 2. (22)
n
By Proposition 4, p1 ±p2 and Let
p3^3 = (tp1 + (1 - t)p2)(^1 + (1 - t)^2), 0 < t < 1.
It is easy to check,
I F(x,y)p3(x)^3(y)dxdy = (1 - 2t + 2t2)A1, ||ps^H = 1 - 2t + 2t2,
thus
[ F(x,y)?P^dxdy = Vie [0,1].
n
This means the functional reaches A1 on B2 along all rays from the origin and intersecting any point of segments joining p1, p2 and
It contradicts the assumption and implies A1 is not supremum of the functional. One may notice, this is true if F(x, y) replaced by FN(x, y) =
N
= F(x,y) - Y A3p^^3 in (21). In particular, if F(x,y) = F(y,x), it
3=1
generates a self-adjoint operator, $ = $*, 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 L2(Q) is strictly and uniformly convex [14,15]. It is believed that the B n Bn possesses the same property. The equations (18) create a supporting tangent hyperplane to Bn in L2 (Q). A conjecture arises that the claim made in Corollary 5 would be true for any finite dimensional underlying spaces Rn. 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 L1 (Q) and in the Banach space of continuous functions C(Q). The question arisen is that the assumption (1) is not enough to ensure the convergence of the infinite sum
A3 except F(x) generates a nuclear gradient operators [16]. However,
3
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) e L2(Q), the series (13) converges uniformly in L1 (Q). If F(x) is continuous on Q and possesses all continuous first partial derivatives in Q, 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 L2 (Q) and /2, the space of square-summable sequences of reals, are isometrically isomorphic. Each element of L2 (Q) has its spectral image in /2 according to bases chosen in each spaces. If one identifies the square of norm of F(x) e L2(Q) with the energy or information it carries, in terminology of physics, the outcome of Theorems presented in
this paper is to assert that for any given F there exists an optimal basis in L2 (Q) which furnishes the element with an image sharply concentrated on a few of spectrum-lines in /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 Rn, xn = F(x\x2xn-1), is needed to be stored, the amount of data is measured as Nn-1 + N, N is the mean number of discrete samplings for each variable. If / terms are taken in (13) to represent F, the amount of data to be stored or processed will be reduced to n/N, a 1/Nn-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 20th century, he called it power series integral equations and imposed severe restriction on the given function. He required F(x1, x2,..., xn) 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 e L2 (Q), 0 < m < F < M < to, the following integral equation
/ n
M=1
= F(s, ¿1, t2, ■ ■ ■ ¿n ) ^(¿1 Mi2) ■ ■ ■ p(tn)dMt2 ■ ■ ■ dtn = A^(s)
possesses at least one positive eigenvalue [22]. Wainberg M.M. had shown that for a totally symmetric F e L2 (Q) all components of the gradient operator $ generated by F are compact, and the functional
11 ^(¿o), is weak continuous respect to p, 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
Mathematics and System Sciences, Chinese Academy of Sciences, for their interest and the time they spent for discussion and checking the proofs of Lemma and Theorems. Their advice and suggestions are extremely valuable for improvement of the paper and helpful for the author to achieve some unexpected results.
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The original manuscript was received by the editors in 8.07.2005
Jian Song (b. 1931) graduated from the Bauman Moscow Higher Technical School. He received degree of D. Sc. (Eng.) in 1990 from the Bauman Moscow State Technical University. Now he is Honorary Chairman of Presidium of Chinese Academy of Engineering, and President of China-Japan Friendship Association.
His academic titles include Academician of both the Chinese Academy of Sciences and the Chinese Academy of Engineering; Honorary Professor of the Academy of Mathematics and System Sciences; Foreign Associate of the US National Academy of Engineering; Foreign Member of the Russian Academy of Sciences, the Royal Swedish Academy of Engineering Sciences, the Yugoslav Academy of Engineering; Corresponding Member of the National Academy of Engineering of Mexico, the National Academy of Engineering of Argentina; Member of the International Astronautic Academy, and Member of the International Euro-Asian Academy of Sciences. He has been conferred numerous honors and awards both at home and abroad: "Most Distinguished Young Scientist" in China, National Award for Scientific and Technological Progress, Albert Einstein Award from the International Association for Mathematical Modeling, China's National Natural Science Prize, National Prize for Excellent Scientific Publications for revising "Engineering Cybernetics", Award for outstanding achievements in science.
He has authored, co-authored, or edited 12 books and has written and published more than 100 scientific articles.
Цзянь Сун родился в 1931г., окончил МВТУ им. Н.Э.Баумана. Д-р техн. наук (МГТУ им. Н.Э. Баумана, в 1990 г.), Почетный председатель Президиума Китайской технической академии, президент Ассоциации дружбы между Китаем и Японией. Действительный член Китайской академии наук, Китайской инженерной академии. Почетный профессор академии математики и системных наук. Иностранный член-корреспондент Национальной инженерной академии США. Иностранный член Российской академии наук, Шведской королевской академии инженерных наук, Югославской инженерной академии. Член-корреспондент национальных инженерных академий Мексики, Аргентины. Академик Международной космической академии и Международной евро-азиатской академии наук. Один из первых в Китае лауреатов премии "Выдающийся молодой ученый". Лауреат премий, присужденных в Китае и заграницей: Государственная премия за прогресс в науке и технике, премия Альберта Эйнштейна Международной ассоциации математического моделирования, Китайская национальная премия в области естественных наук, Государственная премия за научные публикации (переработка книги "Инженерная кибернетика"), премия за выдающиеся научные достижения. Автор более 100 научных работ, в том числе 12 монографий.
