Journal of Inequalities in Pure and
Applied Mathematics
AN APPLICATION OF VAN DER CORPUT’S INEQUALITY
KANTHI PERERA
Department of Engineering Mathematics,
University of Peradeniya,
Sri Lanka.
EMail: kanthi@engmath.pdn.ac.lk
volume 2, issue 1, article 8,
2001.
Received 13 July, 2000;
accepted 31 October 2000.
Communicated by: A.M. Fink
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
P
2
In this note we give a short and elegant proof of the result nt=1 eı(ωt+αt ) = o(n)
for α not a rational multiple of π, uniformly in ω. This was first proved by Hardy
and Littlewood, in 1938. The main ingredient
P of our proof 2is Van der Corput’s
inequality. We then generalize this to obtain nt=1 tβ eı(ωt+αt ) = o(nβ+1 ), where
β is a nonnegative constant.
An Application of Van der
Corput’s Inequality
2000 Mathematics Subject Classification: 42A05
Key words: Van der Corput’s inequality, Hardy and Littlewood.
K. Perera
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
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J. Ineq. Pure and Appl. Math. 2(1) Art. 8, 2001
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1.
Introduction
P
2
Hardy and Littlewood [1] studied the series of the form nt=1 eı(ωt+αt ) and
other similar series associated with the elliptic Theta functions. It was noted
there that the behavior is interesting and difficult when α is not a P
rational multi2
ple of π. The main result proved in [1] can essentially be stated as nt=1 eı(ωt+αt )
= o(n) for α not a rational multiple of π uniformly in ω.
We became interested in this result, rather a generalization of it, while working on a problem of estimation of parameters of a chirp-type statistical model.
Although we hoped to find an easy proof of this result in the literature, we were
unable to find one. The purpose of this note is to
of it. We
Pgive an easy proof
2
then generalize this to obtain a similar result for nt=1 tβ eı(ωt+αt ) , where β is a
positive constant.
An Application of Van der
Corput’s Inequality
K. Perera
Title Page
The main ingredient of our proof is the following remarkable inequality.
Theorem 1.1 (Van der Corput’s Fundamental Inequality). [2, p. 25]:
Let u1 · · · un be complex numbers, and let H be an integer with 1 ≤ H ≤ N .
Then
2
N
N
X
X
2
H un ≤ H(N + H − 1)
|un |2
n=1 n=1
H−1
−h
NX
X
+ 2(N + H − 1)
(H − h) un ūn+h .
h=1
n=1
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2.
Main Result
Theorem 2.1. Let β be a nonnegative real number. Then
o(nβ+1 ), for α not a rational multiple of π, uniformly in ω.
Pn
t=1
tβ eı(ωt+αt
2)
=
2
Proof. By using Van der Corput’s inequality with fixed H and ut = eı(ωt+αt ) ,
we obtain
2
n
n X
X
2
ı(ωt+αt2 ) 2
2
ı(ωt+αt ) (2.1) H e
e
≤ H(n + H − 1)
t=1
t=1
+2
H−1
X
h=1
An Application of Van der
Corput’s Inequality
n−h
X
(n + H − 1)(H − h) ut ūt+h t=1
K. Perera
Title Page
−ıωh−ıαh2 −2ıαth
for all n ≥ H. But ut ūt+h = e
, and so
n−h
n−h
X
X
ut ūt+h = e−2ıαth .
t=1
t=1
Substituting this in the inequality (2.1), we obtain
2
n
H−1
n−h
X
X
X
2
2
ı(ωt+αt )
−2ıαth H e
(n+H −1)(H −h) e
≤ H(n+H −1)n+2
.
t=1
t=1
h=1
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Thus
(2.2)
2
n
1 X
1
1
1
2
ı(ωt+αt ) e
+ −
≤
n t=1
H n nH
+2
n−h
(n + H − 1)(H − h) X −2ıαth e
.
n2 H 2
H−1
X
t=1
h=1
Let
Mn (α, h) =
n−h
X
e−2ıαth .
An Application of Van der
Corput’s Inequality
K. Perera
t=1
Thus if α is not a rational multiple of π we can write Mn (α, h) in the following
form.
sin[(n − h)αh]
Mn (α, h) = e−ıαh(n−h+1)
.
sin(αh)
Then
|Mn (α, h)| ≤
1
.
|sin(αh)|
If h0 is the member of {1, 2, · · · , (H − 1)} for which
|sin(αh0 )| =
min
1≤h≤H−1
|sin(αh)| ,
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then
|Mn (α, h)| ≤
1
.
|sin(αh0 )|
J. Ineq. Pure and Appl. Math. 2(1) Art. 8, 2001
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Substituting this in equation (2.2) we get
2
n
1 X
1
1
1
ı(ωt+αt2 ) e
+ −
≤
n t=1
H n nH
+2
H−1
X
h=1
≤
(n + H − 1)(H − h)
1
2
2
nH
|sin(α, h0 )|
1
1
1
2
+ −
+
.
H n nH n |sin(αh0 )|
Since this is true for all n ≥ H, we obtain
n
2
1 X
1
2
lim eı(ωt+αt ) ≤ .
n→∞ n
H
t=1
Since this is true for all H ≥ 1, it follows that
n
2
1 X
2
eı(ωt+αt ) = 0,
lim n→∞ n
An Application of Van der
Corput’s Inequality
K. Perera
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uniformly in ω. That is, if α is not a rational multiple of π,
(2.3)
n
X
2
eı(ωt+αt ) = o(n),
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t=1
J. Ineq. Pure and Appl. Math. 2(1) Art. 8, 2001
uniformly in ω.
http://jipam.vu.edu.au
Now we will show that
rational multiple of π. Let
Pn
t=1
tβ eı(ωt+αt
2)
= o(nβ+1 ) provided α is not a
Q0 (ω, α) = 0,
and, for n ≥ 1,
Qn (ω, α) =
n
X
2
eı(ωt+αt ) and Sn (ω, α) =
t=1
n
X
2
tβ eı(ωt+αt ) .
t=1
An Application of Van der
Corput’s Inequality
Then
(2.4)
Sn (ω, α) =
n
X
K. Perera
tβ [Qt (ω, α) − Qt−1 (ω, α)]
t=1
β
= n Qn (ω, α) − Q0 (ω, α) −
n−1
X
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β
β
[(t + 1) − t ]Qt (ω, α)
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t=1
β
= n Qn (ω, α) −
n−1
X
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where
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fn (ω, α) = [(n + 1)β − nβ ]Qn (ω, α)
for
n = 1, 2, . . .
By the mean-value theorem we have
(n + 1)β − nβ = β ñβ−1
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where
n ≤ ñ ≤ n + 1.
J. Ineq. Pure and Appl. Math. 2(1) Art. 8, 2001
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If 0 ≤ β ≤ 1, then ñβ−1 ≤ nβ−1 , while if β ≥ 1, then ñβ−1 ≤ (n + 1)β−1 ≤
(2n)β−1 . It follows that, for β ≥ 0,
(2.5)
(n + 1)β − nβ ≤ cβ nβ−1 ,
where cβ is a constant. Hence
|fn (ω, α)| ≤ cβ nβ−1 |Qn (ω, α)| .
But by (2.3), if α is not a rational multiple of π, then Qn (ω, α) = o(n) uniformly
in ω.
Thus if α is not a rational multiple of π, then
An Application of Van der
Corput’s Inequality
K. Perera
|fn (ω, α)| ≤ cβ nβ−1 o(n),
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so
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fn (ω, α) = o(nβ ).
β
uniformly
Pn−1 in ω. However,βfn (ω, α) = o(n ) implies that the mean
1
t=1 ft (ω, α) is o(n ).
n−1
Hence, if α is not a rational multiple of π,
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n−1
X
ft (ω, α) = o(nβ+1 ),
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uniformly in ω. But by (2.4)
Page 8 of 10
β
Sn (ω, α) = n Qn (ω, α) −
n−1
X
t=1
ft (ω, α).
J. Ineq. Pure and Appl. Math. 2(1) Art. 8, 2001
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It follows that Sn (ω, α) = o(nβ+1 ), for α not a rational multiple of π, uniformly
in ω.
This completes the proof of the Theorem.
An Application of Van der
Corput’s Inequality
K. Perera
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References
[1] G.H. HARDY AND J.E. LITTLEWOOD, Some Problems of Diophantine
Approximation. Acta Mathematica Band 37 s. 193-238 [Volume 1 of Collected Works of G.H. Hardy]. (1914).
[2] L. KUIPERS AND H. NIEDERREITER, Uniform Distribution of Sequences, Wiley, New York, (1974).
An Application of Van der
Corput’s Inequality
K. Perera
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