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Markov-Bernstein type inequalities under Littlewood-type coefficient constraints
Indag.
Mathem.,
Ii (2), 159-172
N.S.,
Markov-Bernstein coefficient
type inequalities
under
Littlewood-type
constraints*
by Peter Borwein Depurtment
June 26,200O
and Tam&
r!f Mathemutics
Erdblyi
and Statistics,
Simon Fraser University,
Burnahy,
V5A IS6, B.C.,
Canada emuil: phor~~einb’cecnz.sfu.ccr Department
of Muthematics,
emrril: terdelyi@math.
Communicated
Texas A&M
Universit):
College Station,
Texas 77843, USA
tanwedu
by Prof. J. Korevaar
at the meeting
of October
25, 1999
ABSTRACT
Let 3;, denote
the set of polynomials
the collection
of polynomialsp
p(.x) = 5
j=“,
a,x’5
where nr is an unspecified We establish absolute
of degree at most n with coefficients
I4
= 1.
nonnegative
the right Markov-type
constants
from { -1,O.
1). Let G,,,be
of the form
Ia,15 1.
integer
not greater
inequalities
than n.
for the classes Ffl and Q,, on [O. 11. Namely
there are
Ct > 0 and CZ > 0 such that
Clnlog(n
+ 1) 5
llP’ll[O.I] max 5 Cplog(n Of/JET,’ IIPll,o.,,
lP’(l)l max -5 OfpeT,, llPll,O,,,
+ 1)
and llP’ll[O.,l
b’(l)1 Cln’i2 < max -C max 5 C2rGi2. - o+PtG llPll[O.,~ - OfpEG IIPll[O.I]
It is quite remarkable factor
*Research under doctoral
that the right
Markov
factor
for F,,,. We also show that there are absolute
supported
Grant
in part by the NSERC
No. DMS-9623156
Fellow of the Danish
of Canada
(T. Erdelyi). Research
for & is much larger constants
Council
Research
(P. Borwein) conducted
at University
than the right
Markov
Cl > 0 and C2 > 0 such that
and by the NSF of the USA while an International
of Copenhagen
Post-
(T. Erdelyi).
159
+ 1) 5
Clnlog(n
max -V(l)1 o+PeL IlPll[O.,,So?%,,
IIpl(lio.ll< C*nlog(n + l), llPll,O.,]-
where L, denotes the set of polynomials of degree at most n with coefficients from { -1,l). polynomials p E F := UT=0.F, with (p(O)I = 1 and for y E [0,1) the Bernstein-type inequality
For
is also proved with absolute constants Ct > 0 and C2 > 0 This completes earlier work of the authors where the upper bound in the first inequality is obtained.
Littlewood had a particular fascination with the class of polynomials with coefficients restricted to being in the set { - 1,0, 1). See in particular [22] and the many references included later in the introduction. Many of the problems he considered concerned rates of possible growth of such polynomials in different norms on the unit circle. Others concerned location of zeros of such polynomials. The best known of these is the now solved Littlewood conjecture which asserts that there exists an absolute constant c > 0 such that
ak
exp(i&t)
dt > Clog n
whenever the ]ak] > 1 and the exponents xk are distinct integers. (Here and in what follows the expression ‘absolute constant’ means a constant that is independent of all the variables in the inequality). We are primarily concerned in this paper with establishing the correct Markov-type inequalities on the interval [0, l] for various classes of polynomials related to these Littlewood problems. One of the notable features is that theses bounds are quite distinct from those for unrestricted polynomials. In this paper n always denotes a nonnegative integer. We introduce the following classes of polynomials. Let
Pn:=
p:p(X)=j~oajX’,
I
ajE[w
i
denote the set of all algebraic polynomials of degree at most n with real coefficients. Let pi :=
p : p(x) =
j,$
ajx',
aj
denote the set of all algebraic polynomials coefficients. 160
1
E @
i
of degree at most n with complex
Let JJ :P(X)
.Fn I=
=jeoL7jXj,
Uj E
{-l,O,
1)
(
1
denote the set of polynomials {-l,O,l). Let
of degree at most n with coefficients
Cc,:=
p:p(X)=j$oLZjXj,
from
ajE{-l,l}
i
1
denote the set of polynomials of degree at most n with coefficients from { - 1, 1). (Here we are using 13, in honor of Littlewood.) Let K, be the collection of polynomials p E pf, of the form P(x) = j.$O ajx’3
laOI = 1,
lajlI 1.
Let Gn be the collection of polynomials p E Fn of the form
where m is an unspecified nonnegative integer not greater than n. Obviously
The following two inequalities are well known in approximation theory. See, for example, Duffin and Schaeffer [12], Cheney [9], Lorentz [24], DeVore and Lorentz [ll], Lorentz, Golitschek, and Makovoz [25], Borwein and Erdelyi [4]. Markov Inequality. The inequality
IIP’II~-~,~~ I n211Pll~-1,1~ holds for every p E P,.
Bernstein Inequality. The inequality
[P’(Y)15 J-p--+ IIPll,-,,I] holdsforeveryp
E P,andy
E (-1,l).
In the above two theorems and throughout mum norm on A C R.
the paper 11.IIA denotes the supre-
161
Our intention is to establish the right Markov-type inequalities on [0, I] for the classes &, C,, X,, and &,. We also prove an essentially sharp Bernstein-type inequality on [0, 1) for polynomialsp E 3 := UT=0 3n with (p(O)1= 1. For further motivation and introduction to the topic we refer to Borwein and Erdelyi [5]. This paper is, in part, a continuation of the work presented in [5]. The books by Lorentz, Golitschek, and Makovoz [25], and by Borwein and Erdelyi [4] also contain sections on Markov- and Bernstein-type inequalities for polynomials under various constraints. The classes 3,,, ic,,, and other classes of polynomials with restricted coefficients have been thoroughly studied in many (mainly number theoretic) papers. See, for example, Beck [l], Bloch and Polya [2], Bombieri and Vaaler [3], Borwein, ErdClyi, and Kos [5], Borwein and Ingalls [7], Byrnes and Newman [8], Cohen [IO], Erdiis [ 131,Erdiis and Turan [ 141,Ferguson [15], Hua [i6], Kahane [17] and [18], Konjagin [19], Korner [20], Littlewood [21] and [22], Littlewood and Offord [23], Newman and Byrnes [26], Newman and Giroux [27], Odlyzko and Poonen [28], Salem and Zygmund [29], Schur [30], and Szegii [31]. For several extremal problems the classes 3U tend to behave like G,,,.See, for example, Borwein, Erdelyi, and Kos [5]. It is quite remarkable that as far as the Markov-type inequality on [0, l] is concerned, there is a huge difference between these classes. Compare Theorems 2.1 and 2.4. 2. NEW
RESULTS
Theorem 2.1. There are ubsolute constants
qnlog(n
+ 1) <
q > 0 and c2 > 0 such that
max -IP’(l)l OfPE-%lIPlIp,, < dz%
Theorem 2.2. There are absolute constants
cinlog(n + 1) I
cl > 0 and c2 > 0 such that
max -(p’(l)1 ofPEtfx
IlP’lljO.,]
llPll[O.l] ’ OEzn
Theorem 2.3. There are absolute constants
cinlog(n + 1) I
~
llPlliO.I]
I cznlog(n + 1).
cl > 0 and c2 > 0 such that
max -IP’(l)l OfPEG IlPll[0.,] 5 G%n
Theorem 2.4. There are ubsolute constants
qn”i2 5
lIP’llj0 I] I czn log@ + 1). IIPll[O.l,
d
IIP’II[O.I] 5 cznlog(n + I). llPll,O.,]
___
cl > 0 and Q > 0 such that
IIP’ll[OI]
max [p’(l)1 O#pt& l]Pll[O.,]< o?%x%~
‘/2 I c21E-
The following theorem establishes an essentially sharp Bernstein-type equality on [0,1) for polynomials p E F := Urzo _?jjwith /p(O)1= 1. 162
in-
Theorem 2.5. There are absolute constants cl > 0 and c2 > 0 such that 2 Cl log 1 -Y
( 1
1- Y
< max IIP’ll[O,.r]< cz 10:(125) - ,;;,t,
llPll[o.r] -
for every y E [0,1). Our next result polynomials
is an essentially
sharp
Bernstein-type
inequality
on [0,1) for
p E L := Ur= 0 C,.
Theorem 2.6. There are absolute constants cl > 0 and c2 > 0 such that
.for every y E [0, 1). Our final result is an essentially polynomials p E K := U,“=, Ic,.
sharp
Bernstein-type
inequality
on [0,1) for
Theorem 2.7. There are absolute constants cl > 0 and cp > 0 such that cr log
2 1-Y
( >
I ~~~ l1;6//;>J; < c2lo$+J
1 -Y
0.1 -
for every y E [0, 1). on [0,1) for polynomials p E G := lJ,“=, 8, is also there is a gap between the upper bound in [5] and
A Bernstein-type inequality established in [5]. However, the lower bound 3. LEMMAS
we are able to prove at the moment.
FOR THEOREM
To prove Theorem
2.4
2.4 we need several lemmas.
The first one is a result from [6].
Lemma 3.1. There are absolute constants q > 0 and c4 > 0 such that ew(-c3J;;)
I of;fG
We will also need a corollary
”
IIPII~~.~~ 5 ,j;fs
of the following
Hadamard Three Circles Theorem.
Supposef
n
IIPII~~,~~ 5 ew(-c4JtE).
well known
result.
is regular in
{z E C : rI 5 121< rz}. For r E [rj, rz]. let 163
log(rdr1)
log
M(r)
Note that the conclusion written
I
b(r2lr)
log
of the Hadamard
M(v)
+ hdrlrl)
log
M(r2).
Three Circles Theorem
can be re-
as
log M(r) I
logM(u) +
logcrirl) (i0gM(r2) - logM(rt)).
log(r2~rl)
Corollary 3.2. Let cr E R . Suppose 1 < cy < 2n. Suppose f is regular inside and on the ellipse A,,, with foci at 0 and 1 and with major axis [ - 9, 1 + z]. Let Bn,a be the ellipse with foci at 0 and 1 and with major axis [ - A, 1 + A]. Then there is an absolute constant cg > 0 such that
Proof. This follows from the Hadamard
Three Circles Theorem with the substitution w = i (z + z-‘) + ‘i. See also the remark following the Hadamard Three CirclesTheorem, which is applied with the circles centered at 0 with radii with a suitable and r-2:= 1 + fl cx n, respectively, rl := 1, r := 1 + cdm,
choice of c.
0 cr > log(n + 1). Then there is
Lemma 3.3. Let p E Gn with I[pllro,, i =: exp(-a), an absolute constant cg > 0 such that _y;x (P(z)/ 2 c6 $$
-
“.LI
k(z>l~
where B,,, is the same ellipse as in Corollary 3.2. Proof. Note that jjp/j,0,1I 2 lp( l/4)] 2 4+ for every p E G,,. Therefore (log4)n. Our assumption onp E G,, can be written as ,:$;)I”8 Also,p
[P(z)1 = -o.
E & and z E A,,, imply that logb(z)I
5 log((n+
I)(1 +fjn+‘)
~log(n+l)+(n+1)~5log(n+1)+2aI3c~. Now the lemma
follows from Corollary
3.2.
0
Lemma 3.4. There is an absolute constant CI > 0 such that
IIP’ll[O,l] 5 c7an lIPll,O,,] for everyp E Gnwith ~~p~~~O,ll = exp(-cr) 164
I (n + 1)-l.
a 5
Proof. This follows from Lemma 3.3 and the Cauchy Integral Formula.
Note that for a sufficiently large absolute constant c > 0, the disks centered at y E [0,1] with radius l/(con) are inside the ellipse B,,, (see the definition in Corollary 3.2). 0 Lemma 3.5. There is an absolute constant cs > 0 such that
Proof. Applying Corollary 3.2 with (Y= log(n + 2), we obtain that there is an absolute constant cg > 0 such that
for everyp E Gflwith IIpljro.Ii> (n + 2)-l. To see this note that max log Ip( ZE[O,l]
2 - log(n + 2)
and zy;x log no
Ip(
cc
I log n 1 -t
log(n + 2) n
n
H
<
2 log(n + 2).
Now the Cauchy Integral Formula yields that
IIP’II~~,~~ i clonlodn
+
lM~o,l~
with an absolute constant CIO> 0. Note that for a sufficiently large absolute constant c > 0, the disks centered at y E [0, I] with radius l/(cn log(n + 2)) are inside the ellipse Brt,logcn + 2) ( see the definition in Corollary 3.2). 0 4. PROOF
OF THE
THEOREMS
Proof of Theorem 2.1. The upper bound of the theorem was proved in [5]. It is
sufficient to establish the lower bound of the theorem for degrees of the form N = 16”(n + I), n = 1,2,. . .. This follows from the fact that if we have polynomials PN
E 3N,
N =
16”(n + l),
n== 1,2,...
,
showing the lower bound of the theorem with a constant c > 0, then the polynomials
QN :=
P16”(~+1)
show the lower bound the largest integer for theorem for the values n 2 1 and let T,, be the
E FN,
N = 1,2,...
,
of the theorem with the constant c/1024 > 0, where n is which 16”(n + 1) < N. To show the lower bound of the N = 16”(n + l),n = 1,2,. . ., we proceed as follows. Let Chebyshev polynomial defined by 165
cosize,
T,(x) =
ThenlITnIl,-I.,,=
1. Denote
x =
coso,
f9E [OJ].
the coefficients
of T, by ak = ak,n, that is,
II
T,(x) = c
ak-.xk
k=O
It is well known
that the Chebyshev
polynomials
T,l satisfy the three-term
re-
cursion
T,,(x) = ~xT,_~(x)
- T,,_*(x).
Hence each ak is an integer
and, as a trivial
bound
for the coefficients
of T,, we
have (4.1)
bkl 5
k=O,l,...,
3",
n.
Also, either a0 = 0 or a0 = f 1. Let A := 16” and let
p,,(X)
:=
sign(ak)““fi ’.)k +.i j=O
2
X=0
We will show that P,, gives the required lower bound (with n replaced N := 16”(n + 1) in the theorem). It is straightforward from (4.1) that FN
with
by
N := 16”(n + 1)
(4.21
p, E
Observe
that for x E [0,11,
=
k$osign(ax)l”i’Y$j’ (x’ - 1) j-0
=
k~,sign(ak)xAk(l-x~~~l+x+x2+~ j=l < _
2 k=l
lakl* - lakl 2
Hence, for x E [0, 11,
IP,r(x)(5 IT&“)1 + IPn(x)- G(xA)II 1 + 1 = 2. We conclude (4.3)
that
Ilpnll[o.,] 5 2.
Let Qn,~(x) := m(b). 166
Then
~k,~(l)
= AT:(l)
= An2.
Now
(4.4)
e:‘,(l) = 5 sign(ar)“5’ (Ak +j) !i=l ,j = 0 n
=
Ql,A(l) + C sign(a) k=l
IQ-l
C j
j=o
Thus (4.2), (4.3), and (4.4) give the lower bound
of the theorem.
0
Proof of Theorem 2.2. The upper bound of the theorem follows from Theorem 2.1. To show the lower bound we modify the construction in Theorem 2.1 by replacing the 0 coefficients in P,, by il coefficients with alternating signs. We use the notation of the proof of Theorem 2.1. Let R, E CN be the polynomial arising from P, in this way. Recall that N := 16”(n + 1). Then, using the fact that R,, - P,, is of the form R,(x)
- P,(x)
= ztc (-l)‘xk,, i= I
we have II& - P,,ll~~.,l <_ 1. Combining (4.5)
this with (4.2), we obtain
l1R~ll,0.,] 5 IIfYljo.,, + I/& - W,o., , 5 2 + 1 =
On the other hand, using the I(R,, - P,,)‘(l)\ < N. This, together (4.6)
0 5 k, < k2 < ‘. . < k,, 5 N,
R:,(l)
2 P:,(l)
form of R, - P, with (4.4) gives
- I(Rn - Pn)‘(l)i
for n > 4. Now (4.5) and (4.6) together the lower bound
of the theorem.
> G16”n2
3.
given
above,
we have
- 16”(n + 1) > 616”n’
with R, E CN and N := 16”(n + l), give
0
= 1, the upper bound of the theProof of Theorem 2.3. Since JJpII,o,,I 2 Ip( orem follows as a special case of Lemma 3.5. The lower bound of the theorem follows from the lower bound in Theorem 2.2. 0 Proof of Theorem 2.4. To see the upper bound, note that Lemma 3.1 implies Q < c,n’iz in Lemma 3.4. So the upper bound of the theorem follows from Lemmas 3.4 and 3.5. Now we prove the lower bound of the theorem. Let k be a natural number and let Q,,(x) :=
.%?+‘rk(,@) =
;;$+T b,x’, 167
where Tk is, once again, the Chebyshev polynomial defined by T/&X) = Then
(4.7)
x = cost),
COS no,
0 E [0,7r].
IIQnllp,lI = 1 and Q;(l)=k3+n+1,
while, the coefficients bj of Q,, satisfy (4.8)
j=O,l,...,n+
IbjI < 3k,
1 +k2
(see the argument in the proof of Theorem 2.1). By Lemma 3.1 there is a polynomial R, E Fn such that
IIRnll~o.l~ I ew(-w@ Let ntltk? Pn :=
R, + exp(-c4fi)Qn
ajxj.
C
=:
j=O
Then (4.9)
IIM~o.~15 2expkc4fi).
It follows from (4.8) that (aj( < exp(-c4fi)3k
for eachj > n + 1. From now on
let (4.10)
k:=
s L
, 1
which implies that (UjJ5 1 for each j 2 n + 1, while Uj E { - 1, 0,1) for each j < n. Hence (4.11)
p, E
n +
%+l+kz,
1 + k2 I cqz,
with an absolute constant cs. Note that if n is large enough, then R;(l) = 0. Otherwise, as an integer, IRL(l)/ would be at least 1 and Markov’s inequality would imply that 1 5 IRi(l)I 5
2~211&llio,lI5 2n2exp(--C4\/;;),
which is impossible for a large enough n, say for n > cg. Hence, combining (4.9), (4. lo), and (4.7), we conclude P;(l) = R;(l) +exp(-cdfi)QL(l) 1
ew(-c4J;;)w3/2
2
= exp(-c4fi)(k3
+n+
1)
~n3/211Pn~~10,11
for every n 2 c6 with an absolute constant c-i > 0. This, together with (4.11) 0 finishes the proof of the lower bound of the theorem.
168
Proof of Theorem 2.5. The upper bound of the theorem was proved in [5]. To
show the lower bound we proceed as follows. Let P,, be the same as in the proof of Theorem 2.1. Throughout this proof we will use the notation introduced in the proof of Theorem 2.1. We will show that P,, gives the required lower bound with a suitably chosen even n depending on y. If n := 2m is even, then (4.12)
Ipn(0)I = 1,
and
p, E 3N
with
N := 16”(n + 1).
Recall that by (4.3) we have IIPn 11io,, < 2. As in the proof of Theorem 2.1, let A := 16” and Q”,A(x) := m(xA). F or n 2 no, the Chebyshev polynomial T, has a zero S in [e-*, e-l]. In particular, 7’,(S) 2 n. Let n E N be the largest even integer for which (4.13)
SIIA = b16-” 5 e-16-” < y.
Without loss of generality we may assume that n L no, otherwise p E 3 defined by p(x) := 1 + x shows the lower bound of the theorem. Note that there are absolute constants c3 > 0 and c4 > 0 such that 16” 2 c3(1 - y)-’ and hence
(4.14)
16”n 2
Therefore Q;.A(6”A) = AS CA - ‘)lAT,‘(6) 2 An6 = 16”nS 2 em216”n.
Observe that 6 E [eM2,e-‘1 and 0 Sj 2 lak] - 1 5 A (see the notation troduced in the proof of Theorem 2.1) imply
in-
and
> -2j.
We conclude that (4.15)
I(Ak +j)S (Ak+j- 1)/A _ A/&A”- 1)lAl 5 2j.
Now, using the definition of P,, (see the proof of Theorem 2. l), (4.13), (4.15), and (4.14), we obtain 169
2
( >
ct log
-
1-Y I-Y
>
for every n > 2 with an absolute constant cl > 0. This, together (4.12) gives the lower bound of the theorem. 0
with (4.3) and
Proof of Theorem 2.6. The upper bound of the theorem
follows from Theorem 2.5. To show the lower bound we modify the construction in Theorem 2.5 by replacing the 0 coefficients in P,, by fl coefficients with alternating signs. We use the notation of the proof of Theorems 2.1 and 2.5. Let R, E LN be the polynomial arising from P,, in this way. As in the proof of Theorem 2.2, we have llRn - P,211,0,,l 5 1, and combining this with (4.3), we obtain (4.16)
llR~lII,,, I 5
iV’nll,o.,, + II& - f’nll,o., j I 2 + 1 = 3.
On the other hand, for a E [0, l), we have (4.17)
I(& - P,)‘(a)1
L var;xf,(x),
wherefu(x) := x& ‘. Now it is elementary on [ 1,~) contains two monotone pieces, and
calculus
to show that the graph of,fU
,limX_Mx) = 0.
Hence
with an absolute (4.18)
constant
I(& - C)‘(a)1
Now let y E that
[0,l). By the proof of Theorem
ICKa)l2
with (4.17) yields
< 2.
Ct log
170
c4 > 0. This, together
2
( > ~
1-Y
l-y
2.5, there exists an a E [ 0, y] such
Combining this with (4.18), we deduce
2 ci log ___ l-y L. 1 -y
( > --
c5
c-4
l-y>
2 log ( 1-Y > 1 -Y
for y E [yo, 11, where c5 > 0 and yo E [0,l) are absolute constants. This, together with (4.16) and R, E CN gives the lower bound of the theorem for y E [yo, 11. If y E [0, yo), then the trivial example P(X) := 1 + x shows the lower bound of the theorem. 0 Proof of Theorem 2.7. The upper bound of the theorem was proved in [5]. The
lower bound of the theorem follows from the lower bound in either Theorem 2.5 or Theorem 2.6. 0 5
ACKNOWLEDGMENT
We thank Fedor Nazarov for suggesting the the crucial idea for the proof of the lower bounds in Theorems 2.1 and 2.2. REFERENCES
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15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 2s. 26. 27.
28. 29. 30. 31.
Ferguson, 0. - Le Baron Approximation by Polynomials with Integral Coefficients. Amer. Math. Sot., Rhode Island (1980). Hua, L.K. - Introduction to Number Theory. Springer-Verlag, Berlin-Heidelberg-New York (1982). Kahane, J-P. - Some Random Series of Functions, vol. 5. Cambridge Studies in Advanced Mathematics, Cambridge, Second Edition (1985). Kahane, J-P. Sur les polynomes a coefficients unimodulaires. Bull. London Math. Sot. 12, 321-342 (1980). Konjagin, S. - On a problem of Littlewood. Izv. Acad. Nauk SSSR, ser. mat. 45, 2, 243-265 (1981). Korner, T.W. -On a polynomial of J.S. Byrnes. Bull. London Math. Sot. 12,219-224 (1980). Littlewood, J.E. - On the mean value of certain trigonometric polynomials. J. London Math. Sot. 36,307-334 (1961). Littlewood, J.E. - Some Problems in Real and Complex Analysis. Heath Mathematical Monographs, Lexington, Massachusetts (1968). Littlewood, J.E. and A.C. Offord -On the number of real roots of a random algebraic equation, II. Proc. Cam. Phil. Sot. 35133-148 (1939). Lorentz, G.G. - Approximation of Functions. 2nd ed. Chelsea, New York, NY (1986). Lorentz, G.G., M. von Golitschek and Y. Makovoz - Constructive Approximation, Advanced Problems. Springer, Berlin (1996). Newman, D.J. and J.S. Byrnes The L4 norm of a polynomial with coefficients fl. MAA Monthly 97, 4245 (1990). Newman, D.J. and A. Giroux - Properties on the unit circle of polynomials with unimodular coefficients. In: Recent Advances in Fourier Analysis and its Applications (J.S. Byrnes and J.F. Byrnes, Eds.). Kluwer, pp. 79-81 (1990). Odlyzko, A. and B. Poonen - Zeros of polynomials with 0,l coefficients. Ens. Math. 39, 3177348 (1993). Salem, R. and A. Zygmund - Some properties of trigonometric series whose terms have random signs. Acta Math 91, 254-301 (1954). Schur, I. - Untersuchungen iiber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys.Math. Kl., 4033428 (1933). Szegii, G. - Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys-Math. Kl., 86-98 (1934).
(Received June 1999, revised September 1999)
172
Mathem.,
Ii (2), 159-172
N.S.,
Markov-Bernstein coefficient
type inequalities
under
Littlewood-type
constraints*
by Peter Borwein Depurtment
June 26,200O
and Tam&
r!f Mathemutics
Erdblyi
and Statistics,
Simon Fraser University,
Burnahy,
V5A IS6, B.C.,
Canada emuil: phor~~einb’cecnz.sfu.ccr Department
of Muthematics,
emrril: terdelyi@math.
Communicated
Texas A&M
Universit):
College Station,
Texas 77843, USA
tanwedu
by Prof. J. Korevaar
at the meeting
of October
25, 1999
ABSTRACT
Let 3;, denote
the set of polynomials
the collection
of polynomialsp
p(.x) = 5
j=“,
a,x’5
where nr is an unspecified We establish absolute
of degree at most n with coefficients
I4
= 1.
nonnegative
the right Markov-type
constants
from { -1,O.
1). Let G,,,be
of the form
Ia,15 1.
integer
not greater
inequalities
than n.
for the classes Ffl and Q,, on [O. 11. Namely
there are
Ct > 0 and CZ > 0 such that
Clnlog(n
+ 1) 5
llP’ll[O.I] max 5 Cplog(n Of/JET,’ IIPll,o.,,
lP’(l)l max -5 OfpeT,, llPll,O,,,
+ 1)
and llP’ll[O.,l
b’(l)1 Cln’i2 < max -C max 5 C2rGi2. - o+PtG llPll[O.,~ - OfpEG IIPll[O.I]
It is quite remarkable factor
*Research under doctoral
that the right
Markov
factor
for F,,,. We also show that there are absolute
supported
Grant
in part by the NSERC
No. DMS-9623156
Fellow of the Danish
of Canada
(T. Erdelyi). Research
for & is much larger constants
Council
Research
(P. Borwein) conducted
at University
than the right
Markov
Cl > 0 and C2 > 0 such that
and by the NSF of the USA while an International
of Copenhagen
Post-
(T. Erdelyi).
159
+ 1) 5
Clnlog(n
max -V(l)1 o+PeL IlPll[O.,,So?%,,
IIpl(lio.ll< C*nlog(n + l), llPll,O.,]-
where L, denotes the set of polynomials of degree at most n with coefficients from { -1,l). polynomials p E F := UT=0.F, with (p(O)I = 1 and for y E [0,1) the Bernstein-type inequality
For
is also proved with absolute constants Ct > 0 and C2 > 0 This completes earlier work of the authors where the upper bound in the first inequality is obtained.
Littlewood had a particular fascination with the class of polynomials with coefficients restricted to being in the set { - 1,0, 1). See in particular [22] and the many references included later in the introduction. Many of the problems he considered concerned rates of possible growth of such polynomials in different norms on the unit circle. Others concerned location of zeros of such polynomials. The best known of these is the now solved Littlewood conjecture which asserts that there exists an absolute constant c > 0 such that
ak
exp(i&t)
dt > Clog n
whenever the ]ak] > 1 and the exponents xk are distinct integers. (Here and in what follows the expression ‘absolute constant’ means a constant that is independent of all the variables in the inequality). We are primarily concerned in this paper with establishing the correct Markov-type inequalities on the interval [0, l] for various classes of polynomials related to these Littlewood problems. One of the notable features is that theses bounds are quite distinct from those for unrestricted polynomials. In this paper n always denotes a nonnegative integer. We introduce the following classes of polynomials. Let
Pn:=
p:p(X)=j~oajX’,
I
ajE[w
i
denote the set of all algebraic polynomials of degree at most n with real coefficients. Let pi :=
p : p(x) =
j,$
ajx',
aj
denote the set of all algebraic polynomials coefficients. 160
1
E @
i
of degree at most n with complex
Let JJ :P(X)
.Fn I=
=jeoL7jXj,
Uj E
{-l,O,
1)
(
1
denote the set of polynomials {-l,O,l). Let
of degree at most n with coefficients
Cc,:=
p:p(X)=j$oLZjXj,
from
ajE{-l,l}
i
1
denote the set of polynomials of degree at most n with coefficients from { - 1, 1). (Here we are using 13, in honor of Littlewood.) Let K, be the collection of polynomials p E pf, of the form P(x) = j.$O ajx’3
laOI = 1,
lajlI 1.
Let Gn be the collection of polynomials p E Fn of the form
where m is an unspecified nonnegative integer not greater than n. Obviously
The following two inequalities are well known in approximation theory. See, for example, Duffin and Schaeffer [12], Cheney [9], Lorentz [24], DeVore and Lorentz [ll], Lorentz, Golitschek, and Makovoz [25], Borwein and Erdelyi [4]. Markov Inequality. The inequality
IIP’II~-~,~~ I n211Pll~-1,1~ holds for every p E P,.
Bernstein Inequality. The inequality
[P’(Y)15 J-p--+ IIPll,-,,I] holdsforeveryp
E P,andy
E (-1,l).
In the above two theorems and throughout mum norm on A C R.
the paper 11.IIA denotes the supre-
161
Our intention is to establish the right Markov-type inequalities on [0, I] for the classes &, C,, X,, and &,. We also prove an essentially sharp Bernstein-type inequality on [0, 1) for polynomialsp E 3 := UT=0 3n with (p(O)1= 1. For further motivation and introduction to the topic we refer to Borwein and Erdelyi [5]. This paper is, in part, a continuation of the work presented in [5]. The books by Lorentz, Golitschek, and Makovoz [25], and by Borwein and Erdelyi [4] also contain sections on Markov- and Bernstein-type inequalities for polynomials under various constraints. The classes 3,,, ic,,, and other classes of polynomials with restricted coefficients have been thoroughly studied in many (mainly number theoretic) papers. See, for example, Beck [l], Bloch and Polya [2], Bombieri and Vaaler [3], Borwein, ErdClyi, and Kos [5], Borwein and Ingalls [7], Byrnes and Newman [8], Cohen [IO], Erdiis [ 131,Erdiis and Turan [ 141,Ferguson [15], Hua [i6], Kahane [17] and [18], Konjagin [19], Korner [20], Littlewood [21] and [22], Littlewood and Offord [23], Newman and Byrnes [26], Newman and Giroux [27], Odlyzko and Poonen [28], Salem and Zygmund [29], Schur [30], and Szegii [31]. For several extremal problems the classes 3U tend to behave like G,,,.See, for example, Borwein, Erdelyi, and Kos [5]. It is quite remarkable that as far as the Markov-type inequality on [0, l] is concerned, there is a huge difference between these classes. Compare Theorems 2.1 and 2.4. 2. NEW
RESULTS
Theorem 2.1. There are ubsolute constants
qnlog(n
+ 1) <
q > 0 and c2 > 0 such that
max -IP’(l)l OfPE-%lIPlIp,, < dz%
Theorem 2.2. There are absolute constants
cinlog(n + 1) I
cl > 0 and c2 > 0 such that
max -(p’(l)1 ofPEtfx
IlP’lljO.,]
llPll[O.l] ’ OEzn
Theorem 2.3. There are absolute constants
cinlog(n + 1) I
~
llPlliO.I]
I cznlog(n + 1).
cl > 0 and c2 > 0 such that
max -IP’(l)l OfPEG IlPll[0.,] 5 G%n
Theorem 2.4. There are ubsolute constants
qn”i2 5
lIP’llj0 I] I czn log@ + 1). IIPll[O.l,
d
IIP’II[O.I] 5 cznlog(n + I). llPll,O.,]
___
cl > 0 and Q > 0 such that
IIP’ll[OI]
max [p’(l)1 O#pt& l]Pll[O.,]< o?%x%~
‘/2 I c21E-
The following theorem establishes an essentially sharp Bernstein-type equality on [0,1) for polynomials p E F := Urzo _?jjwith /p(O)1= 1. 162
in-
Theorem 2.5. There are absolute constants cl > 0 and c2 > 0 such that 2 Cl log 1 -Y
( 1
1- Y
< max IIP’ll[O,.r]< cz 10:(125) - ,;;,t,
llPll[o.r] -
for every y E [0,1). Our next result polynomials
is an essentially
sharp
Bernstein-type
inequality
on [0,1) for
p E L := Ur= 0 C,.
Theorem 2.6. There are absolute constants cl > 0 and c2 > 0 such that
.for every y E [0, 1). Our final result is an essentially polynomials p E K := U,“=, Ic,.
sharp
Bernstein-type
inequality
on [0,1) for
Theorem 2.7. There are absolute constants cl > 0 and cp > 0 such that cr log
2 1-Y
( >
I ~~~ l1;6//;>J; < c2lo$+J
1 -Y
0.1 -
for every y E [0, 1). on [0,1) for polynomials p E G := lJ,“=, 8, is also there is a gap between the upper bound in [5] and
A Bernstein-type inequality established in [5]. However, the lower bound 3. LEMMAS
we are able to prove at the moment.
FOR THEOREM
To prove Theorem
2.4
2.4 we need several lemmas.
The first one is a result from [6].
Lemma 3.1. There are absolute constants q > 0 and c4 > 0 such that ew(-c3J;;)
I of;fG
We will also need a corollary
”
IIPII~~.~~ 5 ,j;fs
of the following
Hadamard Three Circles Theorem.
Supposef
n
IIPII~~,~~ 5 ew(-c4JtE).
well known
result.
is regular in
{z E C : rI 5 121< rz}. For r E [rj, rz]. let 163
log(rdr1)
log
M(r)
Note that the conclusion written
I
b(r2lr)
log
of the Hadamard
M(v)
+ hdrlrl)
log
M(r2).
Three Circles Theorem
can be re-
as
log M(r) I
logM(u) +
logcrirl) (i0gM(r2) - logM(rt)).
log(r2~rl)
Corollary 3.2. Let cr E R . Suppose 1 < cy < 2n. Suppose f is regular inside and on the ellipse A,,, with foci at 0 and 1 and with major axis [ - 9, 1 + z]. Let Bn,a be the ellipse with foci at 0 and 1 and with major axis [ - A, 1 + A]. Then there is an absolute constant cg > 0 such that
Proof. This follows from the Hadamard
Three Circles Theorem with the substitution w = i (z + z-‘) + ‘i. See also the remark following the Hadamard Three CirclesTheorem, which is applied with the circles centered at 0 with radii with a suitable and r-2:= 1 + fl cx n, respectively, rl := 1, r := 1 + cdm,
choice of c.
0 cr > log(n + 1). Then there is
Lemma 3.3. Let p E Gn with I[pllro,, i =: exp(-a), an absolute constant cg > 0 such that _y;x (P(z)/ 2 c6 $$
-
“.LI
k(z>l~
where B,,, is the same ellipse as in Corollary 3.2. Proof. Note that jjp/j,0,1I 2 lp( l/4)] 2 4+ for every p E G,,. Therefore (log4)n. Our assumption onp E G,, can be written as ,:$;)I”8 Also,p
[P(z)1 = -o.
E & and z E A,,, imply that logb(z)I
5 log((n+
I)(1 +fjn+‘)
~log(n+l)+(n+1)~5log(n+1)+2aI3c~. Now the lemma
follows from Corollary
3.2.
0
Lemma 3.4. There is an absolute constant CI > 0 such that
IIP’ll[O,l] 5 c7an lIPll,O,,] for everyp E Gnwith ~~p~~~O,ll = exp(-cr) 164
I (n + 1)-l.
a 5
Proof. This follows from Lemma 3.3 and the Cauchy Integral Formula.
Note that for a sufficiently large absolute constant c > 0, the disks centered at y E [0,1] with radius l/(con) are inside the ellipse B,,, (see the definition in Corollary 3.2). 0 Lemma 3.5. There is an absolute constant cs > 0 such that
Proof. Applying Corollary 3.2 with (Y= log(n + 2), we obtain that there is an absolute constant cg > 0 such that
for everyp E Gflwith IIpljro.Ii> (n + 2)-l. To see this note that max log Ip( ZE[O,l]
2 - log(n + 2)
and zy;x log no
Ip(
cc
I log n 1 -t
log(n + 2) n
n
H
<
2 log(n + 2).
Now the Cauchy Integral Formula yields that
IIP’II~~,~~ i clonlodn
+
lM~o,l~
with an absolute constant CIO> 0. Note that for a sufficiently large absolute constant c > 0, the disks centered at y E [0, I] with radius l/(cn log(n + 2)) are inside the ellipse Brt,logcn + 2) ( see the definition in Corollary 3.2). 0 4. PROOF
OF THE
THEOREMS
Proof of Theorem 2.1. The upper bound of the theorem was proved in [5]. It is
sufficient to establish the lower bound of the theorem for degrees of the form N = 16”(n + I), n = 1,2,. . .. This follows from the fact that if we have polynomials PN
E 3N,
N =
16”(n + l),
n== 1,2,...
,
showing the lower bound of the theorem with a constant c > 0, then the polynomials
QN :=
P16”(~+1)
show the lower bound the largest integer for theorem for the values n 2 1 and let T,, be the
E FN,
N = 1,2,...
,
of the theorem with the constant c/1024 > 0, where n is which 16”(n + 1) < N. To show the lower bound of the N = 16”(n + l),n = 1,2,. . ., we proceed as follows. Let Chebyshev polynomial defined by 165
cosize,
T,(x) =
ThenlITnIl,-I.,,=
1. Denote
x =
coso,
f9E [OJ].
the coefficients
of T, by ak = ak,n, that is,
II
T,(x) = c
ak-.xk
k=O
It is well known
that the Chebyshev
polynomials
T,l satisfy the three-term
re-
cursion
T,,(x) = ~xT,_~(x)
- T,,_*(x).
Hence each ak is an integer
and, as a trivial
bound
for the coefficients
of T,, we
have (4.1)
bkl 5
k=O,l,...,
3",
n.
Also, either a0 = 0 or a0 = f 1. Let A := 16” and let
p,,(X)
:=
sign(ak)““fi ’.)k +.i j=O
2
X=0
We will show that P,, gives the required lower bound (with n replaced N := 16”(n + 1) in the theorem). It is straightforward from (4.1) that FN
with
by
N := 16”(n + 1)
(4.21
p, E
Observe
that for x E [0,11,
=
k$osign(ax)l”i’Y$j’ (x’ - 1) j-0
=
k~,sign(ak)xAk(l-x~~~l+x+x2+~ j=l < _
2 k=l
lakl* - lakl 2
Hence, for x E [0, 11,
IP,r(x)(5 IT&“)1 + IPn(x)- G(xA)II 1 + 1 = 2. We conclude (4.3)
that
Ilpnll[o.,] 5 2.
Let Qn,~(x) := m(b). 166
Then
~k,~(l)
= AT:(l)
= An2.
Now
(4.4)
e:‘,(l) = 5 sign(ar)“5’ (Ak +j) !i=l ,j = 0 n
=
Ql,A(l) + C sign(a) k=l
IQ-l
C j
j=o
Thus (4.2), (4.3), and (4.4) give the lower bound
of the theorem.
0
Proof of Theorem 2.2. The upper bound of the theorem follows from Theorem 2.1. To show the lower bound we modify the construction in Theorem 2.1 by replacing the 0 coefficients in P,, by il coefficients with alternating signs. We use the notation of the proof of Theorem 2.1. Let R, E CN be the polynomial arising from P, in this way. Recall that N := 16”(n + 1). Then, using the fact that R,, - P,, is of the form R,(x)
- P,(x)
= ztc (-l)‘xk,, i= I
we have II& - P,,ll~~.,l <_ 1. Combining (4.5)
this with (4.2), we obtain
l1R~ll,0.,] 5 IIfYljo.,, + I/& - W,o., , 5 2 + 1 =
On the other hand, using the I(R,, - P,,)‘(l)\ < N. This, together (4.6)
0 5 k, < k2 < ‘. . < k,, 5 N,
R:,(l)
2 P:,(l)
form of R, - P, with (4.4) gives
- I(Rn - Pn)‘(l)i
for n > 4. Now (4.5) and (4.6) together the lower bound
of the theorem.
> G16”n2
3.
given
above,
we have
- 16”(n + 1) > 616”n’
with R, E CN and N := 16”(n + l), give
0
= 1, the upper bound of the theProof of Theorem 2.3. Since JJpII,o,,I 2 Ip( orem follows as a special case of Lemma 3.5. The lower bound of the theorem follows from the lower bound in Theorem 2.2. 0 Proof of Theorem 2.4. To see the upper bound, note that Lemma 3.1 implies Q < c,n’iz in Lemma 3.4. So the upper bound of the theorem follows from Lemmas 3.4 and 3.5. Now we prove the lower bound of the theorem. Let k be a natural number and let Q,,(x) :=
.%?+‘rk(,@) =
;;$+T b,x’, 167
where Tk is, once again, the Chebyshev polynomial defined by T/&X) = Then
(4.7)
x = cost),
COS no,
0 E [0,7r].
IIQnllp,lI = 1 and Q;(l)=k3+n+1,
while, the coefficients bj of Q,, satisfy (4.8)
j=O,l,...,n+
IbjI < 3k,
1 +k2
(see the argument in the proof of Theorem 2.1). By Lemma 3.1 there is a polynomial R, E Fn such that
IIRnll~o.l~ I ew(-w@ Let ntltk? Pn :=
R, + exp(-c4fi)Qn
ajxj.
C
=:
j=O
Then (4.9)
IIM~o.~15 2expkc4fi).
It follows from (4.8) that (aj( < exp(-c4fi)3k
for eachj > n + 1. From now on
let (4.10)
k:=
s L
, 1
which implies that (UjJ5 1 for each j 2 n + 1, while Uj E { - 1, 0,1) for each j < n. Hence (4.11)
p, E
n +
%+l+kz,
1 + k2 I cqz,
with an absolute constant cs. Note that if n is large enough, then R;(l) = 0. Otherwise, as an integer, IRL(l)/ would be at least 1 and Markov’s inequality would imply that 1 5 IRi(l)I 5
2~211&llio,lI5 2n2exp(--C4\/;;),
which is impossible for a large enough n, say for n > cg. Hence, combining (4.9), (4. lo), and (4.7), we conclude P;(l) = R;(l) +exp(-cdfi)QL(l) 1
ew(-c4J;;)w3/2
2
= exp(-c4fi)(k3
+n+
1)
~n3/211Pn~~10,11
for every n 2 c6 with an absolute constant c-i > 0. This, together with (4.11) 0 finishes the proof of the lower bound of the theorem.
168
Proof of Theorem 2.5. The upper bound of the theorem was proved in [5]. To
show the lower bound we proceed as follows. Let P,, be the same as in the proof of Theorem 2.1. Throughout this proof we will use the notation introduced in the proof of Theorem 2.1. We will show that P,, gives the required lower bound with a suitably chosen even n depending on y. If n := 2m is even, then (4.12)
Ipn(0)I = 1,
and
p, E 3N
with
N := 16”(n + 1).
Recall that by (4.3) we have IIPn 11io,, < 2. As in the proof of Theorem 2.1, let A := 16” and Q”,A(x) := m(xA). F or n 2 no, the Chebyshev polynomial T, has a zero S in [e-*, e-l]. In particular, 7’,(S) 2 n. Let n E N be the largest even integer for which (4.13)
SIIA = b16-” 5 e-16-” < y.
Without loss of generality we may assume that n L no, otherwise p E 3 defined by p(x) := 1 + x shows the lower bound of the theorem. Note that there are absolute constants c3 > 0 and c4 > 0 such that 16” 2 c3(1 - y)-’ and hence
(4.14)
16”n 2
Therefore Q;.A(6”A) = AS CA - ‘)lAT,‘(6) 2 An6 = 16”nS 2 em216”n.
Observe that 6 E [eM2,e-‘1 and 0 Sj 2 lak] - 1 5 A (see the notation troduced in the proof of Theorem 2.1) imply
in-
and
> -2j.
We conclude that (4.15)
I(Ak +j)S (Ak+j- 1)/A _ A/&A”- 1)lAl 5 2j.
Now, using the definition of P,, (see the proof of Theorem 2. l), (4.13), (4.15), and (4.14), we obtain 169
2
( >
ct log
-
1-Y I-Y
>
for every n > 2 with an absolute constant cl > 0. This, together (4.12) gives the lower bound of the theorem. 0
with (4.3) and
Proof of Theorem 2.6. The upper bound of the theorem
follows from Theorem 2.5. To show the lower bound we modify the construction in Theorem 2.5 by replacing the 0 coefficients in P,, by fl coefficients with alternating signs. We use the notation of the proof of Theorems 2.1 and 2.5. Let R, E LN be the polynomial arising from P,, in this way. As in the proof of Theorem 2.2, we have llRn - P,211,0,,l 5 1, and combining this with (4.3), we obtain (4.16)
llR~lII,,, I 5
iV’nll,o.,, + II& - f’nll,o., j I 2 + 1 = 3.
On the other hand, for a E [0, l), we have (4.17)
I(& - P,)‘(a)1
L var;xf,(x),
wherefu(x) := x& ‘. Now it is elementary on [ 1,~) contains two monotone pieces, and
calculus
to show that the graph of,fU
,limX_Mx) = 0.
Hence
with an absolute (4.18)
constant
I(& - C)‘(a)1
Now let y E that
[0,l). By the proof of Theorem
ICKa)l2
with (4.17) yields
< 2.
Ct log
170
c4 > 0. This, together
2
( > ~
1-Y
l-y
2.5, there exists an a E [ 0, y] such
Combining this with (4.18), we deduce
2 ci log ___ l-y L. 1 -y
( > --
c5
c-4
l-y>
2 log ( 1-Y > 1 -Y
for y E [yo, 11, where c5 > 0 and yo E [0,l) are absolute constants. This, together with (4.16) and R, E CN gives the lower bound of the theorem for y E [yo, 11. If y E [0, yo), then the trivial example P(X) := 1 + x shows the lower bound of the theorem. 0 Proof of Theorem 2.7. The upper bound of the theorem was proved in [5]. The
lower bound of the theorem follows from the lower bound in either Theorem 2.5 or Theorem 2.6. 0 5
ACKNOWLEDGMENT
We thank Fedor Nazarov for suggesting the the crucial idea for the proof of the lower bounds in Theorems 2.1 and 2.2. REFERENCES
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(Received June 1999, revised September 1999)
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