On the volume of the Set of Uniformly Bounded Polynomials
E. Gluskin, B.S. Kashin
Institut des Hautes Etudes Scientifiques
35, route de Chartres
91440 - Bures-sur-Yvette (France)
Septembre 1992
IHES/M/92/72
On the volume of the Set of Uniformly Bounded Polynomials
E. Gluskin, B.S. Kashin
Let a; be a positive weight function on the interval [—1,1]. Corresponding orthogonal
polynomials are denoted by pn(w).
Let Wn%w be the following subset of Rn :
n-1
Wn4W =
(xor - • , # n - i ) С
J2 хьрк(и)
k=0
We will estimate the volume of Wn^w.
The set Wn^w and the problem about its volume
appeared in Kashin's paper [K] in connection with his study of the smallest deviation from
zero of polynomials with integer coefficient (see also F. Amoroso [A]). Upr>er and lower
bounds of (volW n ? w ) ' n coinciding up to a multiplicative constant are obtained in this
paper for any weight function w of Szego class. The same lower bound was obtained in
[K] for weight functions bounded away from zero. Note that our method is simpler than
that of [K].
Let ej(j
= l , - " , n ) be the standard basis of R ' V A i be the Euclidean unit ball in
Un; £^o b e ^he space Rn equipped with the norm
Xj6j
i=i
= max \Xj
i<j<n
The fc-dimensional counterpart of the octahedron is denoted by i?*, that is
J9* = c o n v { ± e b - • • , ± e * } .
As the usual 8jj is a Kronecker symbol (6ij = 1 for i = j , otherwise 8ij = 0). Vn will
be linear space of all polynomials of degree < п. с, C\ and so on will be different universal
positive constants.
A linear transform Un —> Шк is identified with its matrix throughout the paper,
Lemma
(Vaaler, Ball). Let T be a linear transform from R n to R*, (к > n) and V =
VT С R n be the following set
1
У={.хеК п :||Т.г-||^ < l } .
Then the following inequalities hold
1 < (det(T*T))^2(vo\VT)1/n
< 2{k~n)l2n.
(1)
T h e left-hand side inequality was proved by Vaaler [V] and the right-hand side one by Ball
[B]. For our purpose the following weaker version of the lemma will suffice :
ci(k/n)-1/2
< ( d e t ( T * r ) ) 1 / 2 ( v o l VT)1/n
<
c2{k/n)
For t h e reader's convenience we will quote a simple reduction (1) to the classical
Santalo a n d inverse Santalo inequalities at the end of the paper.
Theorem.
Lxi-1,1).
Let a weight function w € ( - 1 , 1 ) be a non-negative
and ( l - ^ 2 ) 1 / 2 | l o g w ( i ) | €
Then
lmlw 1,n
^ -TnHI'Mw{t)Vrz¥)v&
Proof.
:
t2
Let /сп(гед) be the leading coefficient of polynomial pn(w).
Hence if w®
IS
another weight function then the set Wn'yW is transformed to WnyWo by a linear operator
with triangle matrix whose diagonal entries axe .K^WQ)/K^W).
SO
we have
If b o t h w a n d WQ satisfy conditions of the theorem, then from Szego's theorem (see [S],
12,1,2) threre e x i s t s . 1
hm
\ = exp I - —
I
log — — d t
T h e two last e q u a l i t i e s reduce the problem to the case of one single weight. We'll deal
with Tchebesheff's Weight function w0(t) = (1 -t2)~ll2.
Henceforth, Pk(w0) will be taken
as pn.
This a s y m p t o t i c formula holds under far weaker conditions in weight functions (see
Mathe, Nevai, Totik [M-N-T]. However, for our purpose Szego's result is sufficient.
2
Being a consequence of Bernstein's inequality,
с
HPIIC[-I,I]
for any polynomial p &Vn
P \ cos g )
^0™f2r
< IHIc[-i,i]
(2)
(see e.g. [Z]).
Vectors (fj G R n , J = 1, • • • ,4n will be defined as follows
"-1
/
о •
fc=0
The matrix with rows </>j will be denoted by T.
Let V С R n be the following set
У = {ж € R n
|Тж||, 4п < 1
The inequality (2) gl*res
Wn,w0 С V С с ^
(3)
л
It follows from (1)
( C l / 2 ) n < ( d e t ( T * T ) ) 1 / 2 vol К < (2c 2 )".
(4)
It is well known that for any p £ Vm the following equality holds
^
'
--x
(
mJ =V Л
P COS
1m
f
,
'
2m
K C S }< =
Г
° * *' V /_,
. .
P(S)
(is
.V /I i-—s ' 2
So for entries of the matrix T*T we have :
An f1
ds
An
In particular T*T = (4п/тг) п and the theorem follows from (3) and (4). 1
P r o o f of t h e inequality (1) We use the following result. Let К be a convex symmetric
body with center of symmetry at zero and Л' 0 be its polar body :
К0 = {х G R n : (х,у)
< 1 for any у G К}.
Then the following inequalities hold
l < I ^ „ ^ <
- volDri
c
vo\Dn
n
(5)
o
v
-
;
This remarkable inequality was obtained by Santalo [S] (left-hand side) and by Bourgain
and Milman [B-M] (right-hand side).
Let A be the collection of all n-element subsets of the set {1, 2, • • • Д } . For a G A let
Ta be a n x n matrix with rows Te^ for г £ a.
By definition of V
VQ = T*5f.
Therefore by Caratheouory's theorem
V° = \J TaB«.
аел
It follows that
max volTaB?
a£A
< vol V° < Y " volTQB?.
*
—J
a EA
From Binet-Cauchy formula
£(уо1ТаВГ)2 = (volBr)2^(det(Ta))2 = ( v o l ^ f det(TT).
Inequality (1) now follows from (5), (6) and standard estimate
f k\
/ei:4
card A = | ' I <
n /
Vn
n
(6)
REFERENCES
[A] F. Amoroso. Sur le diametre transfini entier d'un intervalle reel, Annales de PInstitut
Fourier, vol 40, N 4 (1990), 885-912.
[B] К.М. Ball. Volumes of sections of cubes and related problem, Israel Seminar (G.A.F.A.)
1988, Springer-Verlag, Lecture Notes Ц1376, (1989),251-260.
[B-M] J. Bourgain and V.D. Milman. New volume ratio properties for convex symmetric
bodies in R n , Inventiones Math. 88 (1987), 319-340
[K] B.S. Kashin, Algebraic Polynomials with Integer coefficients Deviating Little from
zero on an Interval, Mat. Zametki, vol. 50, N 3 (1991) 58-67 (Russian) (English
transl. in Math. Notes of Acad. Sci. USSR, 50, N3 (1992), 921-927).
Vf-N-T] A. Mate, P. Nevai, V. Totik, Extension of Szego?s theory of orthogonal polynomials.
II. Constructive Approximation, 1987, 3:51-72.
[S] F. Szego. Orthogonal Polynomials. 1975 (4 th edn), AMS Colloquium Publications^
Vol. 23. Providence, RI : AMS.
[Sa] L. Santalo. Un invariant aim para los cuerpos convexas del espacio de n dimensiones.
Portugal Math. 8 (1949), 155-161.
[V] J.Do Vaaler. A geometric inequality with applications |o linear forms. Pacific J. Math.
83 (1979), 543-553.
[Z] A. Zygmund. Trigonometric series, Vol. II (2nd edn), Cambridge University Press,
1959.
E. Gluskin
The Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, ISRAEL
B.S. Kashin
V.A. Steklov Mathematics Institute
Academy of Science of the USSR
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