ISSN 1064–2269, Journal of Communications Technology and Electronics, 2015, Vol. 60, No. 12, pp. 1441–1444. © Pleiades Publishing, Inc., 2015.
Original Russian Text © V.S. Kozyakin, 2015, published in Informatsionnye Protsessy, 2015, Vol. 15, No. 2, pp. 128–133.
MATHEMATICAL METHODS
OF INFORMATION THEORY
On the Asymptotics of Cosine Series in Several Variables
with Power Coefficients1
V. S. Kozyakin
Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences,
Bolshoj Karetny lane, 19, GSP-4, Moscow, 127994 Russia
e-mail: kozyakin@iitp.ru
Received March 30, 2015
Abstract—The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent
topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied
by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series
in several variables have got less attention. The aim of the work is to find the asymptotics of trigonometric series
in several variables with the terms, having a form of ‘one minus the cosine’ accurate to a decreasing power-like
factor.
Keywords: trigonometric series in several variables, slowly decreasing coefficients, asymptotic behavior at zero,
Hardy type asymptotics.
DOI: 10.1134/S1064226915120165
1. INTRODUCTION
The investigation of the asymptotic behavior of
trigonometric series near the origin is a prominent topic
in mathematical analysis. For trigonometric series in
one variable, this problem was exhaustively studied by
various authors in a series of publications dating back
to the work of G. H. Hardy [4], 1928.
Trigonometric series in several variables have got
less attention. The aim of the work is to partially fill this
gap by finding the asymptotics of trigonometric series
in several variables with the terms, having a form of ‘one
minus the cosine’ accurate to a decreasing power-like
factor.
Given a real number α > 0, consider the function
X
1
F d (θ) :=
(1 − cos⟨z, θ⟩),
(1)
d +α
kzk
z∈Zd \{0}
defined for θ ∈ Rd . Here Zd is the lattice of points from
Rd with integer coordinates, ⟨·, ·⟩ is the standard inner
product and k · k is the max-norm in Rd defined by
⟨x, y⟩ = x 1 y 1 + x 2 y 2 + · · · + x d y d ,
kxk = max{|x 1 |, |x 2 |, . . . , |x d |},
where x = {x 1 , x 2 , . . . , x d }, y = {y 1 , y 2 , . . . , y d }.
The series in (1) is uniformly convergent for any
α > 0 and, therefore, function F d (θ) is non-negative and
1 The article was translated by the author.
continuous, and F d (0) = 0. We will be interested in
study of the asymptotic behavior of F d (θ) as θ → 0.
For d = 1, function F d (θ) can be represented in the
form
F d (θ) = 2Hα (θ),
(2)
where
Hα (θ) =
∞
X
1
(1 − cos nθ),
1+α
n
n=1
θ ∈ R,
and its asymptotics as θ → 0 can be described with
the help of classical results going back to the work of
G. H. Hardy [4] in which it was shown that for the functions
f (θ) =
∞
X
n=1
a n cos nθ,
g (θ) =
∞
X
a n sin nθ,
n=1
where 0 < α < 1 and n α a n → 1, the following asymptotics as θ → 0+ are valid:
³ πα ´
f (θ) ' Γ(1 − α) sin
θ α−1 ,
(3)
2
³ πα ´
g (θ) ' Γ(1 − α) cos
θ α−1 .
(4)
2
Here Γ(·) is the gamma function, and, for functions
h 1 (θ) and h 2 (θ), the notation h 1 (θ) ' h 2 (θ) as θ → θ0
means that h 1 (θ)/h 2 (θ) → 1 as θ → θ0 .
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KOZYAKIN
If α = 1, that is, na n → 1, then instead of (3) and (4)
the following limit relations are valid:
f (θ) ' | ln θ|,
g (θ) →
π
.
2
as θ → 0+ [5]. In [1], it was shown that the asymptotics
(4) is valid also for 0 < α < 2, and, in [11, 12], the asymptotic behavior of function f (θ) for all α > 1 and of function g (θ) for all α ≥ 2 were analyzed.
So, in [1,4,5,11,12], the asymptotic behavior of functions f (θ) and g (θ) has been completely investigated for
all α > 0. In particular, from [11, 12] it follows that
Hα (θ) ' Hα∗ (θ),
2. MAIN RESULTS
For an arbitrary set of numbers θ1 , θ2 , . . . , θd , let us
define the quantity
ω(θ1 , θ2 , . . . , θd ) =
Zθd
=
Zθ2
···
−θd
¯
¯
¯θ 1 + η 2 + · · · + η d ¯α d η 2 . . . d η d
−θ2
and consider the symmetric function
(5)
A d (θ) =
where
X ω(θi 1 , θi 2 , . . . , θi d )
{i j }
Hα∗ (θ) =
³ πα ´
1
|θ|α
Γ(1 − α) cos
α
2
for 0 < α < 2,
1
1
Hα∗ (θ) = θ 2 ln
2
|θ|
θi 2 · · · θi d
,
(6)
where the summation is taken over all permutations {i j }
of coordinates of the vector θ = {θ1 , θ2 , . . . , θd }. If some
of the variables θi in (6) vanishes, then, the value of
A d (θ) under the corresponding values of the argument
is uniquely defined by continuity.
Let |θ| denotes the Euclidean
norm of the vector
q
θ12 + · · · + θd2 .
for α = 2, and
θ = {θ1 , . . . , θd }, that is |θ| :=
1
Hα∗ (θ) = ζ(α − 1)θ 2
2
for α > 2, where
ζ(s) =
Theorem 1. Let d ≥ 2. Then,
∞ 1
X
s
n=1 n
is the Riemann zeta function.
The quantity Γ(1 − α) cos( πα
2 ) in the previous formulas is indeterminate when α = 1 since Γ(0) = ∞ and
cos( π2 ) = 0. This indeterminate form can be resolved by
treating Γ(1 − α) cos( πα
2 ) in the case α = 1 as
³ πα ´ π
lim Γ(1 − α) cos
= .
α→1−
2
2
This indeterminate form can be resolved also with the
help of the identity
³ πα ´
π
.
Γ(1 − α) cos
≡
2
2Γ(α) sin( πα
2 )
Note that, in works [1,4,5,11,12], as well as in subsequent publications [2, 3, 6–8], one can find quite a number of deeper results than those mentioned earlier, part
of which are included in monograph [13, Ch. V].
From (2) and (5) it follows that
F 1 (θ) ' 2Hα∗ (θ)
for d = 1 and each α > 0. For d ≥ 2 the asymptotic behavior of function F d (θ) at zero is much less studied. It
is known [9,10] that function F 2 (θ) has a lower bound of
the order kθkα near the origin.
In connection with this, the aim of the work is to
study the asymptotic behavior of function F d (θ) at zero
for all d ≥ 2.
F d (θ) '
³ πα ´
2
Γ(1 − α) cos
A d (θ)
α
2
for 0 < α < 2,
F d (θ) '
2d −1 (d + 2) 2
1
|θ| ln
3
|θ|
for α = 2 and
Ã
!
∞ (2n + 1)d (n + 1)
1 X
F d (θ) '
|θ|2 −
6 n=1
n d +α−1
Ã
!
∞ (2n − 1)d (n − 1)
1 X
−
|θ|2
6 n=1
n d +α−1
for α > 2.
Function A d (θ) is positive for θ 6= 0 and homogeneous of order α, that is, A d (t θ) ≡ t α A d (θ) for t ≥ 0.
Therefore, for 0 < α < 2, the claim of Theorem 1 can be
represented also in the following form:
F d (θ) '
³ πα ´ ³ θ ´
2
Γ(1 − α) cos
Ad
|θ|α ,
α
2
|θ|
where 0 < c ≤ A d (θ) ≤ C < ∞ for all θ satisfying |θ| = 1,
and hence
c ∗ |θ|α ≤ F d (θ) ≤ c ∗ |θ|α
JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 60 No. 12 2015
ON THE ASYMPTOTICS OF COSINE SERIES IN SEVERAL VARIABLES WITH POWER COEFFICIENTS
for all θ with sufficiently small values of the norm |θ|,
where c ∗ and c ∗ are some positive constants.
Application of the methods of studying the asymptotic behavior of trigonometric series, developed in [1–
8,11,12] for analyzing the one-dimensional case, causes
certain difficulty in passing to dimensions d ≥ 2. Therefore, to prove Theorem 1 we used a method of ‘reduction to dimension one’, which allows to express function
F d (θ) as some explicit combination of one-dimensional
functions F 1 (·), or rather functions Hα (·), and thereby,
to reduce analysis of the case we are interested in to the
one-dimensional case. A similar idea was used in [9,10].
Definition (6) of the function A d (θ) can be found
inconvenient in practical applications, since it eventually requires the calculation of integrals determining the
function ω(θ1 , θ2 , . . . , θd ). Let us simplify the definition
(6) by taking advantage of the fact that the multiple integrals determining ω(θ1 , θ2 , . . . , θd ) can be computed explicitly and expressed via the functions
From here, the next theorem about an alternative representation of function A d (θ) follows.
Theorem 2. For function A d (θ), the following alternative representation holds:
X ν(θi 1 , θi 2 , . . . , θi d )
{i j }
θi 2 · · · θi d
where
1
(θ1 + θ2 )|θ1 + θ2 |α −
θ2
1
− (θ1 − θ2 )|θ1 − θ2 |α +
θ2
1
+ (θ2 + θ1 )|θ2 + θ1 |α −
θ1
1
− (θ2 − θ1 )|θ2 − θ1 |α .
θ1
F̃ 2 (θ) =
Now consider the case when function F (θ) is defined
by a series of more general than (1) form.
Theorem 3. Let 0 < α ≤ 2 and
X
F (θ) =
a z (1 − cos⟨z, θ⟩),
θ ∈ Rd ,
z∈Zd \{0}
where
a z kzkd +α → 1 as kzk → ∞.
(8)
Then, F (θ) ' F d (θ) as θ → 0.
x (m,α) = x m |x|α .
A d (θ) =
1443
,
(7)
where summation is taken over all permutations {i j } of
coordinates of the vector θ = {θ1 , θ2 , . . . , θd }, and function
ν(·) is determined by the equality
1
×
ν(θ1 , θ2 , . . . , θd ) =
(α + 1) · · · (α + d − 1)
X
×
s 2 · · · s d (θ1 + s 2 θ2 + · · · + s d θd )(d −1,α) .
Thus, for 0 < α ≤ 2 the asymptotics of F (θ) can be calculated explicitly by using Theorem 1 for F d (θ). In the case
α > 2, one can only say that F (θ) has the same order of
decrease at zero as function F d (θ), that is F (θ) ∼ F d (θ)
as θ → 0. Here, for functions h 1 (x) and h 2 (x), we write
h 1 (x) ∼ h 2 (x) as x → x 0 if there exist c,C ∈ (0, ∞) such
that c ≤ h 1 (x)/h 2 (x) ≤ C as x → x 0 .
If, in Theorem 3, the condition (8) is replaced by a
less restrictive condition
0 < a ∗ ≤ a z kzkd +α ≤ a ∗ < ∞,
z ∈ Zd \ {0},
then,
a ∗ F d (θ) ≤ F (θ) ≤ a ∗ F d (θ),
and one can only say that the order of decrease of function F (θ) for each α > 0 is the same as the order of
decrease of function F d (θ), that is, in this case, F (θ) ∼
F d (θ) as θ → 0. However, it is difficult to obtain exact
asymptotics for function F (θ) in this case.
s 2 ,...,s d =±1
ACKNOWLEDGMENTS
The resulting expression for function A d (θ) is albeit
more cumbersome than (6) but has the advantage that
it does not require to calculate any integrals.
Let us present an example of calculating the asymptotic behavior of function F d (θ) for d = 2 and 0 < α < 2
with the help of Theorems 1 and 2.
The research was carried out at the Institute for Information Transmission Problems, Russian Academy of
Science, at the expense of the Russian Foundation for
Sciences (project no. 14-50-00150).
REFERENCES
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F 2 (θ) '
³ πα ´
2
Γ(1 − α) cos
F̃ 2 (θ),
α(α + 1)
2
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Translated by V. Kozyakin
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