On the Dirichlet Kernels with respect to certain special representative product systems
The Fourier analysis uses the calculations with kernel functions from the very beginning. The maximal values of the n th Dirichlet kernels divided by n for the Walsh–Paley, “classical” Vilenkin, and some other systems are 1. In the paper, we deal with some more general systems and use the accumulate...
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Інститут математики НАН України
2014
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| Цитувати: | On the Dirichlet Kernels with respect to certain special representative product systems / I. Blahota // Український математичний журнал. — 2014. — Т. 66, № 11. — С. 1578–1584. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860211239746011136 |
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| author | Blahota, I. |
| author_facet | Blahota, I. |
| citation_txt | On the Dirichlet Kernels with respect to certain special representative product systems / I. Blahota // Український математичний журнал. — 2014. — Т. 66, № 11. — С. 1578–1584. — Бібліогр.: 4 назв. — англ. |
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| description | The Fourier analysis uses the calculations with kernel functions from the very beginning. The maximal values of the n th Dirichlet kernels divided by n for the Walsh–Paley, “classical” Vilenkin, and some other systems are 1. In the paper, we deal with some more general systems and use the accumulated results to develop the methods aimed at determination of the properties of specific systems. In these cases, the situation with Dnn may be different.
Аналіз Фур'є використовує розрахунки з ядерними Функціями з самого початку. Максимальні значення n-х ядер Діріхлє, поділених на n, для систем Уолша-Пейлі, „класичних" систем Віленкіна та деяких інших систем дорівнюють 1. Ми розглядаємо більш загальні системи і, використовуючи результати, що отримані, розробляємо методи, призначені для визначення властивостей конкретних систем. У цих випадках ситуація з відношенням Dnn може бути іншою.
|
| first_indexed | 2025-12-07T18:14:17Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.5
I. Blahota (College Nyı́regyháza, Inst. Math. and Comput. Sci., Hungary)
ON THE DIRICHLET KERNELS WITH RESPECT
TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS*
ПРО ЯДРА ДIРIХЛЕ ВIДНОСНО ДЕЯКИХ СПЕЦIАЛЬНИХ
РЕПРЕЗЕНТАТИВНИХ СИСТЕМ ДОБУТКIВ
The Fourier analysis uses the calculations with kernel functions from the beginning. The maximal values of the nth
Dirichlet kernels divided by n for the Walsh – Paley, for the “classical” Vilenkin, and some other systems are 1. In the paper
we deal with some more general systems and, from the results, we develop methods aimed at assigning the properties of
specific systems. In these cases, the situation with
Dn
n
can be different.
Аналiз Фур’є використовує розрахунки з ядерними функцiями з самого початку. Максимальнi значення n-х ядер
Дiрiхле, подiлених на n, для систем Уолша – Пейлi, „класичних” систем Вiленкiна та деяких iнших систем дорiв-
нюють 1. Ми розглядаємо бiльш загальнi системи i, використовуючи результати, що отриманi, розробляємо методи,
призначенi для визначення властивостей конкретних систем. У цих випадках ситуацiя з вiдношенням
Dn
n
може
бути iншою.
1. Introduction. Let m := (m0,m1, . . . ) be a sequence of positive integers not less than 2. Denote
N by the set of nonnegative integers and P by the set of positive ones. Denote by Gk a finite group
(not necessarily Abelian) with order mk, k ∈ N. Define the measure on Gk as follows:
µk({j}) :=
1
mk
, j ∈ Gk, k ∈ N.
Let G be the complete direct product of the sets Gk, with the product of the topologies and measures
(denoted by µ). This product measure is a regular Borel one on G with µ(G) = 1. If the sequence
m is bounded, then G compact totally disconnected group is called a bounded group, otherwise it is
an unbounded one. The elements of G can be represented by sequences x := (x0, x1, . . . ). It is easy
to give a neighbourhood base of G :
I0(x) := G,
In(x) := {y ∈ G|y0 = x0, . . . , yn−1 = xn−1}
for x ∈ G, n ∈ P. Define the well-known generalized number system in the usual way. If M0 := 1,
Mk+1 := mkMk, k ∈ N, then every n ∈ N can be uniquely expressed as n =
∑∞
j=0
njMj , where
0 ≤ nj < mj , j ∈ N, and only a finite number of nj’s differ from zero. Let |n| := max{k ∈ N :
nk 6= 0} (that is, M|n| ≤ n < M|n|+1) if n ∈ P, and |0| := 0. Let n(k) :=
∑∞
j=k
njMj .
Denote by Σk the dual object of the group Gk, k ∈ N. So each σ ∈ Σk is a set of continuous
irreducible unitary representations of Gk which are equivalent to some fixed representation U (σ).
* The paper was supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051.
c© I. BLAHOTA, 2014
1578 ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
ON THE DIRICHLET KERNELS WITH RESPECT TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS 1579
Let dσ be the dimension of its representation space and let {ζ1, . . . , ζdσ} be a fixed, but arbitrary
orthonormal basis in the representation space. The functions
u
(σ)
i,j (x) := 〈U (σ)
x ζi, ζj〉, i, j ∈ {1, . . . , dσ}, x ∈ G,
are called the coordinate functions of U (σ) and {ζ1, . . . , ζdσ} is the basis. Thus for each σ ∈ Σk we
get d2σ coordinate functions, mk number of functions for the whole dual object of Gk in all.
Let {ϕsk : 0 ≤ s < mk} be a system of all normalized coordinate functions of the group Gk. We
do not decide the order of the system ϕ now, but we suppose that ϕ0
k is always the character 1. So
for every 0 ≤ s < mk there exists a σ ∈ Σk, i, j ∈ {1, . . . , dσ}, such that
ϕsk(x) =
√
dσu
(σ)
i,j (x), x ∈ G.
Let ψ be the product system of ϕsk, namely
ψn(x) :=
∞∏
k=0
ϕnkk (xk), x ∈ G, n ∈ N.
We say that ψ is the representative product system of ϕ. The Weyl – Peter’s theorem (see [3]) ensures
that the system ψ is orthonormal and complete on L2(G).
Let f : G→ C an integrable function. Let us define the Fourier coefficients and partial sums by
f̂k :=
∫
G
fψk, k ∈ N, Snf :=
n−1∑
k=0
f̂kψk, n ∈ P.
We define the Dirichlet kernels in this way
Dn(y, x) :=
n−1∑
k=0
ψk(y)ψk(x), n ∈ P, D0 := 0.
We notice that in most of restricted systems (denoted by ϑ now) Dirichlet kernel functions depend
only on one element of the domain. The “one way” connection between the two conceptions is
Dn(y, x) = Dϑ
n(y − x).
It is easy to see that
Snf(x) =
∫
G
f(y)Dn(x, y)dµ(y).
This shows the importance of the Dirichlet kernels in the study of the convergence of Fourier series.
For more on representative product systems see, e.g., [2].
The representative product systems are generalizations of the known Walsh – Paley and Vilenkin
systems. Indeed, we obtain the Vilenkin systems (which are generalizations of the Walsh – Paley
system) if the sequence m is an arbitrary sequence of integers greater than 1 and Gk = Zmk , the
cyclic group of order mk for all k ∈ N. The characters of Zmk are the generalized Rademacher
functions:
ϕsk(x) = exp(2πısx/mk), s ∈ {0, . . . ,mk − 1}, x ∈ G, ı2 = −1.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1580 I. BLAHOTA
Finally, let us define the maximal value sequence of Dirichlet kernels in the following way
Dn := sup
x,y∈G
|Dn(y, x)|, n ∈ N.
In the cases of the original (commutative) Vilenkin systems (and Walsh – Paley system) n = Dn
holds for every n ∈ N, because of n = Dn(0) ≥ |Dn(x)| for all x ∈ G and n ∈ N in those systems.
In the case of general representative product systems the situation can be different.
2. Auxiliary results.
Lemma 1 [2]. Let x, y ∈ G, n ∈ N. Then
DMn(y, x) =
Mn if y ∈ In(x),
0 if y 6∈ In(x).
Lemma 2 [2]. Let x, y ∈ G, n ∈ N. Then
Dn(y, x) =
∞∑
i=0
DMi(y, x)
ni−1∑
j=0
ϕji (yi)ϕ̄
j
i (xi)
ψn(i+1)(y)ψ̄n(i+1)(x).
The next result helps us in the counting of the Dirichlet kernels.
Lemma 3 [4]. Let n ∈ N. Then
Dn = sup
x∈G
Dn(x, x).
Lemma 4 [1]. Let n ∈ P. Then
1 ≤ Dn
n
≤ m|n|.
The study of quotients
Dn
n
is important in order to estimate the Dirichlet kernels. As it was
mentioned these quotients equal 1 in the commutative cases, but they can be unbounded if the
dimensions of the representations appeared in the finite groups are also unbounded (for more see [4]).
On the other hand the order of the systems plays an important role in the value of the quotient
Dn
n
.
The statement of the following lemma is necessary to the proof of one of my new result.
Lemma 5 [1]. Sequence Dn is monotonically increasing.
3. Results on special systems. If the system is based on the product of same groups, and the
generalized Rademacher functions are also the same on them, we get an interesting property of the
Dirichlet kernel functions.
Lemma 6. Let x ∈ I|n|(y)\I|n|+1(y), where x, y ∈ G, n ∈ N and let z̆ := (z1, z2, . . . ) for any
z ∈ G. If mk = p and ϕsk(x) = ϕs(x) for all x ∈ G, k ∈ N, s ∈ {0, . . . , p− 1}, where 2 ≤ p ∈ N
is fixed, then
Dpn(y, x) = pDn(y̆, x̆).
Proof. From Lemmas 1 and 2 we obtain, that if x ∈ I|n|(y)\I|n|+1(y), where x, y ∈ G and
n ∈ N, then
Dn(y, x) =
|n|∑
i=0
Mi
ni−1∑
j=0
ϕji (yi)ϕ̄
j
i (xi)
ψn(i+1)(y)ψ̄n(i+1)(x)
holds.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
ON THE DIRICHLET KERNELS WITH RESPECT TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS 1581
In this way, because of |pn| = |n|+ 1 and (pn)i = ni−1 we get
Dpn(y, x) =
|pn|∑
i=0
Mi
(pn)i−1∑
j=0
ϕj(yi)ϕ̄
j(xi)
ψ(pn)(i+1)(y)ψ̄(pn)(i+1)(x) =
=
|n|+1∑
i=0
Mi
ni−1−1∑
j=0
ϕj(yi)ϕ̄
j(xi)
ψ(pn)(i+1)(y)ψ̄(pn)(i+1)(x) = (∗).
If |(pn)(i+1)| = 0 then |n(i)| = 0. Otherwise |(pn)(i+1)| = |pn|. Let k− := k− 1. Using (pn)(i+1) =
= pn(i) and zk = z̆k− it is easy to see that
ψ(pn)(i+1)(z) =
|pn|∏
k=0
ϕ((pn)(i+1))k(zk) =
|n|+1∏
k=0
ϕ(pn(i))k(zk) =
=
|n|+1∏
k=1
ϕ(n(i))k−1(zk) =
|n|∏
k−=0
ϕ(n(i))k− (z̆k−) = ψn(i)(z̆).
So
(∗) =
|n|∑
i−=0
Mi−+1
ni−−1∑
j=0
ϕj(yi−+1)ϕ̄
j−+1(xi−+1)
ψ
n(i−+1)(y̆)ψ̄
n(i−+1)(x̆) =
= p
|n|∑
i−=0
Mi−
ni−−1∑
j=0
ϕj(y̆i−)ϕ̄j(x̆i−)
ψ
n(i−+1)(y̆)ψ̄
n(i−+1)(x̆) = pDn(y̆, x̆).
Lemma 6 is proved.
Corollary 1. If mk = p and ϕsk(x) = ϕs(x) for all x ∈ G, k ∈ N, s ∈ {0, . . . , p − 1}, where
2 ≤ p ∈ N is fixed, then
Dpn = pDn
holds.
Proof. Considering the definition of Dn, it is an obvious consequence of Lemmas 6 and 3.
This Corollary 1 can account for the fractal-like, self-similar structure of the graph of Dn in the
case of any regular order of the system for S3 (see, e.g., Fig. 1 or [4]) and in the case of other systems,
as for Q2 or for U4. For more see [4].
On the other hand, Corollary 1 could help us counting preciser estimates to expressions containing
Dn. For example Lemma 4 gives us a rough upper estimate as
Dn
n
≤ p in this special case, but using
Corollary 1 we can verify the next theorem, which is a good tool to get a better estimation.
Theorem 1. If mk = p and ϕsk(x) = ϕs(x) for all x ∈ G, k ∈ N, s ∈ {0, . . . , p − 1}, where
2 ≤ p ∈ N is fixed, then
Dn
n
< e
1
(p−1)pr−1 max
k∈{pr−1+1,...,pr}
Dk
k
for all r, n ∈ P.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1582 I. BLAHOTA
ON THE DIRICHLET KERNELS WITH RESPECT TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS 5
63
63
64
64
Figure 1. Dn on the complete product of S3.
=
Dn[j+1]
n[j + 1]
n[j + 1]
n[j + 1] − 1
≤
Dn[j+1]
n[j + 1]
�
1 +
1
p|n|−j−1
�
.
From this inequality we obtain
Dn
n
≤ max
k∈{pr−1+1,...,pr}
Dk
k
|n|−r�
j=0
�
1 +
1
p|n|−j−1
�
< max
k∈{pr−1+1,...,pr}
Dk
k
∞�
i=0
�
1 +
1
pr−1+i
�
,
and from the arithmetic-geometric mean inequality
n�
i=0
�
1 +
1
pr−1+i
�
≤
n +
n�
i=0
1
pr−1+i
n
n
<
<
1 +
∞�
i=0
1
pr−1+i
n
n
n→∞−−−→ e
1
(p−1)pr−1 .
Theorem 1 is proved.
Since lim
r→∞
e
1
(p−1)pr−1 = 1, Theorem 1 give us a relative error easily. In this way we can
approximate supn∈P
Dn
n
arbitrary.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 711
Fig. 1. Dn on the complete product of S3.
Proof. If n < p, then the proof is trivial. Otherwise let us define sequences n[k] and l[k] in the
following way. Let n[0] := n and n[k] := p(n[k + 1] − 1) + l[k], where l[k] ∈ {0, . . . , p − 1} and
n[k] ∈ {p|n|−k + 1, . . . , p|n|+1−k}. Using Corollary 1 and Lemma 5 we have
Dn[j]
n[j]
=
Dp(n[j+1]−1)+l[j])
p(n[j + 1]− 1) + l[j]
≤
Dpn[j+1]
p(n[j + 1]− 1)
=
Dn[j+1]
n[j + 1]− 1
=
=
Dn[j+1]
n[j + 1]
n[j + 1]
n[j + 1]− 1
≤
Dn[j+1]
n[j + 1]
(
1 +
1
p|n|−j−1
)
.
From this inequality we obtain
Dn
n
≤ max
k∈{pr−1+1,...,pr}
Dk
k
|n|−r∏
j=0
(
1 +
1
p|n|−j−1
)
< max
k∈{pr−1+1,...,pr}
Dk
k
∞∏
i=0
(
1 +
1
pr−1+i
)
,
and from the arithmetic-geometric mean inequality
n∏
i=0
(
1 +
1
pr−1+i
)
≤
n+
∑n
i=0
1
pr−1+i
n
n
<
<
1 +
∑∞
i=0
1
pr−1+i
n
n
n→∞−−−→ e
1
(p−1)pr−1 .
Theorem 1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
ON THE DIRICHLET KERNELS WITH RESPECT TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS 1583
Table 1. A possible system for S3
e (12) (13) (23) (123) (132)
ϕ0 1 1 1 1 1 1
ϕ1
√
2
√
2 −
√
2
2
−
√
2
2
−
√
2
2
−
√
2
2
ϕ2
√
2 −
√
2
√
2
2
√
2
2
−
√
2
2
−
√
2
2
ϕ3 1 −1 −1 −1 1 1
ϕ4 0 0 −
√
6
2
√
6
2
√
6
2
−
√
6
2
ϕ5 0 0 −
√
6
2
√
6
2
−
√
6
2
√
6
2
Since limr→∞ e
1
(p−1)pr−1 = 1, Theorem 1 give us a relative error easily. In this way we can
approximate supn∈P
Dn
n
arbitrary.
Now let us investigate a concrete system. Namely, let us see the complete product of S3, which is
the symmetric group on 3 elements. It means that mk = 6 holds for all k ∈ N. S3 has two characters
and a 2-dimensional representation. The values of the system ϕ obtained from the 2-dimensional
representation depend on the chosen basis. Table 1 contains the values of a possible system ϕ (for
details see [4]). You can see a part of Dn sequence from this system in Fig. 1.
Corollary 2. Let the system be the described one in Table 1 for S3. In this case
Dn
n
< 2.04
for all n ∈ P.
Proof. With some manual counting we verified that
max
k∈{2,...,6}
Dk
k
=
D3
3
=
5
3
.
Using Lemma 6 with p = 6 and r = 1 we have
Dn
n
<
5
3
e
1
5 < 2.04.
In this situation this estimate is definitely better than the 6, what we got from Lemma 4. Enlarging
value of r we can obtain even better upper estimates.
Corollary 3. Let the system be the described one in Table 1 for S3. In this case
1.92303 < sup
n∈P
Dn
n
< 1.92309.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
1584 I. BLAHOTAON THE DIRICHLET KERNELS WITH RESPECT TO SOME SPECIAL REPRESENTATIVE PRODUCT SYSTEMS 7
1
63
2
64
Figure 2. 1 ≤ Dn
n
< 1.92309 on the complete product of S3.
In the end, maximizing
Dk
k
(with the help of a computer program) on the set {6r−1 +1, . . . , 6r},
where r ∈ {1, . . . , 6} we get
5
3
,
30
16
,
180
94
,
1080
562
,
6480
3370
,
38880
20218
, respectively. It is easy to find a
formula for this finite sequence, it is
5 · 6r−1
br
, where b1 = 3 and br = 6br−1 − 2. If this idea also
worked for the following members of the sequence, we would find the exact upper limit easily.
Conjecture. Let the system be the described one in Table 1 for S3. In this case
sup
n∈P
Dn
n
=
25
13
.
1. Blahota I. On the maximal value of Dirichlet and Fejér kernels with respect to the Vilenkin-like space // Publ. Math.
Debrecen. – 2012. – 80, № 3, 4. – P. 503 – 513.
2. Gát G., Toledo R. Lp-norm convergence of series in compact totally disconnected groups // Anal. Math. – 1996. –
22. – P. 13 – 24.
3. Hewitt E., Ross K. Abstract harmonic analysis, Heidelberg: Springer-Verlag, 1963.
4. Toledo R. On the maximal value of Dirichlet kernels with respect to representative product systems // Rend. Circ.
mat. Palermo. Ser. II. – 2010. – № 82. – P. 431 – 447.
Received 17.05.11,
after revision — 19.08.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 711
Fig. 2. 1 ≤ Dn
n
< 1.92309 on the complete product of S3.
Proof. With similar method and using a computer program to the mechanical counting it is easy
to show that the exact upper limit is between 1.92303 <
38880
20218
and e
1
38880
38880
20218
< 1.92309 (see
Fig. 2).
Of course we can use this method to estimate for other systems, too.
In the end, maximizing
Dk
k
(with the help of a computer program) on the set {6r−1 + 1, . . . , 6r},
where r ∈ {1, . . . , 6} we get
5
3
,
30
16
,
180
94
,
1080
562
,
6480
3370
,
38880
20218
, respectively. It is easy to find a
formula for this finite sequence, it is
5 · 6r−1
br
, where b1 = 3 and br = 6br−1 − 2. If this idea also
worked for the following members of the sequence, we would find the exact upper limit easily.
Conjecture. Let the system be the described one in Table 1 for S3. In this case
sup
n∈P
Dn
n
=
25
13
.
1. Blahota I. On the maximal value of Dirichlet and Fejér kernels with respect to the Vilenkin-like space // Publ. Math.
Debrecen. – 2012. – 80, № 3, 4. – P. 503 – 513.
2. Gát G., Toledo R. Lp-norm convergence of series in compact totally disconnected groups // Anal. Math. – 1996. –
22. – P. 13 – 24.
3. Hewitt E., Ross K. Abstract harmonic analysis. – Heidelberg: Springer-Verlag, 1963.
4. Toledo R. On the maximal value of Dirichlet kernels with respect to representative product systems // Rend. Circ.
mat. Palermo. Ser. II. – 2010. – № 82. – P. 431 – 447.
Received 17.05.11,
after revision — 19.08.14
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 11
|
| id | nasplib_isofts_kiev_ua-123456789-166316 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T18:14:17Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Blahota, I. 2020-02-18T17:46:19Z 2020-02-18T17:46:19Z 2014 On the Dirichlet Kernels with respect to certain special representative product systems / I. Blahota // Український математичний журнал. — 2014. — Т. 66, № 11. — С. 1578–1584. — Бібліогр.: 4 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166316 517.5 The Fourier analysis uses the calculations with kernel functions from the very beginning. The maximal values of the n th Dirichlet kernels divided by n for the Walsh–Paley, “classical” Vilenkin, and some other systems are 1. In the paper, we deal with some more general systems and use the accumulated results to develop the methods aimed at determination of the properties of specific systems. In these cases, the situation with Dnn may be different. Аналіз Фур'є використовує розрахунки з ядерними Функціями з самого початку. Максимальні значення n-х ядер Діріхлє, поділених на n, для систем Уолша-Пейлі, „класичних" систем Віленкіна та деяких інших систем дорівнюють 1. Ми розглядаємо більш загальні системи і, використовуючи результати, що отримані, розробляємо методи, призначені для визначення властивостей конкретних систем. У цих випадках ситуація з відношенням Dnn може бути іншою. The paper was supported by project TAMOP-4.2.2.A-11/1/KONV-2012-0051 en Інститут математики НАН України Український математичний журнал Короткі повідомлення On the Dirichlet Kernels with respect to certain special representative product systems Про ядра Діріхлє відносно деяких спеціальних репрезентативних систем добутків Article published earlier |
| spellingShingle | On the Dirichlet Kernels with respect to certain special representative product systems Blahota, I. Короткі повідомлення |
| title | On the Dirichlet Kernels with respect to certain special representative product systems |
| title_alt | Про ядра Діріхлє відносно деяких спеціальних репрезентативних систем добутків |
| title_full | On the Dirichlet Kernels with respect to certain special representative product systems |
| title_fullStr | On the Dirichlet Kernels with respect to certain special representative product systems |
| title_full_unstemmed | On the Dirichlet Kernels with respect to certain special representative product systems |
| title_short | On the Dirichlet Kernels with respect to certain special representative product systems |
| title_sort | on the dirichlet kernels with respect to certain special representative product systems |
| topic | Короткі повідомлення |
| topic_facet | Короткі повідомлення |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/166316 |
| work_keys_str_mv | AT blahotai onthedirichletkernelswithrespecttocertainspecialrepresentativeproductsystems AT blahotai proâdradíríhlêvídnosnodeâkihspecíalʹnihreprezentativnihsistemdobutkív |