On the norms of the means of spherical Fourier sums
The spherical Fourier sums of a periodic functions in m variables, the strong means and the strong integral means of these sums for p ≥ 1 are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for m ≥ 2 the norms of the corresponding operators in the space L∞ are no...
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| Цитувати: | On the norms of the means of spherical Fourier sums / O.I. Kuznetsova // Труды Института прикладной математики и механики. — Донецьк: ІПММ, 2015. — Т. 29. — С. 95-99. — Бібліогр.: 9 назв. — англ. |
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irk-123456789-1242322017-09-23T03:03:58Z On the norms of the means of spherical Fourier sums Kuznetsova, O.I. The spherical Fourier sums of a periodic functions in m variables, the strong means and the strong integral means of these sums for p ≥ 1 are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for m ≥ 2 the norms of the corresponding operators in the space L∞ are not bounded. The sharp order of growth of these norms is found. The upper and lower bounds differ by a factor depending only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given. Сферическая сумма Фурье периодической функции m переменных, ее сильные средние и сильные интегральные средние рассмотрены при p ≥ 1. В отличие от одномерного случая, рассмотренного Харди и Литвудом, при m ≥ 2 нормы соответствующих операторов в пространстве L∞ не ограничены. Найден точный порядок роста этих норм. Оценки сверху и снизу различаются на коэффициенты, зависящие лишь от размерности . Получено достаточное условие на функцию, обеспечивающее равномерную сильную суммируемость ее ряда Фурье. The present paper is the talk represented in International Conference «Harmonic analysis and approximation, VI», 12–18 September, 2015, Tsaghkadzor, Armenia. 2015 Article On the norms of the means of spherical Fourier sums / O.I. Kuznetsova // Труды Института прикладной математики и механики. — Донецьк: ІПММ, 2015. — Т. 29. — С. 95-99. — Бібліогр.: 9 назв. — англ. 1683-4720 http://dspace.nbuv.gov.ua/handle/123456789/124232 531.35 en Труды Института прикладной математики и механики Інститут прикладної математики і механіки НАН України |
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The spherical Fourier sums of a periodic functions in m variables, the strong means and the strong integral means of these sums for p ≥ 1 are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for m ≥ 2 the norms of the corresponding operators in the space L∞ are not bounded. The sharp order of growth of these norms is found. The upper and lower bounds differ by a factor depending only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given. |
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Kuznetsova, O.I. |
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Kuznetsova, O.I. On the norms of the means of spherical Fourier sums Труды Института прикладной математики и механики |
| author_facet |
Kuznetsova, O.I. |
| author_sort |
Kuznetsova, O.I. |
| title |
On the norms of the means of spherical Fourier sums |
| title_short |
On the norms of the means of spherical Fourier sums |
| title_full |
On the norms of the means of spherical Fourier sums |
| title_fullStr |
On the norms of the means of spherical Fourier sums |
| title_full_unstemmed |
On the norms of the means of spherical Fourier sums |
| title_sort |
on the norms of the means of spherical fourier sums |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/124232 |
| citation_txt |
On the norms of the means of spherical Fourier sums / O.I. Kuznetsova // Труды Института прикладной математики и механики. — Донецьк: ІПММ, 2015. — Т. 29. — С. 95-99. — Бібліогр.: 9 назв. — англ. |
| series |
Труды Института прикладной математики и механики |
| work_keys_str_mv |
AT kuznetsovaoi onthenormsofthemeansofsphericalfouriersums |
| first_indexed |
2025-07-09T01:05:31Z |
| last_indexed |
2025-07-09T01:05:31Z |
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1837129411304882176 |
| fulltext |
ISSN 1683-4720 Труды ИПММ. 2015. Том 29
UDK 531.35
c©2015. O. I. Kuznetsova
ON THE NORMS OF THE MEANS OF SPHERICAL fOURIER SUMS1
The spherical Fourier sums SR(f, x) =
∑
‖k‖�R
f̂(k) eik·x of a periodic functions in m variables, the strong
means
(
1
n
n−1∑
j=0
|Sj(f, x)|p
)1/p
and the strong integral means
((
R∫
0
|sr(f, x)|pdx
)
/R
)1/p
of these sums
for p ≥ 1 are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for
m ≥ 2 the norms of the corresponding operators in the space L∞ are not bounded. The sharp order of
growth of these norms is found. The upper and lower bounds differ by a factor depending only on the
dimension m.
A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series
is given.
Keywords: Multiple Fourier series, spherical sums, strong means.
1. Introduction. Given a function f integrable on the cube Tm = [−π, π]m, the
spherical partial sum SR(f, x) of the Fourier series of f , has the form (hereafter k ∈ Zm)
SR(f, x) =
∑
‖k‖�R
f̂(k) eik·x, where f̂(k) =
1
(2π)m
∫
Tm
f(u) e−ik·udu.
This is the convolution f with the spherical Dirichlet kernal
DR(x) =
1
(2π)m
∑
‖k‖�R
e−ik·x.
The L1-norm
LR =
∫
Tm
|DR(x)| dx = sup
|f |�1
|SR(f, 0)|
of this kernel is called the Lebesgue constant. In the multidimensional case (m > 1) the
two-sided bound AmR
m−1
2 � LR � BmR
m−1
2 is valid for R � 1 (see [1], [2]).
The strong spherical means of the Fourier series are defined as follows (hereafter p � 1
and n ∈ N)
Hn,p(f, x) =
( 1
n
n−1∑
j=0
|Sj(f, x)|p
) 1
p =
1
n
1
p
sup
|ε|q�1
∣∣∣∣n−1∑
j=0
εjSj(f, x)
∣∣∣∣.
Here, supremum is taken over all collections ε = {ε0, . . . , εn−1} of real numbers satisfying
the condition
|ε|q =
(|ε0|q + . . . + |εn−1|q
) 1
q � 1
1The present paper is the talk represented in International Conference «Harmonic analysis and
approximation, VI», 12–18 September, 2015, Tsaghkadzor, Armenia.
95
O. I. Kuznetsova
(q is the conjugate exponent: 1
p + 1
q = 1).
Our purpose is to estimate the norms of the corresponding operators, i.e., the quantities
Hn,p = sup
|f |�1
Hn,p(f, 0) =
1
n
1
p
sup
|ε|q�1
∫
Tm
∣∣∣∣n−1∑
j=0
εjDj(x)
∣∣∣∣dx.
Note that, by Hölder’s inequality, the means Hn,p(f), and hence the norms Hn,p,
increase with p.
The notion of strong summability of Fourier series was introduced by Hardy and
Littlewood [3] in the one-dimensional case more one hundred years ago. They proved
that for m = 1 and a fixed p, the norms Hn,p are bounded.
In the multidimensional case, the situation is different. For m � 3 the norms Hn,p
not bounded, being of order n
m−1
2
−min{ 1
2
, 1
p
} (см. [4], [5]). In the two-dimensional case,
the results were not complete: this two-sided bound still holds true for any fixed p > 2,
while for p ∈ [1, 2] only the upper bound Hn,p � c
√
ln(n+ 1) was known [see 6,7].
2. The main result for Hn,p. We prove [8] that 2
Hn,p �
⎧⎪⎪⎨⎪⎪⎩
n
m−1
2
−min{ 1
2
, 1
p
} if m � 3, p � 1;
n
1
2
− 1
p min
1
p
{
ln(n+ 1), 1
p−2
}
if m = 2, p � 2;√
ln(n+ 1) if m = 2, p ∈ [1, 2].
Note that in the two-dimensional case, for “large” p (with p − 2 greater that a fixed
positive number) the factor min
1
p
{
ln(n+ 1), 1
p−2
}
can be replaced by 1.
3. The main lemma.
Lemma 1. Let a1, . . . , an be nonnegative numbers. Then
sup
εj=±1
∫ π
0
∣∣∣∣ n∑
j=1
εjaj cos
(
jt+
π
4
)∣∣∣∣ dt√t �
√
lnn
n
n∑
j=1
aj .
With aj =
√
j this immediately implies the desired relation Hn,1 � √
lnn.
4. The norms of the integral means of spherical Fourier sums. In addition
to the means Hn,p(f), it is also natural to consider their integral analogs defined by the
equality
HR,p(f, x) =
( 1
R
∫ R
0
|Sr(f, x)|pdr
) 1
p
.
Note that averaging over the radius was used systematically in many papers, for example,
in the study of the Riesz sums of multiple Fourier series.
Our goal is to estimate of the norms the corresponding operators, i.s., the quantities
2The notion αn � βn means that αn = O(βn) and βn = O(αn) simultaneously. Instead of αn =
O(βn) we also write αn � βn. Hereafter, the constants in the corresponding inequalities valid for all n
may depend only on the dimension m.
96
On the norms of the means of spherical Fourier sums
HR,p = sup
|f |�1
HR,p(f, 0).
In view of Hölder’s inequality, the means HR,p(f)p increase with p and, therefore, so
do the norms HR,p. In [7] they were estimated from above. It is easy to see that, to do
this, it suffices to consider HR,p only for integer values R.
We prove [9] that the norms Hn,p и Hn,p have the same order. More precisely,
Hn,p � Hn,p �
⎧⎪⎪⎨⎪⎪⎩
n
m−1
2
−min{ 1
2
, 1
p
} if m � 3, p � 1;
n
1
2
− 1
p min
1
p
{
ln(n+ 1), 1
p−2
}
if m = 2, p > 2;√
ln(n+ 1) if m = 2, p ∈ [1, 2].
5. Estimating Hn,p from below. We begin with two-dimensional case. For 1 �
p � 2
Hn,p = Hn,p +O(1) � αm
√
ln(n+ 1).
In addition, for p > 2 the relations
Hn,p � n
1
2
− 1
p min
1
p
{
ln(n+ 1), 1
p−2
}
и Hn,p = Hn,p +O
(
n
1
2
− 1
p
)
imply
Hn,p � βmn
1
2
− 1
p min
1
p
{
ln(n+ 1), 1
p−2
}
at least for p, sufficiently close to 2.
We need following result for p � 2.
Further, R > 1, 0 < δ � 1; fR,δ is a function, equal to cos
(
R‖x‖ − π
4 (m + 1)
)
for
‖x‖ � δ and zero for other x ∈ Tm; B(r) is the ball of radius r centered at zero.
Lemma 2. If δ is sufficiently small and the product Rδ is sufficiently large (the
boundaries depend only the dimension m), then the function fR,δ satisfies the inequality
Sr(fR,δ, 0) =
∫
B(δ)
fR,δ(x)Dr(x) dx � cm (rδ)
m−1
2 for all r, |r −R| � 1/δ.
For the coefficient cm we can take the fraction π
m−1
2 /2Γ(1 + m
2 ).
In view [1] |Sr(f, 0)| �
∫
Tm |Dr(x)| dx = O
(
r
m−1
2
)
, if |f | � 1, then this implies
that, for fR,δ,not only the spherical sum SR(fR,δ, 0), but also all the sums Sr(fR,δ, 0) for
R− 1/δ � r � R+ 1/δ are the largest possible (among bounded functions).
For p�2 the lemma 2 (for R=n and small δ) immediately yields the lower bound:
Hn,p = sup
|f |�1
(
1
n
∫ n
0
∣∣Sr(f, 0)∣∣pdr) 1
p
� 1
n
1
p
∫ n
n−1
Sr(fn,δ, 0) dr � cmδ
m−1
2 n
m−1
2
− 1
p .
In the two-dimensional case, for “large” p > 2 (the difference p − 2 is bounded away
from zero), the above relations are equivalent to the inequality
Hn,p � βmn
1
2
− 1
p min
1
p
{
ln(n+ 1),
1
p− 2
}
,
97
O. I. Kuznetsova
which, for “small” p > 2, was established at the beginning. Thus, for p > 2 the required
lower bound for the norms Hn,p is obtained for all m = 2, 3, . . .. Recall that, at the
beginning of this section, the lower bound was also obtained for m = 2 and 1 � p � 2.
6. On p-summation of the Fourier series in the strong sense. The estimates
of the norms HR,p, established above, by standard arguments lead to condition on a
continuous periodic function f that ensures the uniform strong p-summability of the
Fourier series, i.e., the uniform (with respect to x ∈ Rm) convergence to zero as R→ +∞
of the quantities
1
R
∫ R
0
|Sr(f, x) − f(x)|pdr.
To do this, we shall need the notion of best uniform approximation of a function f , which
is defined by the equality
ER(f) = min
M
‖f −M‖C , where M(x) =
∑
‖k‖�R
ck e
ik·x.
Corollary. Let f be a continuous and 2π-periodic (in each variable) function in Rm
such thah HR,pER(f) −→
R→+∞
0. Then
max
x
1
R
∫ R
0
|Sr(f, x) − f(x)|pdr −→
R→+∞
0.
By Jackson’s theorem, the best approximation ER(f) is majorized by the modulus
of smoothness of the functions f . Therefore, in the two-dimensional, for 1 � p � 2, the
condition
ωf (t) = o(1/
√
| ln t|)
on the modulus of continuity of the function f is sufficient for the uniform strong p-
summability of the spherical sums and, for a fixed p > 2, it ensures the more restrictive
Lipschitz condition
ωf (t) = o(|t| 12− 1
p ).
1. Babenko K.I. On the mean convergence of multiple Fourier series and the asymptotic behavior of the
Dirichlet kernel of the spherical means, Preprint № 52, Inst. Prikl. Mat. Akad. Nauk SSSR (1971).
2. Il’in V.A. Localization and convergence problem for Fourier series in fundamental function systems
of Laplace’s operator // Uspehi Mat. Nauk. – 1968. – 23:2. – P. 61–120.
3. Hardy G.H., Littlewood J.E. Sur la serie de Fourier d’une function á carré summable // CR. – 1913.
– 156. – P. 1303–1309.
4. Kuznetsova O.I. Strong means and the convergence in L multiple trigonometric series // Ukr. Math.
Bull. – 2006. – 3:1. – P. 46–63.
5. Kuznetsova O.I. Strong means and the convergence of multiple trigonometric series in L // Dokl.
Akad. Nauk. – 2003. – 391:3. – P. 303–305.
6. Kuznetsova O.I. On the problem of strong summation over disks // Ukrainian Math. J. – 1996. –
48:5. – P. 629–634.
7. Kuznetsova O.I. Strong spherical means of multiple Fourier series // J. Contemp. Math. Anal.,
Armen. Acad. Sci. – 2009. – 44:4. – P. 219–229.
98
On the norms of the means of spherical Fourier sums
8. Kuznetsova O.I., Podkorytov A.N. On strong averages of spherical Fourier sums // – Algebra Anal.
– 2013. – 25:3. – P. 447–453.
9. Kuznetsova O.I., Podkorytov A.N. On the norms of the integral means of spherical Fourier sums
// Mathematical Notes. – 2014. – 96:5. – P. 55–62.
О.И. Кузнецова
О нормах средних сферических сумм Фурье.
Сферическая сумма Фурье SR(f, x) =
∑
‖k‖�R
f̂(k) eik·x периодической функции m переменных, ее
сильные средние и сильные интегральные средние рассмотрены при p ≥ 1. В отличие от одномер-
ного случая, рассмотренного Харди и Литвудом, при m ≥ 2 нормы соответствующих операторов в
пространстве L∞ не ограничены. Найден точный порядок роста этих норм. Оценки сверху и снизу
различаются на коэффициенты, зависящие лишь от размерности . Получено достаточное условие
на функцию, обеспечивающее равномерную сильную суммируемость ее ряда Фурье.
Ключевые слова: Кратные ряды Фурье, сферические суммы, сильные средние
Ин-т прикл. математики и механики, Донецк
kuznets@iamm.su
Received 02.10.15
99
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