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 Kashin, 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 a n d S. J. Szarek 2
Received May 2003
Abstract—Let AN, N — 1,2,..., be the set of the Gram matrices of systems {ej}f=l formed
by vectors ej of a Hilbert space H with norms ||е^||# < 1, j = 1,..., N. Let BN(K) be the set of
the Gram matrices of systems {fj}f=i formed by functions fj G L°°(0,1) with Ц/^Цх^од) < if,
j = 1,... , iV. It is shown that, for any K, the set BN{K) is narrower than AN as N -> oo.
More precisely, it is proved that not every matrix A in AN can be represented as A = В -f Д,
where В G BN{K) and Д is a diagonal matrix.
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 G N, is the iV-dimensional Euclidean space with the inner product (•, •) and
Euclidean norm | • |. Let
BN = {xeRN:
S1*-1 = {x£RN:
\x\ < 1},
\x\ = 1}.
For a set of vectors Z = {zi}pi=l С М^, denote by Gz the Gram matrix
Gz =
{(zi,zk)}lkz=1.
Let 0N be the rotation group of R-^. Clearly, for a G 0N and aZ = {<T£i}f=i>
we
^ave
GaZ = Gz.
(1)
2
Below, we consider the Hilbert space I/ (0,1) and the smaller space L°°(0,1). Denote by (•, •) the
inner product
l
/,#GL2(0,1),
(f,g) = ffgdu,
о
where /x is the Lebesgue measure on (0,1).
Finally, for N = 1, 2 , . . . , we introduce an N~dimensional space of piecewise constant functions:
DN = {fE
L 2 (0,1): f(x) = const =
Cj
for x G ( ^
, j^j , 1 < j < N^ .
The following problem arises naturally in analysis, geometry, and applied mathematics: Given
a set
Z = {zu...,zN}cBN
(2)
Steklov Institute of Mathematics, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.
E-mail: [email protected]
2
Universite 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
KASHIN, SZAREK
of N vectors in RN (JV = 1,2,...), determine whether there exists a set {fj}f=1 С L°°(0,1) of
functions such that their uniform norms are bounded by an absolute constant and
(fj,fk)
= (zj,zk),
l<3,k'<N.
(3)
Using the well-known inequality
x
Ml^xeS»
: \\x\\E„>
Nl/2
j<-^,
where С > 0 is an absolute constant, M is the normalized Lebesgue measure on the sphere SN~l,
and ||a;||^jv = maxi<i<7v \x(i)\ for x = {E(»)}£LI S M.N, we can immediately conclude that, for any
set of the form (2), there exists a rotation a € 0N such that
\°*j\\t»<
OnWjN
»n/,
#1/2
+ l)
'.
J = l,2,...,iV.
(4)
Due to the obvious relationship between the Euclidean spaces RN and DJV, we directly find from (4)
that, for any set of vectors of the form (2), there exist functions {fj}f-i С Djy С L°°(0,1) such
that (3) holds and
max ||/7-||L°O < Cln 1 / 2 (iV + 1).
l<j<N
J
~
(5)
'
Estimate (5) is order-sharp. More precisely, if the vectors {zi}fL± form a 1-net (with respect to the
norm |-|) on a sphere of a subspace L C l ^ of dimension dimL > с In TV, then each function set
{fj}f=i w itli property (3) satisfies
max ||/7-||LOO >ciln 1 / 2 7V
(6)
(see, e.g., Lemma 3 in [1]).
It is of interest to find conditions on the set (2) under which inequality (5) can be replaced by
a stronger estimate
Ш1ь~<Я",
j- = l,2,...,JV,
(7)
where К is an absolute constant.
The first partial results of this kind were obtained by Menchoff as early as the 1930s in connec­
tion with extending a system of functions defined on the interval [—1,0] to a uniformly bounded
orthonormal system on [—1,1] (see [2]). Note that the following question, raised by Olevskii [3,
p. 58], still remains open: Does there exist a function system {fj}f=i with properties (3) and (7)
if the Gram matrix of the set (2) satisfies
IN
~ Gz > 0
(here, IN is the identity matrix of order iV, and В > 0 means that the quadratic form generated
by a symmetric matrix В is positive semidefinite)?
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 imply
that, for any set Z of the form (2), there exists a function system T = {/^}^=1 with Ц/JHL00 ^ v W ^
for j = 1 , . . . , N and such that
GZ = GT-A*
A>0
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(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 = Gjr — Д,
Д is a diagonal matrix,
where the functions in T 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 {/jHLi with the property
(*i,*k) = (fj,fk)
for
l<j<k<N
(8)
satisfies the inequality
i m^||/ j |Uco>c(ln^)
1
/'*,
where с > 0 is an absolute constant.
Proof. For N = 1,2,..., define Kp? as the infimum of the numbers К such that, for any set
of vectors of the form (2), there exist functions {/j}jLi w ^ h ll/jlU00 < K, j = 1?2,... ,7V, that
satisfy (8). We need to verify that K^ > c(lnTV)1/4. For N = 1, 2 , . . . , we also define Мдг as the
infimum of the numbers M such that, for any set of vectors of the form (2), there exist functions
{/j}jLi with \\fj\\Loo < M, j = 1 , . . . , N, that satisfy (8) and
(*ji*i)<Ujifj),
i = l,2,...,JV.
(9)
Let us verify the inequality
K2N>MN,
N = 1,2,...,
(10)
and then show that
Miv^c^lniV) 1 / 4 ,
c'X),
iV = l , 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 = (WU...,W2N)
С BN С B2N
= (zuzuz2,z2,...,zN,zN)
and (see the definition of KN), for a given e > 0, find functions p i , . . . ,g2N such that {gi^gk) —
(wi,Wk) if г ф к and lift Hz,*» < K2N + e for i = 1 , . . . , 2N. Then, in particular, we have
Ы
= (ZfaZk) = (w2k-l,W2k) = (S2fe-1,P2A;) < l|p2fc-l||L2 ' II#2/C||L2
for к = 1,2,... ,iV; i.e.,
max{||p2/c-i|lL2,||P2fc||L2} > \zk\,
к = 1,2, ...,N.
Hence, taking g2k-i от g2t as Д , we ensure that condition (9), together with (8) and the estimate
ll/fcllb00 ^ К2м + £, is satisfied. Thus, we have proved inequality (10).
Let us prove (11). Below, for a set of vectors {VJ}^=1 in a Euclidean space, we denote by
spanjt'i,... ,i>N} and {i>i,... ,^Ar} x the linear span and its orthogonal complement, respectively.
We fix a set of the form (2) (arbitrary for the time being) and an e € (0,1) and choose functions
{fi)U
with
ll/illb- <MN + e,
j = l,...,N,
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KASHIN, SZAREK
so that (8) and (9) hold. Due to the canonical isometric embeddings MN С R2N and BN С B2N,
we can find vectors UJ, j = 1 , . . . , iV", in Е2ЛГ such that
UJ E {zUy.,zN}L,
j = l,...,iV, •
(щ,щ)=0
for гф]
(13)
and
(UJ.UJ)
+ (zj,Zj) = ( / , , / , ) ,
j = 1 , . . . ,N.
Setting WJ = Zj + iij, j = 1 , . . . , iV, and taking into account (13), we obtain
{wi,v}j) = (fiJj)
V(i,j),
(14)
which implies that the mapping
R2N D s p a n { ^ , j = 1 , . . . , iV} А 8рап{/7-, j = 1 , . . . , N} С L 2 (0,1)
is well defined by the equalities <p(wj) = fj, j = 1,...,N, and is an isometry. Obviously, this
mapping can be extended to an isometry (also denoted by <p) between Ш2М and a 2iV-dimensional
subspace of L2(0,1).
Let
9j = <P(ZJ),
hj = <p(uj),
j =
l,...,N.
Then,
g^hjEL2,
й
+ ^ = /;еГ(0,1),
j = l,...,N.
(15)
Moreover,
(9i,hj) = 0 V(z,i),
(/*Л-)=0
if*#J.
(16)
Let I/ = span{^i,... , длг}. Then, </? isometrically maps the subspace
J57 = s p a n { z i , . . . , ^ } СЖ2АГ
onto L. The set {zJ}j^=1 С BN С J32iV has been arbitrary thus far. Now, we specify Zj1 j = 1 , . . . , JV,
assuming (without loss of generality) that N is sufficiently large.
Let E С R^ С R2N be a subspace of dimension £ = [colniV]. Here, the absolute constant
Co > 0 is chosen so small that there exists a 1-net V = {v^YJLi o n the sphere SE (with respect to
the Euclidean norm | • |), where the number of elements UQ satisfies
i/o < 3^ < Nl'A
(17)
([ж] stands for the integer part of x). Then, for the convex hull, we have
c o n v K } ? = 1 ^>\BE = \ (B2N П E).
(.18)
The required set Z = {ZJ}JL1 of type (2) is constructed by repeating each element vv of V s times,
s = [iV3/4], 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
conv{9j}?=1
D±{f
EL: \\f\\L2 < 1} = \ B% = \ (B2 П L),
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where B2 is the unit ball in L2(0,1). Moreover, relations (12), (15), and (16) guarantee that
^ • c ( M J V + e)P L (B 0 0 ),
j = l,...,N,
(20)
where P/, is the orthogonal projection from L2(0,1) onto L and
S o o = { / G L 2 ( 0 , l ) : ||/!U~ < 1}.
It follows from (20) and (19) that
B%c2(MN
+ e)PL(B00).
(21)
In turn, (21) implies that
\\9hi<\\9\\L><4MN
(22)
+ e)\\g\\Ll
for any g € L (the left inequality in (22) is true for any g G L2). Indeed (see (21)), if g £ L with
=
WQWL2
1 c a n be represented as
g = f + h,
||/||Loo < 2{MN + s),
h± g,
then
0 = ( 5 ,/i) = ( 5 , 5 - / ) = l - ( 5 , / ) .
This implies that ||#||Li • |I/IU°° > 1- Hence, ||#|| L i > [2(MN + e)]" 1 , which proves (22).
Thus, we have a subspace L of I/ 2 (0,1) with dimL = £ = [CQ InN] such that
(i) if g E L, then (22) is true;
(ii) for each element ш of the 1-net ft on the sphere S'f = {g £ L: \\д\\ь2 = 1}, there exist
functions hj!,..., hjs G LL satisfying (/ij , /ij ) = 0 for p ф q and such that
. \\w + hjp\\Loo < MN + e,
(23)
p = l,....,s.
Moreover, for p = 1 , . . . , s, we have
114 Hi* = II" + 4 Hi' - IMlb < (MN + e)2 - 1 < (M* + e)2
(24)
(the set </?(V) should be used as ft). It follows from (23) that
W
1
< Mjv + £,
+ T ( 4 + --- + 4 )
(25)
L°°
while the pairwise orthogonality of the functions hj and estimate (24) imply
(4 + --- + 4 )
L
2
<
MN + e
(26)
It now follows from (25) and (26) that
UE(MN+
e)B'o0 + M^±l
B2
(27)
Vs
for all ш G O. Inclusion (27) can be transferred to the convex hull of fi. Finally, we have (see
also (22))
\g\\Li >[2{Мн + е)]-г\\д\\Ь2,
9 €L,
\\g\\L2 < 1
g G 2{MN + e)B M + 2 ^ L ± £ £ 2 .
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KASHIN, SZAREK
Let us derive a lower estimate for MJJ from (*). To this end, we need the following elementary
and easy-to-check result.
Lemma. Let g G L2(0,1) and f 6 L°°(0,1); let a > 0 and /3 > 0 be such that
* < ЫЬ < Ы\ь* < i.
P<
II/IILI
< ll/IU~ < i.
Set
A = A(g,a) = {t: f < \g(t)\ < ^
,
В = B(f,/3) = {t: | / ( t ) | > \ } .
Then,
/Ы<^>|,
/х(А) = ш е а з Л > ( | )
,
(28)
A
measB > —
~ 2
/l/l^>f,
Let - 0 1 , . . . , ярi be an arbitrary orthonormal basis in L. Applying the above lemma with a =
_1
[2(MJV + e)]
to the functions фг, r = 1,... , £, from £ 2 (0,1), we find the corresponding sets A r ,
r = 1,2,...,£ (see (28)). Let
l
where x ^ denotes the characteristic function of a set A. Then,
l
Applying the second part of the lemma to the function / = Ф[6(М/у + e)l}~1 and /3 =
[36(Мдг + e ) 2 ] ~ \ we find a set Б С (0,1) with m e a s # > [72(MN + e)2}~1 such that
1
y u (
t
) i > _ i
I
(29)
for t € В and, moreover,
J52\fl>r(t)\dtl>
В
12(MN + e) '
^
It follows from (29) that there exists a subset BQ С В with
measJ5 0 > 2 - m m e a s B > 2~ f [6(MN + e)] 2
and a set of signs er = ± 1 , r = 1 , . . . , £, such that
1
-=^£r^.(i)
V ** -»•—-1
£
щет-
teBo
<30)
-
Let
1
£
^ Э ^ ) = - 7 =Х) е ^(*).
v«r=i
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Note that, by virtue of (30), every function / e 2(MJV + е)Дх> satisfies
W) - №\ > 12{MN
+ £)
- 2(MN + e) = 7
for t € So-
(31)
Now, employing (31), we show that the inequality
(MN + e) < 1 [ c 0 \nN}^ = 1 / V 4
(32)
contradicts (*); hence,
MN>^[cQlnN}^\
N>NQ,
which completes the proof of estimate (11). Indeed, inequality (32) implies that 7 > \ [colnTV]1/4.
Therefore, for each / e 2(MN + £)Дх>,
\\9 ~ / l b > 7(measB0)^ > \ 2~^ > 2ЛЦ+А .
(33)
This contradicts the second relation in (*). While verifying the last inequality in (33), we took into
account that s = [N3/% 2l < 3£ < N1/4 (see (17)), and, hence (see also (5)),
s > 20(Cln 1 / 2 N)2 • 2£ > [A(MN + e)]2 • 2l
if the number N is sufficiently 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 Scientific
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. Menchoff, D., Sur les series de fonctions orthogonales bornees dans leur ensemble, Mat. 56., 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