Proceedings of the Steklov Institute of Mathematics, Vol. 243, 2003, pp. 227{233.
Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, Vol. 243, 2003, pp. 237{243.
Original Russian Text Copyright c 2003 by B.S. Kashin, S.J. Szarek.
English Translation Copyright c 2003 by MAIK \Nauka/Interperiodica" (Russia).
On the Gram Matrices of Systems
of Uniformly Bounded Functions
B. S. Kashin
1
2
and S. J. Szarek
Received May 2003
Abstract|Let A , N = 1 2 : : : , be the set of the Gram matrices of systems fe g =1 formed
N
j
N
j
by vectors e of a Hilbert space H with norms ke k
1, j = 1 : : : N . Let B (K ) be the set of
the Gram matrices of systems ff g =1 formed by functions f 2 L1 (0 1) with kf k 1(0 1) K ,
j = 1 : : : N . It is shown that, for any K , the set B (K ) is narrower than A as N ! 1.
More precisely, it is proved that not every matrix A in A can be represented as A = B + ,
where B 2 B (K ) and is a diagonal matrix.
j
j
j
H
N
N
j
j
j
N
L
N
N
N
In this paper we prove a theorem announced in 1]. First, we recall some well-known concepts
and problems and introduce the notation used in what follows.
As usual, RN , N 2 N , is the N -dimensional Euclidean space with the inner product h i and
Euclidean norm j j. Let
B N = fx 2 RN : jxj 1g
S N 1 = fx 2 RN : jxj = 1g:
For a set of vectors Z = fzi gpi=1 RN , denote by GZ the Gram matrix
GZ = fhzi zk igpi k=1 :
Let ON be the rotation group of RN . Clearly, for 2 ON and Z = fzi gpi=1 , we have
GZ = GZ :
(1)
Below, we consider the Hilbert space L2 (0 1) and the smaller space L1(0 1). Denote by ( ) the
inner product
Z1
f g 2 L2(0 1)
(f g) = fg d
0
where is the Lebesgue measure on (0 1).
Finally, for N = 1 2 : : : , we introduce an N -dimensional space of piecewise constant functions:
D = f 2 L2(0 1): f (x) = const = c for x 2 j q 1 j 1 j N :
N
j
N
N
The following problem arises naturally in analysis, geometry, and applied mathematics: Given
a set
Z = fz1 : : : zN g B N
(2)
1 Steklov
Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.
E-mail: [email protected]
2 Universit
e Paris VI, 4 Place Jussieu, 75252 Paris, France Case Western Reserve University, 10900 Euclid Avenue,
Cleveland, Ohio 44106, USA.
E-mail: [email protected]
227
228
B.S. KASHIN, S.J. SZAREK
of N vectors in RN (N = 1 2 : : : ), determine whether there exists a set ffj gNj=1 L1 (0 1) of
functions such that their uniform norms are bounded by an absolute constant and
(fj fk ) = hzj zk i
1 j k N:
(3)
Using the well-known inequality
(
)
1=2
C
ln
N
N
1
M x 2 S : kxk`N1 12
1
=
2
N
N
where C > 0 is an absolute constant, M is the normalized Lebesgue measure on the sphere S N 1 ,
and kxk`N1 = max1iN jx(i)j for x = fx(i)gNi=1 2 RN , we can immediately conclude that, for any
set of the form (2), there exists a rotation 2 ON such that
kzj k`N1 C ln1=2 (N + 1)
N 1=2
j = 1 2 : : : N:
(4)
Due to the obvious relationship between the Euclidean spaces RN and DN , we directly nd from (4)
that, for any set of vectors of the form (2), there exist functions ffj gNj=1 DN L1 (0 1) such
that (3) holds and
max kfj kL1 C ln1=2 (N + 1):
(5)
1j N
Estimate (5) is order-sharp. More precisely, if the vectors fzi gNi=1 form a 1-net (with respect to the
norm j j) on a sphere of a subspace L RN of dimension dim L c ln N , then each function set
ffj gNj=1 with property (3) satises
max kfj kL1 c1 ln1=2 N
(6)
1j N
(see, e.g., Lemma 3 in 1]).
It is of interest to nd conditions on the set (2) under which inequality (5) can be replaced by
a stronger estimate
kfj kL1 K
j = 1 2 ::: N
(7)
where K is an absolute constant.
The rst partial results of this kind were obtained by Mencho as early as the 1930s in connection with extending a system of functions dened on the interval q1 0] to a uniformly bounded
orthonormal system on q1 1] (see 2]). Note that the following question, raised by Olevskii 3,
p. 58], still remains open: Does there exist a function system ffj gNj=1 with properties (3) and (7)
if the Gram matrix of the set (2) satises
IN q GZ 0
(here, IN is the identity matrix of order N , and B 0 means that the quadratic form generated
by a symmetric matrix B is positive semidenite)?
The problems considered here are directly related to the classical results on the factorization of
linear operators in functional analysis. For example, the Grothendieck and Pietsch theorems p
imply
that, for any set Z of the form (2), there exists a function system F = ffj gNj=1 with kfj kL1 =2
for j = 1 : : : N and such that
GZ = GF q 0
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229
(for more detail, see 4, Ch. 5] and, in particular, Theorem 5.10 there). In view of this result, it is
natural to consider whether the Gram matrix of an arbitrary set (2) can be represented in the form
GZ = GF q is a diagonal matrix
where the functions in F satisfy estimate (7). This question was also raised by Megretski in
connection with applied problems in quadratic programming (see 5]). It turns out that the answer
to this question is negative.
Theorem. For N = 1 2 : : : there exists a set Z = Z (N ) of the form (2) such that any
function system ffj gNj=1 with the property
hzj zk i = (fj fk )
satis es the inequality
for 1 j < k N
(8)
max kfj kL1 c(ln N )1=4
1j N
where c > 0 is an absolute constant.
Proof. For N = 1 2 : : : , dene KN as the inmum of the numbers K such that, for any set
of vectors of the form (2), there exist functions ffj gNj=1 with kfj kL1 K , j = 1 2 : : : N , that
satisfy (8). We need to verify that KN c(ln N )1=4 . For N = 1 2 : : : , we also dene MN as the
inmum of the numbers M such that, for any set of vectors of the form (2), there exist functions
ffj gNj=1 with kfj kL1 M , j = 1 : : : N , that satisfy (8) and
hzj zj i (fj fj )
Let us verify the inequality
and then show that
K2N MN
MN c0(ln N )1=4
j = 1 2 : : : N:
(9)
N = 1 2 :::
(10)
c0 > 0 N = 1 2 : : : :
(11)
This will prove the theorem. To verify (10), for a given set Z of the form (2), we construct a set
of vectors
W = (w1 : : : w2N ) = (z1 z1 z2 z2 : : : zN zN ) B N B 2N
and (see the denition of KN ), for a given " > 0, nd functions g1 : : : g2N such that (gi gk ) =
hwi wk i if i 6= k and kgi kL1 K2N + " for i = 1 : : : 2N . Then, in particular, we have
jzk j2 = hzk zk i = hw2k
1
w2k i = (g2k
1
g2k ) kg2k 1 kL2 kg2k kL2
for k = 1 2 : : : N i.e.,
max kg2k 1 kL2 kg2k kL2 jzk j
k = 1 2 : : : N:
Hence, taking g2k 1 or g2k as fk , we ensure that condition (9), together with (8) and the estimate
kfk kL1 K2N + ", is satised. Thus, we have proved inequality (10).
Let us prove (11). Below, for a set of vectors fvj gNj=1 in a Euclidean space, we denote by
spanfv1 : : : vN g and fv1 : : : vN g? the linear span and its orthogonal complement, respectively.
We x a set of the form (2) (arbitrary for the time being) and an " 2 (0 1) and choose functions
ffj gNj=1 with
j = 1 ::: N
(12)
kfj kL1 MN + "
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B.S. KASHIN, S.J. SZAREK
so that (8) and (9) hold. Due to the canonical isometric embeddings RN R2N and B N B 2N ,
we can nd vectors uj , j = 1 : : : N , in R2N such that
uj 2 fz1 : : : zN g? j = 1 : : : N
and
huj uj i + hzj zj i = (fj fj )
hui uj i = 0 for i 6= j
(13)
j = 1 : : : N:
Setting wj = zj + uj , j = 1 : : : N , and taking into account (13), we obtain
hwi wj i = (fi fj )
8(i j )
(14)
which implies that the mapping
R2N spanfwj
'
j = 1 : : : N g q!
spanffj j = 1 : : : N g L2 (0 1)
is well dened by the equalities '(wj ) = fj , j = 1 : : : N , and is an isometry. Obviously, this
mapping can be extended to an isometry (also denoted by ') between R2N and a 2N -dimensional
subspace of L2 (0 1).
Let
gj = '(zj )
hj = '(uj )
j = 1 : : : N:
Then,
Moreover,
gj hj 2 L2
gj + hj = fj 2 L1(0 1)
(gi hj ) = 0 8(i j )
j = 1 : : : N:
(15)
(hi hj ) = 0 if i 6= j:
(16)
Let L = spanfg1 : : : gN g. Then, ' isometrically maps the subspace
E = spanfz1 : : : zN g R2N
onto L. The set fzj gNj=1 B N B 2N has been arbitrary thus far. Now, we specify zj , j = 1 : : : N ,
assuming (without loss of generality) that N is suciently large.
Let E RN R2N be a subspace of dimension ` = c0 ln N ]. Here, the absolute constant
c0 > 0 is chosen so small that there exists a 1-net V = fv g0=1 on the sphere SE (with respect to
the Euclidean norm j j), where the number of elements 0 satises
0 3` N 1=4
(17)
( x] stands for the integer part of x). Then, for the convex hull, we have
convfv g0=1 12 BE 21 (B 2N \ E ):
(18)
The required set Z = fzj gNj=1 of type (2) is constructed by repeating each element v of V s times,
s = N 3=4 ], and by supplementing (if necessary) the resulting set in an arbitrary manner to obtain
a system of N elements in BE . Using the notation introduced earlier, by virtue of (18), we conclude
that
(19)
convfgj gNj=1 12 f 2 L : kf kL2 1 21 B2L 12 (B2 \ L)
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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where B2 is the unit ball in L2 (0 1). Moreover, relations (12), (15), and (16) guarantee that
gj (MN + ")PL (B1)
j = 1 ::: N
(20)
where PL is the orthogonal projection from L2 (0 1) onto L and
B1 = f 2 L2(0 1): kf kL1 1 :
It follows from (20) and (19) that
B2L 2(MN + ")PL (B1):
(21)
In turn, (21) implies that
(22)
kgkL1 kgkL2 2(MN + ")kgkL1
for any g 2 L (the left inequality in (22) is true for any g 2 L2 ). Indeed (see (21)), if g 2 L with
kgkL2 = 1 can be represented as
g =f +h
kf kL1 2(MN + ") h ? g
then
0 = (g h) = (g g q f ) = 1 q (g f ):
This implies that kgkL1 kf kL1 1. Hence, kgkL1 2(MN + ")] 1 , which proves (22).
Thus, we have a subspace L of L2 (0 1) with dim L = ` = c0 ln N ] such that
(i) if g 2 L, then (22) is true
(ii) for each element ! of the 1-net on the sphere S2L = fg 2 L : kgkL2 = 1g, there exist
functions hj1 : : : hjs 2 L? satisfying (hjp hjq ) = 0 for p 6= q and such that
k! + hjp kL1 MN + "
p = 1 : : : s:
(23)
Moreover, for p = 1 : : : s, we have
khjp k2L2 = k! + hjp k2L2 q k!k2L2 (MN + ")2 q 1 (MN + ")2
(24)
(the set '(V ) should be used as ). It follows from (23) that
1 q
! + hj1 + : : : + hjs MN + "
(25)
s
L1
while the pairwise orthogonality of the functions hj and estimate (24) imply
1 q
hj1 + : : : + hjs MNp + " :
(26)
s
s
L2
It now follows from (25) and (26) that
! 2 (MN + ")B1 + MNps+ " B2
(27)
for all ! 2 . Inclusion (27) can be transferred to the convex hull of . Finally, we have (see
also (22))
8
1
>
< kgkL1 2(MN + ")] kgkL2
()
g 2 L kgkL2 1 =) >
: g 2 2(MN + ")B1 + 2 MNp + " B2 :
s
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B.S. KASHIN, S.J. SZAREK
Let us derive a lower estimate for MN from (). To this end, we need the following elementary
and easy-to-check result.
Lemma. Let g 2 L2(0 1) and f 2 L1(0 1) let > 0 and > 0 be such that
kgkL1 kgkL2 1
kf kL1 kf kL1 1:
Set
A = A(g ) = t : 3 jg(t)j 3
B = B (f ) = t : jf (t)j 2 :
Then,
Z
A
Z
B
jgj d 2
(A) meas A 3
jf j d 2
meas B 2 :
3
(28)
Let 1 : : : ` be an arbitrary orthonormal basis in L. Applying the above lemma with =
2(MN + ")] 1 to the functions r , r = 1 : : : `, from L2 (0 1), we nd the corresponding sets Ar ,
r = 1 2 : : : ` (see (28)). Let
X̀
= jr j Ar
r=1
where A denotes the characteristic function of a set A. Then,
Z1
1
kkL 6(MN + ")`
d 6(M ` + ") :
0
N
Applying the second part of the lemma to the function f = 6(MN + ")`]
36(MN + ")2 ] 1 , we nd a set B (0 1) with meas B 72(MN + ")2 ] 1 such that
X̀
`
jr (t)j 12(
M
N + ")
r=1
for t 2 B and, moreover,
Z X̀
`
jr (t)j d :
12(
M
N + ")
r=1
1
and =
(29)
B
It follows from (29) that there exists a subset B0 B with
meas B0 2 `+1 meas B 2 ` 6(M 1+ ")]2
N
and a set of signs "r = 1, r = 1 : : : `, such that
p
`
p1 X̀ "r r (t) t 2 B0 :
12(MN + ")
` r=1
Let
X̀
B2L 3 g(t) = p1 "r r (t):
(30)
` r=1
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
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2003
GRAM MATRICES OF SYSTEMS OF UNIFORMLY BOUNDED FUNCTIONS
Note that, by virtue of (30), every function f 2 2(MN + ")B1 satises
p
`
jg(t) q f (t)j 12(M + ") q 2(MN + ") for t 2 B0 :
233
(31)
N
Now, employing (31), we show that the inequality
1 c ln N ]1=4 = 1 `1=4
(MN + ") 10
0
10
contradicts () hence,
N N0
MN 101 c0 ln N ]1=4
(32)
which completes the proof of estimate (11). Indeed, inequality (32) implies that 21 c0 ln N ]1=4 .
Therefore, for each f 2 2(MN + ")B1 ,
2(MpN + ") :
kg q f kL2 (meas B0 )1=2 1 2 `=2 >
2
s
(33)
This contradicts the second relation in (). While verifying the last inequality in (33), we took into
account that s = N 3=4 ], 2` < 3` N 1=4 (see (17)), and, hence (see also (5)),
s > 20(C ln1=2 N )2 2` 4(MN + ")]2 2`
if the number N is suciently large.
The theorem is proved.
ACKNOWLEDGMENTS
The results of this paper were obtained during the visit of B.S. Kashin to the University Paris VI,
which was supported by the CNRS (France). B.S. Kashin also acknowledges the support of the Russian Foundation for Basic Research (project no. 02-01-00315) and the programs \Leading Scientic
Schools of the Russian Federation" (project no. NSh-1549.2003.1) and \Contemporary Problems
of Theoretical Mathematics" of the Russian Academy of Sciences, and S. Szarek acknowledges the
support of the National Science Foundation (USA).
REFERENCES
1. Kashin, B.S. and Szarek, S.J., The Knaster Problem and the Geometry of High-Dimensional Cubes, C. R. Acad.
Sci. Paris, Ser. 1, 2003, vol. 336, pp. 931{936.
2. Mencho, D., Sur les series de fonctions orthogonales bornees dans leur ensemble, Mat. Sb., 1938, vol. 3,
pp. 103{120.
3. Olevskii, A.M., Fourier Series with Respect to General Orthogonal Systems, Berlin: Springer, 1975.
4. Pisier, G., Factorization of Linear Operators and Geometry of Banach Spaces, Providence: Am. Math. Soc.,
1986 (CBMS Reg. Conf. Ser. Math., vol. 60).
5. Megretski, A., Relaxation of Quadratic Programs in Operator Theory and System Analysis, in Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Basel: Birkhauser, 2001, pp. 365{392
(Oper. Theory Adv. Appl., vol. 129).
Translated by I. Ruzanova
PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS
Vol. 243
2003