JOURNAL OF COMPLEXITY
ARTICLE NO.
12, 47–57 (1996)
0006
About Widths of Wiener Space in the Lq-Norm
V. E. MAIOROV*
Department of Mathematics, Technion, Haifa, Israel
Received October 31, 1991
We calculate the average Kolmogorov and linear n-widths of the Wiener space
in the Lq-norm. For the case 1 # q , y, the n-widths dn decrease asymptotically
as n21/2.  1996 Academic Press, Inc.
1. INTRODUCTION
In this paper we consider average n-widths of the Wiener space C in the
Lq-norm. We study two kinds of average n-widths which describe best
and best linear approximation of the Wiener space over all n-dimensional
subspaces. The approximation problem on spaces with Wiener measure in
the L2-norm was investigated in the book of Traub et al. (1988). Papers
concerned with average n-widths in Banach spaces include Buslaev (1988),
Heinrich (1990), Maiorov (1990, 1993a,b), Maiorov and Wasilkowski (1996),
Mathe (1990), Ritter (1990), Pietsch (1980), and Traub et al. (1988). The
approximation of the functionals on Banach space with measure can be
found in Lee and Wasilkowski, (1986). The problem of computing n-widths
is closely connected with the problem of complexity of approximation
(Traub et al., 1988).
The asymptotic n-widths dn(C, Lq , g) ³ n21/2 were calculated in Maiorov
(1990) for the case 2 # q , y, and in Maiorov (1993a) for the case
q 5 y. In this work we prove that the same estimate is also true for the
case 1 # q , 2. Note that the estimate for n-widths coincides with the
estimate for the average spline-approximation of Wiener space in the Lqnorm for all 1 # q # y (Ritter, 1990). The probabilistic n-widths of the space
* The research was supported by The Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.
47
0885-064X/96 $18.00
Copyright  1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
48
V. E. MAIOROV
C r in the Ly-norm with respect to r-fold Wiener measure were calculated in
Maiorov and Wasilkowski (1996).
Average n-widths are defined as follows. Let X be a set, equipped with
a probability measure e that is defined on Borel sets from X. Let X , Y,
where Y is a separable Banach space. We denote by Ln(X, Y ) the set of
all mappings or (nonlinear) operators L with range X in a linear subspace
of dimension at most n. Let also LLn(X, Y ) be the subset of Ln(X, Y )
consisting of all linear operators. Let
E
e(X, L, Y, e) 5
X
ix 2 LxiY e(dx)
be the average error in approximating the set X with the operator
L , Ln(X, Y ).
We now define the average n-width of the set X in the space Y for
measure e to be
dn(X, Y, e) 5
inf
L,Ln(X,Y )
e(X, L, Y, e).
Consider also the average linear n-width of set X given by
dn(X, Y, e) 5
inf
L,LLn(X,Y )
e(X, L, Y, e).
We now recall the definition of the Wiener space. Let C be the space of
all continuous functions x(?) defined on [0, 1] such that x(0) 5 0. The
measure g in C is defined as follows:
n
g(B9) 5 p [2f(ti 2 ti21)]21/2
i51
E
B
SO
D
2(ui 2 ui21)2
du1 ? ? ? dun , (1)
i51 2(ti 2 ti21)
n
exp
for every n $ 1, Borel set B , Rn, and 0 5 t0 , ? ? ? , tn # 1 with
u(0) 5 0; we set B9 5 hx [ C: (x(t1), . . . , x(tn)) [ Bj.
Let Lq , 1 # q , y, be the usual Banach space consisting of all functions
x(?), with norm
ixiLq 5
SE
1
0
ux(t)u q dt
D
1/q
.
Let c, ci , i 5 0, 1, . . . , be positive constants depending just upon the
parameters q (and p in Section 2). For two positive sequences fn and gn ,
WIENER SPACE WIDTHS IN THE
Lq-NORM
49
n 5 0, 1, . . . , we write fn ! gn or fn ³ gn if there exist constants c1 and
c2 such that fn # c1 gn or c2 # fn /gn # c1 for all n.
THEOREM 1. Let 1 # q , y be any number. Then the average n-widths
satisfy the asymptotic estimate
dn(C, Lq , g) ³ dn(C, Lq , g) ³ n21/2.
2. ESTIMATIONS OF n-WIDTHS ON FINITE DIMENSIONAL SETS
Let l pm, 1 # p # y, be the m-dimensional linear normed space of the
vectors x 5 (x1 , . . . , xm) from Rm with norm
ixi l pm 5 ixi p 5
SO D
m
i51
uxi u p
1/p
.
Let B pm( r) 5 hx [ l pm : ixi p # rj be the ball with radius r in the space
and let B pm 5 B pm(1). By Vm(G) we denote m-dimensional Euclidean
volume for any Borel set G , Rm.
We consider the Lebesgue measure l in the space l pm, with normalizing
condition l(B pm) 5 1. This measure is equal to l(G) 5 Vm(G)/Vm(B pm). For
the function f (x) defined on B pm, we let
l pm,
I[ f (?)] 5
E
B pm
f (x)l(dx).
THEOREM 2. Let 1 # p # y, 1 # q , y be any numbers. There exists
c0 , a constant depending just on p and q, such that for all integers m, n with
m $ c0 n . 0, the asymptotic estimate
dn(B pm, l qm, l) ³ m1/q21/p
holds.
Proof. (1) We first prove the upper bound. Let dn ; dn(B pm, l qm, l) and
Vm ; Vm(B pm). We have
d nq # I(i?iq)q # I(i?i qq) 5 mI[w (?)],
(2)
where w(x) 5 ux1u q. Let G(u) and B(u, v) be the gamma and beta functions. Then
50
V. E. MAIOROV
21
I[w (?)] 5 V m
21
5 Vm
E
E
B pm
1
21
ux1u q dx
21
ux1u qVm21[B m
((1 2 ux1u 1/p)] dx1
p
21
5 Vm
Vm21
E
1
21
ux1u q(1 2 ux1u p)(m21)/p dx1
that is,
I[w (?)] 5
S
D
q11 m1p21
2 21
V m Vm21 B
,
.
p
p
p
(3)
From here with q 5 0 we have
Vm 5
S
D
1 m1p21
2
.
Vm21 B
,
p
p
p
(4)
We use the known property of gamma functions
B(u, v) 5
G(u)G(v)
,
G(u 1 v)
and Stirling’s formula G(u) 5 u u21/2e uÏ2f (1 1 o(1)). Then
B
S
D@ S
D
S D@ S
D
q11 m1p21
,
p
p
m1p
5G
p
B
1 m1p21
,
p
p
m1p1q
G
³ m2q/p.
p
(5)
From relations (3), (4), and (5) we conclude that
I[w (?)] ³ mq/p.
Then the upper estimate for dn follows from (2).
(2) To prove the lower bound for dn , we use the method of estimating
n-widths with the help of «-cardinality numbers of ball B pm.
We need supplementary definitions and statements. Let « . 0 be any
number. We define the average «-cardinality of ball B pm in the space l qm for
measure l to be
WIENER SPACE WIDTHS IN THE
H
Lq-NORM
F
51
G J
N« (B pm, l qm, l) 5 min N: z1 , . . . , zN [ Rm, I min i(?) 2 zi i q # « .
1#i#N
LEMMA 1. Let 1 # q , p, and a 5 1/q 2 1/p. Then for any « . dn we have
S
N« (B pm, l qm, l) # c1 1 1
D
4ma n
.
« 2 dn
Proof. Let Qn be an n-dimensional linear subspace of Rm, such that
F
G
I min i(?) 2 yiq 5 dn(B pm, l qm, l).
y[Qn
Consider the set G 5 2maB pm > Qn . Let G 5 hz1 , . . . , zN j be the minimal
d-net of the set G in the l pm-norm. Then for N, we have the estimate
S
N# 11
D
4ma
d
n
(see [9, 11]).
From the trivial inequality dn # ma, it follows that for any x [ B pm, the
set h y [ Qn : ix 2 yi q # dnj is a subset of G. Therefore we conclude that
F
G F
G
I min i(?) 2 zi i q # I min min (i(?) 2 yiq 1 i y 2 zi i q)
1#i#N
1#i#N y[G
F
F
G
# I min i(?) 2 yi q 1 d 5 I min i(?) 2 yi q 1 d
y[G
y[Qn
5 dn 1 d.
Hence for « 5 dn 1 d we obtain the statement of the lemma.
LEMMA 2. If p $ q then the inequality
N« (B pm, l qm, l) $
holds.
Proof. We claim that
S D
c2ma
«
m
G
52
V. E. MAIOROV
N« (B pm, l qm, l) $
l(B pm)
.
3l(2«B qm)
(6)
Indeed, suppose that inequality (6) is not correct. Then for N ;
N« (B pm, l qm, l) and some points z91 , . . . , z9N we have
«$
E
S U<
N
min ix 2 z9i iq l(dx) $ 2«l B pm
B pm 1#i#N
i 51
$ 2«[l(B pm) 2 Nl(2«B qm)] $
D
hz9i 1 B qm(2«)j
4«
,
3
a contradiction. From the inequality (6) it follows that
Vm(B pm)
.
3(2«)mVm(B qm)
N« (B pm, l qm, l) $
(7)
Using the relation (3) and properties of Euler functions we have
(c3m)2m/p ! Vm (B pm) ! (c3m)2m/p.
(8)
From estimates (7), (8) for c2 5 c3 /(3c4), the lemma follows.
We now continue the proof of the lower estimate for the n-widths dn .
First, suppose that p $ q. Then from Lemma 1 and Lemma 2 it follows that
S
c1 1 1
D S D
4ma
« 2 dn
n
$
c2ma
«
m
.
Let « 5 5dn . Taking logarithms in the last inequality we have for some
constants c0 , c5 , and all m, n (with m $ c0n) the estimate
dn (B pm, l qm, l) $ c5ma.
(9)
If p , q then using Holder’s inequality and estimate (9) for p 5 q
we obtain
dn (B pm, l qm, l) $ madn(B pm, l pm, l) $ c5ma.
This completes the proof of Theorem 2.
COROLLARY 1. If 1 # p # y, 1 # q , y, and m $ c0n . 0, then the
linear n-width has the asymptotic estimate
WIENER SPACE WIDTHS IN THE
Lq-NORM
53
dn (B pm, l qm, l) ³ m1/q21/p.
Next we study average n-width of the space Rm equipped with Gaussian
measure by the formula
c(B) 5 cm(B) 5 (2fm)2m/2
E
B
S
exp 2
D
x 21 1 ? ? ? 1 x 2m
dx1 ? ? ? dxm ,
m
for every Borel set B in Rm. We have c(Rm) 5 1.
LEMMA 3. For any 1 # q , y,
dn(Rm, l qm, c) 5 dn(B2m, l qm, l).
(10)
Proof. Indeed, let L be the operator such that
e(Rm, L, l qm, c) 5 dn(Rm, l qm, c).
Obviously L is a homogeneous operator. Then using polar coordinates
we have
E
Rm
ix 2 Lxiqc(dx)
5 (2fm)2m/2
E
y
0
S D
r2
r m exp 2
dr
2
E
Sm21
(11)
ix 2 Lxiq dx,
where Sm21 is the unit sphere in l 2m and dx is the Lebesgue measure on
Sm21. Taking into account
(2fm)2m/2
E
y
0
S D
r m exp 2
r2
21
dr 5 Vm
(B2m)
2
Er
and equality (11) we have Lemma 3.
From Theorem 2 and (10) we derive
COROLLARY 2. Let 1 # q , y and m $ c0n. Then
dn (R m, l qm, c) ³ m1/q21/2.
1
0
m
dr
54
V. E. MAIOROV
3. ESTIMATIONS
n-WIDTHS
OF
FOR
WIENER SPACE
In this section we prove Theorem 1. First, we need two auxiliary lemmas.
LEMMA 4. Let n, nk (k 5 0, 1, . . .) be any natural numbers such that
y
ok50 nk # n, and nk # 2 k. Then the average n-widths of Wiener space in
the Lq-norm satisfy the inequality
dn(C, Lq , g) #
O2
y
2(k11)/q
k50
k
k
dnk (R2 , l q2 , c2k).
Proof. Let x(?) [ C be any function. Build a sequence of spline functions
(k 5 0, 1, . . .)
xk (t) 5
O x(i2
2k21
2k
k51
)xik (t),
where xik is the characteristic function of [i/2 k, (i 1 1)/2 k ).
Consider the linear operator Pk x 5 xk11 2 xk . The function x is supposed
to be continuous. Therefore we can write x as a uniformly convergent series
x5
O P x.
y
(12)
k
k50
k
k
Let Lk be the extremal operator for the n-width dnk (R2 , l q2 , c2k), that is,
k
k
k
k
e(R2 , Lk , l q2 , c2k) 5 dnk (R2 , l q2 , c2k).
(13)
We consider the points on [0, 1) of the form tik 5 i/2 k11, i 5 0, . . . ,
2 . For all even i we have ( Pk x)(tik ) 5 0. Consider the isomorphism from
k
the space Ck 5 Pk C to R2 :
k11
Dk Pk x 5 y 5 ( y1 , . . . , y2k ),
where yi 5 ( Pk x)(t2i21,k ) 5 x(t2i21,k ) 2 x(t2i22,k ). We have
iPk x 2 Dk21Lk Dk Pk xiLq 5 22(k11)/qi y 2 Lk yi l 2qk .
Therefore using the definition of Wiener measure (1) we have
WIENER SPACE WIDTHS IN THE
Lq-NORM
k
55
k
e(Ck , Dk21Lk Dk , Lq , g) 5 22(k11)/qe(R2 , Lk , l q2 , c2k).
(14)
Consider the operator
OD
y
L5
21
k L k Dk
k50
in the space Lq . Then the range can be estimated by
dim L #
O dim L # O n # n.
y
y
k
k50
k
k50
Therefore from (12), (14), and (13) we conclude that
dn(C, Lq , g) # e(C, L, Lq , g)
#
O e(C , D
y
21
k Lk Dk ,
k
k50
Lq , g) #
O2
y
2(k11)/q
k50
k
k
dnk (R2 , l q2 , c2k).
LEMMA 5. For all integers m, n (with m . n) we have the inequality
m21dn(Rm, l 1m, cm) ! dn(C, L1 , g).
Proof. Let « be any positive number and L be an operator such that
dn(C, L1 , g) $ e(C, L, L1 , g) 2 «.
(15)
Consider the operator Tx 5 x 2 Lx. We have
e(C, L, L1 , g) 5
5
E E u(Tx)(t)u dt g(dx)
1
C
0
EE
1/(m11)
C
0
O U(Tx) St 1 m 1i 1DU dt g(dx).
m
i50
From here and Fubini’s theorem we derive
e(C, L, L1 , g) 5
E
1/(m11)
0
E
C
O U(Tx) St 1 m 1i 1DU dt g(dx).
m
(16)
i50
For every t [ [0, 1) and i 5 1, . . . , m we define the linear functional
Dti 5 x(t 1 i/(m 1 1)) 2 x(t 1 (i 2 1)/(m 1 1)). Then we have
56
V. E. MAIOROV
O U(Tx) St 1 m 1i 1DU $ 12 O uD Txu 5 12 O uD x 2 D Lxu.
m
m
m
ti
i50
ti
i51
ti
i51
Using the definition of Wiener measure (1), we obtain
E
C
O uD x 2 D Lxug(dx) $ 2 1 e d (R , l , c ).
m
ti
ti
Ï
i51
m
n
m
1
m
(17)
Summarizing the relations (16), (17) we conclude that
e(C, L, L1 , g) $
c
dn(Rm, l 1m, cm).
m11
(18)
From inequalities (15), (18) for « R 0 we have
m21 dn(Rm, l 1m, cm) ! dn(C, L1 , g),
which completes the proof.
Proof of Theorem 1. Consider the integer sequence nk 5 2 k for
k # k9, and nk 5 0 for k . k9, where k 5 0, 1, . . . , k9 5 log2 n 2 1. Then
y
ok50 nk # n. Using Lemma 4 and Corollary 2 we have
O2
!O2
dn(C, Lq , g) #
y
2(k11)/q
k50
k.k9
2k/2
k
k
dnk (R2 , l q2 , c2k)
! n21/2.
Using Lemma 5 and Corollary 2 we derive the lower bound
dn(C, Lq , g) $ dn(C, L1 , g) $ cn21 dn(R2n, l 12n, c2n) ³ n21/2.
Hence Theorem 1 is proved.
Remark 1. The optimal method, which provided the upper bound for
the Kolmogorov n-widths in Theorem 1, is linear. Therefore for the average
linear n-widths of Wiener space C in Lq-norm (1 # q , y) the asymptotic estimate
dn(C, Lq , g) ³ n21/2
holds.
WIENER SPACE WIDTHS IN THE
Lq-NORM
57
REFERENCES
BUSLAEV, A. P. (1988), On best approximation of the random functions and functionals. Third
Saratov Winter School 2 (1988), 14–17.
HEINRICH, S. (1990), Probabilistic complexity analysis for linear problems in bounded domains,
J. Complexity 6, 231–255.
LEE D., AND WASILKOWSKI, G. W. (1986), Approximation of linear functionals on a Banach
space with a Gaussian measure, J. Complexity 2, 12–43.
MAIOROV, V. E. (1990), ‘‘About Widths of Space Equipped with a Measure,’’ pp. 91–105,
Yaroslav. State University, Yaroslavle.
MAIOROV, V. E. (1993a), Average n-widths of the Wiener space in the Ly-norm, J. Complexity
9, 222–230.
MAIOROV, V. E. (1993b), Kolmogorov (n, d )-widths of the spaces of the smooth functions,
Math. Sb. 184, 49–70.
MAIOROV, V. E., AND WASILKOWSKI, G. W. (1996), Probabilistic and average linear widths
in Ly-norm with respect to r-fold Wiener measure, J. Approx. Theory 84, 37–40.
MATHE, P. (1990), s-numbers in information-based complexity, J. Complexity 6, 41–66.
MITIAGIN, B. S. (1961), Approximative dimension of basic in kernel spaces, Uspekhi. Mat.
Nauk 16, No. 4, 63–132.
PIETSCH, A. (1980), ‘‘Operator Ideals,’’ North-Holland, Amsterdam.
RITTER, K. (1990), Approximation and optimization on Wiener space, J. Complexity 6,
337–364.
TRAUB, J. F., WASILKOWSKI, G. W., AND WOZNIAKOWSKI, H. (1988), ‘‘Information-Based
Complexity,’’ Academic Press, New York.
VORONIN, S. M., AND TEMIRGALIEV, N. (1984), On some applications of Banach measure,
Izv. Akad. Nauk Kaz. SSR, Ser. Phys.-Math. 5, 8–11.