LOWER BOUNDS FOR THE MERIT
FACTORS OF TRIGONOMETRIC
POLYNOMIALS FROM LITTLEWOOD CLASSES
Peter Borwein and Tamás Erdélyi
Abstract. With the notation K := R (mod 2π),
kpkLλ (K) :=
Z
K
|p(t)|λ dt
1/λ
and
Mλ (p) :=
1
2π
1/λ
Z
K
|p(t)|λ dt
we prove the following result.
Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with
real coefficients that satisfies
kpkL2 (K) ≤ An1/2
kp0 kL2 (K) ≥ Bn3/2 .
and
Then
M4 (p) − M2 (p) ≥ εM2 (p)
with
ε :=
1
111
B 12
A
We also prove that
p
M∞ (1 + 2p) − M2 (1 + 2p) ≥ (
and
.
4/3 − 1)M2 (1 + 2p)
M2 (p) − M1 (p) ≥ 10−31 M2 (p)
for every p ∈ An , where An denotes the collection of all trigonometric polynomials
of the form
p(t) := pn (t) :=
n
X
aj cos(jt + αj ) ,
j=1
aj = ±1 ,
αj ∈
R.
1991 Mathematics Subject Classification. Primary: 41A17.
Key words and phrases. trigonometric polynomials, merit factor, unimodular trigonometric
polynomials, Littlewood class..
Research of P. Borwein is supported by MITACS and by NSERC of Canada. Research of T.
Erdélyi is supported, in part, by NSF under Grant No. DMS–0070826.
Typeset by AMS-TEX
1
2
PETER BORWEIN AND TAMÁS ERDÉLYI
Introduction
We give shorter and more direct proofs of some of the main results from Littlewood’s papers [Li-61], [Li-62], [Li-66a], [Li-66b], and [Li-68]. There are two reasons
for doing this. First our approaches are, we believe, much easier, and secondly
they lead to explicit constants. Littlewood himself remarks that his methods were
“extremely indirect.” Motivation and discussion of these types of results may be
found in [Bo-02]. Kahane’s paper [Ka-85] is also central among those related to the
subject of this paper.
2. New Results
We use the notation K := R (mod 2π). Let
Z
kpkLλ(K) :=
K
1/λ
λ
|p(t)| dt
and Mλ (p) :=
1
2π
Z
K
λ
|p(t)| dt
1/λ
.
Theorem 1. Assume that p is a trigonometric polynomial of degree at most n with
real coefficients that satisfies
kpkL2(K) ≤ An1/2
(1)
and
kp0 kL2 (K) ≥ Bn3/2 .
(2)
Then
M4 (p) − M2 (p) ≥ εM2 (p)
with
ε :=
1
111
B
A
12
.
Let the Littlewood class An be the collection of all trigonometric polynomials of
the form
p(t) := pn (t) :=
n
X
aj cos(jt + αj ) ,
j=1
Note that for the Littlewood class An we have
B
A
12
= 3−6 .
aj = ±1 ,
αj ∈ R .
LOWER BOUNDS FOR THE MERIT FACTORS
Corollary 2. We have
M4 (p) − M2 (p) ≥
3
M2 (p)
80920
for every p ∈ An . The merit factor
−1
4
M4 (p)
−1
M24 (p)
is bounded above by 20230 for every p ∈ An .
If Qn is a polynomial of degree n of the form
n
X
Qn (z) =
ak z k ,
ak ∈ C ,
k=0
and the coefficients ak of Qn satisfy
ak = an−k ,
k = 0, 1, . . . n ,
then we call Qn a conjugate-reciprocal polynomial of degree n. We say that the
polynomial Qn is unimodular, if |ak | = 1 for each k = 0, 1, 2, . . . , n. Note that if
p ∈ An , then
1 + 2p(t) = eint Q2n (eit )
with a conjugate-reciprocal unimodular polynomial Q2n of degree exactly 2n. One
can ask how flat a conjugate reciprocal unimodular polynomial can be. Here we
reprove a result of Erdős [Er-62]. His proof is much longer and his constant ε > 0
is unspecified. This result has already been recorded in [Er-01].
Theorem 3. Let ∂D denote the unit circle. Let P be a conjugate reciprocal unimodular polynomial of degree n. Then
√
max |P (z)| ≥ (1 + ε) n + 1
z∈∂D
p
with ε := 4/3 − 1. As a consequence, we have
p
M∞ (1 + 2p) − M2 (1 + 2p) ≥ ( 4/3 − 1)M2 (1 + 2p)
for every p ∈ An .
In our next theorem we give the numerical value of an unspecified constant
appearing in another main result of Littlewood. In the proof we will need to refer
to only a two-page-long (very clever) piece of Littlewood’s paper [Li-66a].
Theorem 4. We have
M2 (p) − M1 (p) ≥ 10−31 M2 (p)
for every p ∈ An .
Based on the fact that for a fixed trigonometric polynomial p the function
λ → λ log(Mλ (p))
is a convex function on [0, ∞), we can state explicit numerical values of certain
unspecified constants in some other related Littlewood results. For example, as a
consequence of Theorem 4, we have
4
PETER BORWEIN AND TAMÁS ERDÉLYI
Theorem 5. We have
log(Mλ (p)) − log(M2 (p)) ≥
λ−2
log
λ
1
1 − 10−31
,
λ > 2,
and
log(M2 (p)) − log(Mλ (p)) ≥
2−λ
log
λ
1
1 − 10−31
,
1 ≤ λ < 2,
for every p ∈ An .
3. Proofs
Proof of Theorem 1. For the sake of brevity let µn := µn (p) = M2 (p). Note that
Bernstein’s inequality in L2 (K) implies B ≤ A. Without loss of generality we may
assume that
kpk4L4 (K) ≤ 2π
(3)
33 4
µ .
32 n
Then by the Bernstein Inequality for trigonometric polynomials in L4 (K) we can
deduce that
1/4
1/4
33
33
0
1/4
−1/4
(2π) µn ≤ π
An3/2 .
kp kL4 (K) ≤ nkpkL4(K) ≤ n
32
64
Hence, combining this with (2) and Hölder’s Inequality, we obtain
B 2 n3 ≤
Therefore
that is
Z
2/3
K
4/3
2/3
|p0 (t)|2 dt ≤ kp0 kL1 (K) kp0 kL4 (K) ≤ kp0 kL1 (K) π −1/3
π 1/3
π 1/2
64 1/3
33
A4/3
64 1/2
33
A2
B2
B3
33
64
1/3
2/3
n ≤ kp0 kL1 (K) ,
n3/2 ≤ kp0 kL1 (K) .
Combining this with (1), we have
1/2 3
π 1/2 64
B 3/2
A √
33
n≤
n
≤ kp0 kL1 (K)
γnµn ≤ γn
A2
(2π)1/2
with
γ :=
Now let
E := E(n, p, γ) :=
128 1/2
33
A3
πB 3
.
γµ 2 n
.
t ∈ [0, 2π) : (|p(t)| − µn )2 ≥
16
A4/3 n2 .
LOWER BOUNDS FOR THE MERIT FACTORS
5
Then using Hölder’s Inequality and then Bernstein’s Inequality for trigonometric
polynomials in L2 (K), we can deduce that
Z
γnµn ≤
2π
0
|p0 (t)| dt ≤
Z
[0,2π]\E
|p0 (t)| dt +
Z
E
|p0 (t)| dt
Z
1/2
p
γ
nµn + m(E)
|p0 (t)|2 dt
2
E
E
Z 2π
1/2
p
p
γ
γ
≤ nµn + m(E) n
|p(t)|2 dt
≤ nµn + m(E) n(2π)1/2 µn .
2
2
0
≤ 2 · (2n) ·
Hence
2γ
µn +
16
Z
|p0 (t)| dt ≤
p
γ
nµn ≤ m(E)n(2π)1/2 µn ,
2
that is
(4)
β :=
γ2
≤ m(E) .
8π
So we have
2π(M4 (p)4 − M2 (p)4 ) = kpk4L4 (K) − 2πµ4n
Z 2π
=
(p(t)2 − µ2n )2 dt
0
γµ 2
γ 2 γµn 2 2
n
≥ m(E)
µ2n ≥
µn
16
8π 16
2 12
1
128
B
= 11
π4
µ4n .
2 π 33
A
Combining this with (3) we obtain
M4 (p) − M2 (p) ≥ 2−14
B
A
12 128
33
2
π 2 M2 (p) ,
and the theorem is proved Here we used
M4 (p)4 − M2 (p)4 ≥ (M4 (p) − M2 (p))4M2 (p)3 ,
which is a consequence of the Mean Value Theorem. Note that
1
≤ 2−14
111
128
33
2
π2 ≤
1
.
110
Proof of Theorem 3. Let P be a conjugate reciprocal unimodular polynomial of
degree n. To prove the statement, observe that Malik’s inequality [MMR, p. 676]
gives
n
max |P 0 (z)| ≤ max |P (z)| .
z∈∂D
2 z∈∂D
6
PETER BORWEIN AND TAMÁS ERDÉLYI
(Note that the fact that P is conjugate reciprocal improves the Bernstein factor for
P on ∂D from n to n/2.) Using the fact that each coefficient of P is of modulus 1,
then applying Parseval’s formula and Malik’s inequality, we obtain
2π
n2 (n + 1)
n(n + 1)(2n + 1)
≤ 2π
=
3
6
Z
and
max |P (z)| ≥
follows.
∂D
p
z∈∂D
|P 0 (z)|2 |dz| ≤ 2π
n 2
2
max |P (z)|2 ,
z∈∂D
√
4/3 n + 1
Proof of Theorem 4. Let p ∈ An . For the sake of brevity let µn := µn (p) = M2 (p).
Let N (p, v) be the number of real roots of p − vµn = 0 in (−π, π). Littlewood
proves (see Theorem 1 (i) of [Li-66a]) that if p ∈ An and
1
2π
then
Z
2π
0
|p(t)| dt = cµn ,
N (p, v) ≥ 2−16 c11 n ,
|v| ≤ 2−5 c3 .
The reader may wish to find this lower bound hidden in the proof of Theorem 1
(i) of Littlewood’s paper [Li-66a]. Hence, by estimating the total variation in the
usual way, γnµn ≤ kp0 kL1 (K) with γ := 2−20 c14 . If c ≤ 2−1/14 , then the proof of
the theorem is finished. If c ≥ 2−1/14 , then γ ≥ 2−21 , so in the sequel we may
assume that γ ≥ 2−21 holds. Now let
E := E(n, p, γ) :=
γµ 2 n
.
t ∈ [0, 2π) : (|p(t)| − µn )2 ≥
16
Estimating the total variation of p on [0, 2π] \ E, hen using Hölder’s Inequality and
then Bernstein’s Inequality for trigonometric polynomials in L2 (K), we can deduce
that
Z
γnµn ≤
0
2π
|p0 (t)| dt ≤
Z
[0,2π]\E
|p0 (t)| dt +
Z
E
|p0 (t)| dt
Z
1/2
p
γ
nµn + m(E)
|p0 (t)|2 dt
2
E
E
Z 2π
1/2
p
p
γ
γ
≤ nµn + m(E) n
|p(t)|2 dt
≤ nµn + m(E) n(2π)1/2 µn .
2
2
0
≤ 2 · (2n) ·
Hence
2γ
µn +
16
Z
|p0 (t)| dt ≤
p
γ
nµn ≤ m(E)n(2π)1/2 µn ,
2
that is
β :=
γ2
≤ m(E) .
8π
LOWER BOUNDS FOR THE MERIT FACTORS
7
So we have
4π((M2 (p))2 − M2 (p)M1 (p)) =
Z
0
2π
(|p(t)| − µn )2 dt
≥ m(E)
γµ 2
n
≥
γ 2 γµn 2
8π 16
16
γ4 2
= 11 µn ≥ 2−95 π −1 µ2n .
2 π
This implies
M2 (p) − M1 (p) ≥ 2−97 π −2 M2 (p) ,
and the theorem is proved.
Acknowledgements
The authors wish to thank the referee, Szilárd Révész, for his careful reading of
the original version of this manuscript. The constant 1/111 in Theorem 1 would
not be the same without him.
References
[Bo-02]
P. Borwein, Computational Excursions in Analysis and Number Theory, SpringerVerlag, New York, 2002.
[Er-01]
T. Erdélyi The phase problem of ultraflat unimodular polynomials: the resolution of
the conjecture of Saffari, Math. Annalen 321 (2001), 905–924.
[Er-62]
P. Erdős, An inequality for the maximum of trigonometric polynomials, Annales Polonica
Math. 12 (1962), 151–154.
[Ka-85]
J-P. Kahane, Sur les polynômes á coefficients unimodulaires, Bull. London Math. Soc
12 (1980), 321–342.
[Li-61]
J.E. Littlewood, On the mean values of certain trigonometrical polynomials, J. London
Math. Soc. 36 (1961), 307–334.
[Li-64]
J.E. Littlewood, On the real roots of real trigonometrical polynomials (II), J. London
Math. Soc. 39 (1964), 511–552.
[Li-66a]
J.E. Littlewood, The real zeros and value distributions of real trigonometrical polynomials, J. London Math. Soc. 41 (1966), 336-342.
[Li-66b]
J.E. Littlewood, On polynomials
Soc. 41 (1966), 367–376.
[Li-68]
J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical
Monographs, Lexington, Massachusetts, 1968.
[MMR]
Milovanović, G.V., D.S. Mitrinović, & Th.M. Rassias, Topics in Polynomials: Extremal
Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
P ±z m
and
P eα
mi
z m , z = eθi , J. London Math.
Department of Mathematics and Statistics, Simon Fraser University, Burnaby,
B.C., Canada V5A 1S6 (P. Borwein)
8
PETER BORWEIN AND TAMÁS ERDÉLYI
E-mail address: pborwein@cecm.sfu.ca (Peter Borwein)
Department of Mathematics, Texas A&M University, College Station, Texas
77843 (T. Erd
elyi)
E-mail address: terdelyi@math.tamu.edu (Tamás Erdélyi)