Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series
We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L...
Збережено в:
| Дата: | 2016 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/1824 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507691433066496 |
|---|---|
| author | Nagy, K. Нагі, К. |
| author_facet | Nagy, K. Нагі, К. |
| author_sort | Nagy, K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:28:56Z |
| description | We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L_p$, we mean $C$, i.e., the collection of continuous functions.
Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz –Norlund means is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz –Norlund means. Earlier results for one-dimensional N¨orlund means of the Walsh – Fourier series was given by M´oricz and Siddiqi [J.
Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). – 2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz –N¨orlund means by the author [Georg. Math. J. –2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by M´oricz and Rhoades [J. Approxim. Theory. –
1987. – 50. – P. 341 – 358]. |
| first_indexed | 2026-03-24T02:13:20Z |
| format | Article |
| fulltext |
UDC 517.5
K. Nagy (Inst. Math. and Comput. Sci., College of Nyı́regyháza, Hungary)
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL
PARTIAL SUMS OF DOUBLE WALSH – KACZMARZ-FOURIER SERIES*
НАБЛИЖЕННЯ СЕРЕДНIМИ НОРЛУНДА КВАДРАТИЧНИХ
ЧАСТКОВИХ СУМ ПОДВIЙНИХ РЯДIВ УОЛША – КАЧМАРЖА – ФУР’Є
We discuss the Nörlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in Lp. We
investigate the rate of approximation by this means, in particular, in Lip(\alpha , p), where \alpha > 0 and 1 \leq p \leq \infty . For p = \infty ,
by Lp, we mean C, i.e., the collection of continuous functions.
Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz – Nörlund means
is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz – Nörlund means.
Earlier results for one-dimensional Nörlund means of the Walsh – Fourier series was given by Móricz and Siddiqi [J.
Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). –
2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz – Nörlund means by the author [Georg. Math. J. –
2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by Móricz and Rhoades [J. Approxim. Theory. –
1987. – 50. – P. 341 – 358].
Розглядаються середнi Норлунда для квадратичних часткових сум рядiв Уолша – Качмаржа – Фур’є функцiї з про-
стору Lp. Вивчено швидкiсть наближення цими середнiми, зокрема, в Lip(\alpha , p), де \alpha > 0 та 1 \leq p \leq \infty . Для
p = \infty пiд Lp ми розумiємо C, тобто набiр всiх неперервних функцiй.
Основна теорема у цiй статтi стверджує, що апроксимацiйна поведiнка таких двовимiрних середнiх Уолша –
Качмаржа – Норлунда так само гарна, як i апроксимацiйна поведiнка одновимiрних середнiх Уолша та Уолша –
Качмаржа – Норлунда.
Ранiше результати для одновимiрних середнiх Норлунда рядiв Уолша – Фур’є були отриманi Морiчем та Сiддiкi
[J. Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] та Фрiдлi, Манчанда i Сiддiкi [Acta Sci. Math. (Szeged). –
2008. – 74. – P. 593 – 608]. Для одновимiрних середнiх Уолша – Качмаржа – Норлунда вiдповiднi результати були
отриманi автором [Georg. Math. J. – 2011. – 18. – P. 147 – 162]. Випадок двовимiрних тригонометричних систем
було розглянуто Морiчем i Роадсом [J. Approxim Theory. – 1987. – 50. – P. 341 – 358].
1. Nörlund means. Let \{ qk : k \geq 1\} be a sequence of nonnegative numbers. The Nörlund means
and kernels of the Walsh – (Kaczmarz) – Fourier series are defined by
t\alpha n(f, x) :=
1
Qn
n - 1\sum
k=1
qn - kS
\alpha
k (f, x), L\alpha
n(x) :=
1
Qn
n - 1\sum
k=1
qn - kD
\alpha
k (x),
where Qn :=
\sum n - 1
k=1
qk, n \geq 1, and \alpha is the Walsh system in the Paley or Kaczmarz enumeration.
We always assume that q1 > 0 and
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Qn = \infty .
In this case, the summability method generated by \{ qk\} is regular (see [17]) if and only if
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
qn - 1
Qn
= 0.
* This paper supported by project TÁMOP-4.2.2.A-11/1/KONV-2012-0051.
c\bigcirc K. NAGY, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1 87
88 K. NAGY
In particular case t\alpha n are the Fejér means (for all k set qk = 1) and t\alpha n are the (C, \beta )-means\Biggl(
qk := A\beta
k :=
\biggl(
\beta + k
k
\biggr)
for k \geq 1 and \beta \not = - 1, - 2, . . .
\Biggr)
.
In the paper [17] the rate of the approximation by Nörlund means for Walsh – Fourier series of
a function in Lp (in particular, in Lip (\alpha , p), where \alpha > 0 and 1 \leq p \leq \infty ) was studied. In case
p = \infty , by Lp we mean C, the collection of the continuous functions. As special cases Móricz and
Siddiqi obtained the earlier results by Yano [32], Jastrebova [14] and Skvortsov [27] on the rate of
the approximation by Cesàro means. The approximation properties of the Cesàro means of negative
order was studied by Goginava in 2002 [10]. In 2008 Fridli, Manchanda and Siddiqi generalized the
result of Móricz and Siddiqi for homogeneous Banach spaces and dyadic Hardy spaces [3]. Recently,
Tephnadze discussed some new aspect of the Nörlund means [29, 30].
The case when qk = 1/k is not discussed in the paper of Móricz and Siddiqi, in this case t\alpha n
are called the Nörlund logarithmic means. It was studied for Walsh system by Gát, Goginava and
Tkebuchava earlier [5, 9], for unbounded Vilenkin system by Blahota and Gát [2].
The Nörlund means and kernels of cubical partial sums of the two-dimensional Walsh – (Kaczmarz) –
Fourier series are defined by
\bft \alpha n(f, x
1, x2) :=
1
Qn
n - 1\sum
k=1
qn - kS
\alpha
k,k(f, x
1, x2), \scrL \alpha
n(x
1, x2) :=
1
Qn
n - 1\sum
k=1
qn - kD
\alpha
k (x
1)D\alpha
k (x
2).
\bft \alpha n is called the nth Nörlund mean of quadratical partial sums or the nth Nörlund mean of
Marcinkiewicz type. The approximation behaviour of this Nörlund means of Marcinkiewicz type
of Walsh – Fourier series was treated by the author [19] in 2010. We mention that the case that
qk := 1/k was not included in that paper. For Walsh system this case is discussed by Gát and
Goginava in [6], they investigated the uniform and L-convergence of the Nörlund logarithmic means
of Marcinkiewicz type. If we choose qk := A\beta
k =
\biggl(
\beta + k
k
\biggr)
(for k \geq 1 and \beta \not = - 1, - 2, . . .), then
we get the (C, \beta )-means of Marcinkiewicz type which was discussed by Goginava [11, 13] with
respect to double Walsh – Fourier series, for \beta = 1 we get the Marcinkiewicz means [20].
In 1948 \u Sneider [28] introduced the Walsh – Kaczmarz system and showed that the inequality
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow \infty
D\kappa
n(x)
\mathrm{l}\mathrm{o}\mathrm{g} n
\geq C > 0
holds a.e. In 1974 Schipp [24] and Young [33] proved that the Walsh – Kaczmarz system is a
convergence system. Skvortsov in 1981 [26] showed that the Fejér means with respect to the Walsh –
Kaczmarz system converge uniformly to f for any continuous functions f. Gát [4] proved, for any
integrable functions, that the Fejér means with respect to the Walsh – Kaczmarz system converge
almost everywhere to the function. Gát’s result was generalized by Simon [25] in 2004. Recently, the
approximation behavior of the Walsh – Kaczmarz – Nörlund means in Lp, 1 \leq p \leq \infty , [18] and the
rate of the approximation of the Cesàro means of negative order in Lp was discussed by the author
[21, 22].
In 2003 the uniform and L-convergence of double Walsh – Kaczmarz – Fourier series was dis-
cussed by Goginava [12]. In 2006 the almost everywhere convergence of the Walsh – Kaczmarz –
Marcinkiewicz means of integrable functions was proved by the author [20] (see also [7]).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 89
2. Walsh system. Now, we give a brief introduction to the Walsh – Fourier analysis [1, 23].
Let us denote by \BbbZ 2 the discrete cyclic group of order 2, the group operation is the modulo 2 addition
and the topology is the discrete topology. The normalized Haar measure on \BbbZ 2 is given in the way
that \mu (\{ 0\} ) = \mu (\{ 1\} ) = 1/2. Let G :=
\infty
\times
k=0
\BbbZ 2, which is called the Walsh group. The elements of G
are sequences x = (x0, x1, . . . , xk, . . .) with coordinates xk \in \{ 0, 1\} , k \in \BbbN .
The group operation on G is the coordinate-wise addition (denoted by +), the normalized Haar
measure (denoted by \mu ) is the product measure, the topology is the product topology. A base for the
neighbourhoods are given by
I0(x) := G, In(x) := \{ y \in G : y = (x0, . . . , xn - 1, yn, yn+1, . . .)\}
for x \in G,n \in \BbbP (\BbbP := \BbbN \setminus \{ 0\} ), they are called dyadic intervals. Let 0 = (0 : i \in \BbbN ) \in G denote the
null element of G and In := In(0) for n \in \BbbN . Set ei := (0, . . . , 0, 1, 0, . . .), where the ith coordinate
is 1 the rest are 0.
Let Lp denote the usual Lebesgue spaces on G (with the corresponding norm \| .\| p). For the
sake of brevity in notation, we agree to write L\infty instead of C and set \| f\| \infty := \mathrm{s}\mathrm{u}\mathrm{p}\{ | f(x)| :
x \in G\} .
For x \in G we define | x| by | x| :=
\sum \infty
j=0
xj2
- j - 1, for x = (x1, x2) \in G2 by | x| 2 := | x1| 2+| x2| 2.
Next, we define the modulus of continuity of a function f \in Lp, 1 \leq p \leq \infty , by
\omega p(\delta , f) := \mathrm{s}\mathrm{u}\mathrm{p}
| t| <\delta
\| f(.+ t) - f(.)\| p, \delta > 0.
We define the mixed modulus of continuity as follows:
\omega p
1,2(\delta 1, \delta 2, f) :=
:= \mathrm{s}\mathrm{u}\mathrm{p}\{ \| f(.+ x1, .+ x2) - f(.+ x1, .) - f(., .+ x2) + f(., .)\| p : | x1| \leq \delta 1, | x2| \leq \delta 2\} ,
where \delta 1, \delta 2 > 0.
The Lipschitz classes in Lp for each \alpha > 0 are defined by
Lip(\alpha , p) := \{ f \in Lp : \omega p(\delta , f) = O(\delta \alpha ) as \delta \rightarrow 0\} .
The Rademacher functions are defined as
rk(x) := ( - 1)xk , x \in G, k \in \BbbN .
Each natural number n can be uniquely expressed as n =
\sum \infty
i=0
ni2
i, ni \in \{ 0, 1\} , i \in \BbbN , where only
a finite number of ni’s different from zero. Let us define the order | n| of n > 0 by | n| := \mathrm{m}\mathrm{a}\mathrm{x} \{ j \in \BbbN :
nj \not = 0\} . The Walsh functions can be enumerated in Paley enumeration as follows, w0 = 1 and for
n \geq 1
wn(x) :=
\infty \prod
k=0
(rk(x))
nk = r| n| (x)( - 1)
\sum | n| - 1
k=0 nkxk .
(Simply we say Walsh – Paley functions, Walsh – Paley system.) The Walsh functions can be given
in other enumerations, the most investigated is the Kaczmarz rearrengement. The Walsh – Kaczmarz
functions are defined by \kappa 0 = 1 and for n \geq 1
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
90 K. NAGY
\kappa n(x) := r| n| (x)
| n| - 1\prod
k=0
(r| n| - 1 - k(x))
nk = r| n| (x)( - 1)
\sum | n| - 1
k=0 nkx| n| - 1 - k .
The set of Walsh – Kaczmarz functions (denoted by \kappa ) and the set of Walsh – Paley functions (denoted
by w) are equal in each dyadic block. Skvortsov [26] gave a relation between the Walsh – Kaczmarz
functions and the Walsh – Paley functions by the transformation \tau A : G \rightarrow G defined by
\tau A(x) := (xA - 1, xA - 2, . . . , x1, x0, xA, xA+1, . . .)
for A \in \BbbN . By the definition of \tau A, we have
\kappa n(x) = r| n| (x)wn - 2| n| (\tau | n| (x)), n \in \BbbN , x \in G.
Moreover, it is showed that the transformation \tau A is measure-preserving. The Dirichlet kernels are
defined by
D\alpha
n :=
n - 1\sum
k=0
\alpha k,
where \alpha n = wn(n \in \BbbP ) or \kappa n (n \in \BbbP ), D\alpha
0 := 0. The 2n th Dirichlet kernels have a closed form
(see, e. g., [23])
Dw
2n = D\kappa
2n = D2n(x) =
\left\{ 2n, x \in In,
0, otherwise (n \in \BbbN ).
(1)
The nth Fejér mean and the nth Fejér kernel of the Fourier series of a function f is defined by
\sigma \alpha
n(f ;x) :=
1
n
n\sum
k=0
S\alpha
k (f ;x), K\alpha
n (x) :=
1
n
n\sum
k=0
D\alpha
k (x), x \in G,
where \alpha := w or \kappa and K\alpha
0 = 0.
On G2 we consider the two-dimensional system as \{ \alpha n1(x1) \times \alpha n2(x2) : n := (n1, n2) \in \BbbN 2\} .
The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series and
Dirichlet kernels are defined in the usual way. Let us define the nth Marcinkiewicz kernel \scrK \alpha
n by
\scrK \alpha
n(x
1, x2) :=
1
n
n\sum
k=0
D\alpha
k (x
1)D\alpha
k (x
2) (x = (x1, x2) \in G2, \alpha = w or \kappa ).
Recently, the almost everywhere convergence of the Walsh – Kaczmarz – Marcinkiewicz means of
integrable functions was discussed by the author [20] and later by Gát, Goginava and the author [7].
3. The rate of the approximation. Now, we decompose the Walsh – Kaczmarz – Nörlund
kernels \scrL \kappa
n. The following lemma is the two-dimensional analogue of the decomposition lemmas in
[17, 18].
Lemma 1. Let | n| = A \geq 1, then
Qn\scrL \kappa
n(x
1, x2) = Qn - 2A - 1+1D2A(x
1)D2A(x
2) -
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 91
- D2A(x
1)w2A - 1(x
2)
2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)jK
w
j (\tau A - 1(x
2)) -
- D2A(x
2)w2A - 1(x
1)
2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)jK
w
j (\tau A - 1(x
1)) -
- qn - 2A - 12A - 1D2A(x
1)w2A - 1(x
2)Kw
2A - 1(\tau A - 1(x
2)) -
- qn - 2A - 12A - 1D2A(x
2)w2A - 1(x
1)Kw
2A - 1(\tau A - 1(x
1))+
+w2A - 1(x
1)w2A - 1(x
2)
2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)j\scrK w
j (\tau A - 1(x
1), \tau A - 1(x
2))+
+qn - 2A - 12A - 1w2A - 1(x
1)w2A - 1(x
2)\scrK w
2A - 1(\tau A - 1(x
1), \tau A - 1(x
2)))+
+
2A - 1 - 2\sum
j=1
(qn - j - qn - j - 1)j\scrK \kappa
j (x
1, x2)+
+qn - 2A - 1+1(2
A - 1 - 1)\scrK \kappa
2A - 1 - 1(x
1, x2)+
+Qn - 2AD2A(x
1)rA(x
2)Lw
n - 2A(\tau A(x
2))+
+Qn - 2AD2A(x
2)rA(x
1)Lw
n - 2A(\tau A(x
1))+
+Qn - 2ArA(x
1)rA(x
2)\scrL w
n - 2A(\tau A(x
1), \tau A(x
2)).
Proof. During the proof of Lemma 1 we use the following equations:
D\kappa
2A+j(x) = D2A(x) + rA(x)D
w
j (\tau A(x)), j = 0, 1, . . . , 2A - 1, (2)
D\kappa
2A - j(x) = D2A(x) - w2A - 1(x)D
w
j (\tau A - 1(x)), j = 0, 1, . . . , 2A - 1. (3)
Let | n| = A, then we write
Qn\scrL \kappa
n(x
1, x2) =
2A\sum
k=1
qn - kD
\kappa
k(x
1)D\kappa
k(x
2) +
n - 1\sum
k=2A+1
qn - kD
\kappa
k(x
1)D\kappa
k(x
2) =: I + II.
By the help of (2), we decompose II:
II =
n - 2A - 1\sum
j=1
qn - 2A - jD
\kappa
2A+j(x
1)D\kappa
2A+j(x
2) =
= D2A(x
1)D2A(x
2)
n - 2A - 1\sum
j=1
qn - 2A - j +D2A(x
1)rA(x
2)
n - 2A - 1\sum
j=1
qn - 2A - jD
w
j (\tau A(x
2))+
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
92 K. NAGY
+D2A(x
2)rA(x
1)
n - 2A - 1\sum
j=1
qn - 2A - jD
w
j (\tau A(x
1)) + rA(x
1)rA(x
2)Qn - 2A\scrL w
n - 2A(\tau A(x
1), \tau A(x
2)) =
= Qn - 2AD2A(x
1)D2A(x
2) +D2A(x
1)rA(x
2)Qn - 2AL
w
n - 2A(\tau A(x
2))+
+D2A(x
2)rA(x
1)Qn - 2AL
w
n - 2A(\tau A(x
1)) + rA(x
1)rA(x
2)Qn - 2A\scrL w
n - 2A(\tau A(x
1), \tau A(x
2)).
We write for I that
I =
2A - 1\sum
j=0
qn - 2A+jD
\kappa
2A - j(x
1)D\kappa
2A - j(x
2) =
=
2A - 1\sum
j=0
qn - 2A+jD
\kappa
2A - j(x
1)D\kappa
2A - j(x
2) +
2A - 1\sum
j=2A - 1+1
qn - 2A+jD
\kappa
2A - j(x
1)D\kappa
2A - j(x
2) =: I1 + I2.
We use (3) and Abel’s transformation for the term I1:
I1 = D2A(x
1)D2A(x
2)(Qn - 2A - 1+1 - Qn - 2A) -
- D2A(x
1)w2A - 1(x
2)
\left( 2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)jK
w
j (\tau A - 1(x
2))+
+qn - 2A - 12A - 1Kw
2A - 1(\tau A - 1(x
2))
\Biggr)
-
- D2A(x
2)w2A - 1(x
1)
\left( 2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)jK
w
j (\tau A - 1(x
1))+
+qn - 2A - 12A - 1Kw
2A - 1(\tau A - 1(x
1))
\Biggr)
+
+w2A - 1(x
1)w2A - 1(x
2)
\left( 2A - 1 - 1\sum
j=1
(qn - 2A+j - qn - 2A+j+1)j\scrK w
j (\tau A - 1(x
1), \tau A - 1(x
2))+
+qn - 2A - 12A - 1\scrK w
2A - 1(\tau A - 1(x
1), \tau A - 1(x
2))
\Biggr)
.
To discuss the expression I2, we set s = 2A - j and use Abel’s transformation:
I2 =
2A - 1 - 1\sum
s=1
qn - sD
\kappa
s (x
1)D\kappa
s (x
2) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 93
=
2A - 1 - 2\sum
s=1
(qn - s - qn - s - 1)s\scrK \kappa
s (x
1, x2) + qn - 2A - 1+1(2
A - 1 - 1)\scrK \kappa
2A - 1 - 1(x
1, x2).
Lemma 1 is proved.
By the help of this lemma we have our main theorem, which states that the approximation
behavior of the two-dimensional Walsh – Kaczmarz – Nörlund means of Marcinkiewicz type is as
good as the approximation behavior of the one-dimensional Walsh – Nörlund means. The last one
was investigated by Móricz and Siddiqi [17] and recently by Fridli, Manchanda and Siddiqi [3].
Moreover, the rate of the approximation of Nörlund means of Marcinkiewicz type are close to each
other for both rearrengement of the Walsh system (see also [19]).
Theorem 1. Let f \in Lp, 1 \leq p \leq \infty (with the notation L\infty = C), | n| = A \geq 1 and \{ qk :
k \geq 1\} be a sequence of nonnegative numbers.
If \{ qk\} is nondecreasing, in sign \uparrow , then
\| \bft \kappa n(f) - f\| p \leq
c
Qn
A - 1\sum
l=0
qn - 2l2
l\omega p(2
- l, f) +O(\omega p(2
- A, f)).
If \{ qk\} is nonincreasing, in sign \downarrow , such that
n
Q2
n
n - 1\sum
k=1
q2k = O(1), (4)
then
\| \bft \kappa n(f) - f\| p \leq
c
Qn
A - 1\sum
l=0
qn - 2l2
l\omega p(2
- l, f) +O(\omega p(2
- A, f)).
To prove our theorem we need the following lemmas proved by Schipp, Móricz [16], Yano [31],
Simon [25], Glukhov [8] and Gát, Goginava, Nagy [7].
Lemma 2 [16]. If the condition (4) is satisfied, then there exists a constant C such that \| Lw
n \| 1 \leq
\leq C, n \geq 1.
Lemma 3 [31]. Let n \geq 1, then \| Kw
n \| 1 \leq 2.
Lemma 4 [25]. There is a constant C such that \| K\kappa
n\| 1 \leq C, n \geq 1.
Lemma 5 [8]. Let \alpha 1, . . . , \alpha n be real numbers. Then
1
n
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
n\sum
k=1
\alpha kD
w
k \otimes Dw
k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1
\leq c\surd
n
\Biggl(
n\sum
k=1
\alpha 2
k
\Biggr) 1/2
,
where c is an absolute constant.
As corollary of the lemma of Glukhov, we get that there exists a constant C such that \| \scrK w
n \| 1 \leq C,
n \geq 1, and the fact that condition (4) implies \| \scrL w
n \| 1 \leq C, n \geq 1), where C is an absolute constant.
Lemma 6 [7]. There exists a constant C such that
\| \scrK \kappa
n\| 1 \leq C, n \geq 1.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
94 K. NAGY
Proof of Theorem 1. Clearly, condition (4) implies the regularity of the summability method.
We make the proof for 1 \leq p < \infty , for p = \infty the proof is analogous (where L\infty = C ), thus we
omit to write it.
Let n \in \BbbN be fixed and set | n| = A. By Lemma 1 and Minkowski inequality we may write that
Qn\| \bft \kappa n(f) - f\| p \leq
\leq Qn - 2A - 1+1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
1)D2A(x
2)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j\times
\times
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
1)w2A - 1(x
2)Kw
j (\tau A - 1(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j\times
\times
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)w2A - 1(x
1)Kw
j (\tau A - 1(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+qn - 2A - 12A - 1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
1)w2A - 1(x
2)Kw
2A - 1(\tau A - 1(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+qn - 2A - 12A - 1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)w2A - 1(x
1)Kw
2A - 1(\tau A - 1(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j\times
\times
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))w2A - 1(x
1)w2A - 1(x
2)\scrK w
j (\tau A - 1(x
1), \tau A - 1(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+ qn - 2A - 12A - 1
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))w2A - 1(x
1)w2A - 1(x
2)\scrK w
2A - 1(\tau A - 1(x
1), \tau A - 1(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 95
+
2A - 1 - 2\sum
j=1
| qn - j - qn - j - 1| j
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))\scrK \kappa
j (x)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+ qn - 2A - 1+1(2
A - 1 - 1)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))\scrK \kappa
2A - 1 - 1(x)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+ Qn - 2A
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
1)rA(x
2)Lw
n - 2A(\tau A(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+ Qn - 2A
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)rA(x
1)Lw
n - 2A(\tau A(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
+
+Qn - 2A
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))rA(x
1)rA(x
2)\scrL w
n - 2A(\tau A(x
1), \tau A(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=:
=:
12\sum
i=1
An,i.
Now, we discuss the expression An,1. By (1) and generalized Minkowski inequality we find\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
1)D2A(x
2)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
\int
I2A
D2A(x
1)D2A(x
2)
\left( \int
G2
| f(y + x) - f(y)| pd\mu (y)
\right) 1/p
d\mu (x) \leq c\omega p(2
- A, f).
Thus, we immediately have
An,1 \leq cQn - 2A - 1+1\omega p(2
- A, f).
To discuss An,2, An,3, An,4, An,5, for any \varepsilon \in G, y \in G2 and A \in \BbbP we write the following:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA(\varepsilon )\times IA
(f(y + x) - f(y))rA(x
1)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA(\varepsilon )\times IA
f(y + x)rA(x
1)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA+1(\varepsilon )\times IA
f(y + x)rA(x
1)d\mu (x) +
\int
IA+1(\varepsilon +eA)\times IA
f(y + x)rA(x
1)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
96 K. NAGY
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA+1(\varepsilon )\times IA
f(y + x) - f(y + x+ e1A)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\int
IA+1(\varepsilon )\times IA
| f(y + x) - f(y + x+ e1A)| d\mu (x), (5)
where e1A := (eA, 0) (and e2A := (0, eA) we will use it later). Now, for any j \leq 2A - 1 we write that
BA - 1
j :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)w2A - 1(x
1)Kw
j (\tau A - 1(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2A
\int
IA - 1(\varepsilon )\times IA
(f(.+ x) - f(.))rA - 1(x
1)w2A - 1 - 1(x
1)Kw
j (\tau A - 1(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
.
The function w2A - 1 - 1(x
1)Kw
j (\tau A - 1(x
1)) is constant on the sets IA - 1(\varepsilon ) (\varepsilon \in G, | j| \leq A - 1). Thus,
the method of (5) and Lemma 3 imply
BA - 1
j =
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2Aw2A - 1 - 1(\varepsilon )K
w
j (\tau A - 1(\varepsilon ))
\int
IA - 1(\varepsilon )\times IA
(f(.+ x) - f(.))rA - 1(x
1)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2A| Kw
j (\tau A - 1(\varepsilon ))| \times
\times
\left( \int
G2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA - 1(\varepsilon )\times IA
(f(y + x) - f(y))rA - 1(x
1)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p
d\mu (y)
\right)
1/p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2A| Kw
j (\tau A - 1(\varepsilon ))| \times
\times
\left( \int
G2
\left( \int
IA(\varepsilon )\times IA
| f(y + x) - f(y + x+ e1A - 1)| d\mu (x)
\right)
p
d\mu (y)
\right)
1/p
\leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 97
\leq
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2A| Kw
j (\tau A - 1(\varepsilon ))| \times
\times
\int
IA(\varepsilon )\times IA
\left( \int
G2
| f(y + x) - f(y + x+ e1A - 1)| pd\mu (y)
\right) 1/p
d\mu (x) \leq
\leq c
1\sum
\varepsilon i=0
i\in \{ 0,1,...,A - 2\}
2A| Kw
j (\tau A - 1(\varepsilon ))| \omega p(2
- A+1, f)
\int
IA(\varepsilon )\times IA
d\mu (x) \leq
\leq c\omega p(2
- A+1, f)\| Kw
j \circ \tau A - 1\| 1 \leq
\leq c\omega p(2
- A+1, f)\| Kw
j \| 1 \leq c\omega p(2
- A+1, f). (6)
This yields that
An,4, An,5 \leq qn - 2A - 12A - 1BA - 1
2A - 1 \leq cqn - 2A - 12A - 1\omega p(2
- A+1, f),
An,2, An,3 =
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| jBA - 1
j \leq
\leq c
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j\omega p(2
- A, f).
If qk \uparrow , we get that
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j \leq 2A - 1qn - 2A - 1 -
2A - 1 - 1\sum
j=1
qn - 2A+j \leq 2A - 1qn - 2A - 1 (7)
and
An,3 \leq c2A - 1qn - 2A - 1\omega p(2
- A+1, f),
while in the case when qk \downarrow
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j =
2A - 1 - 1\sum
j=1
qn - 2A+j - (2A - 1 - 1)qn - 2A - 1 \leq
\leq Qn - 2A - 1 - Qn - 2A+1 (8)
and
An,3 \leq c(Qn - 2A - 1 - Qn - 2A+1)\omega p(f, 2
- A).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
98 K. NAGY
Now, we introduce the notation \~BA
j (see inequality (6)) to discuss An,10, An,11. First, let qk \downarrow
\~BA
j :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)rA(x
1)Lw
j (\tau A(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
,
for | j| \leq A. The method presented for inequality (6) (we note that | n - 2A| \leq A - 1) and Lemma 2
imply
\~BA
n - 2A \leq c\omega p(2
- A, f)\| Lw
n - 2A \circ \tau A\| 1 \leq c\omega p(2
- A, f)\| Lw
n - 2A\| 1 \leq c\omega p(2
- A, f)
and
An,10, An,11 \leq cQn - 2A\omega p(2
- A, f).
Now, let qk \uparrow . We use Abel’s transformation for the expression Qn - 2AL
w
n - 2A
:
Qn - 2AL
w
n - 2A =
n - 2A - 2\sum
j=1
(qn - 2A - j - qn - 2A - j - 1)jK
w
j + q1(n - 2A - 1)Kw
n - 2A - 1,
and define \~\~BA
j by
\~\~BA
j :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))D2A(x
2)rA(x
1)Kw
j (\tau A(x
1))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
, | j| \leq A.
The method of the discussion BA
j (see inequality (6)), Lemma 3 and the fact that the transformation
\tau A is measure-preserving [26] immediately give that
\~\~BA
j \leq c\omega p(2
- A, f)\| Kw
j \circ \tau A\| 1 \leq c\omega p(2
- A, f)\| Kw
j \| 1 \leq c\omega p(2
- A, f)
and
An,10, An,11 \leq c\omega p(2
- A, f)
\left( n - 2A - 2\sum
j=1
| qn - 2A - j - qn - 2A - j - 1| j + q1(n - 2A - 1)
\right) \leq
\leq c\omega p(2
- A, f)
\left( n - 2A - 2\sum
j=1
qn - 2A - j + q1(n - 2A - 1)
\right) \leq
\leq c\omega p(2
- A, f)(Qn - 2A + q1(n - 2A - 1)).
We note that Qn \geq (n - 1)q1 for increasing sequence \{ qk\} .
Now, we discuss the terms An,8 and An,9. Let us set j < 2A - 1. We use Lemma 6 of Gát,
Goginava and the author and the fact that the functions \scrK \kappa
j are constant on the sets I| j| (\varepsilon ) \times I| j| (\rho )
for any \varepsilon , \rho \in G:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 99
Ej :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))\scrK \kappa
j (x)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\sum
\varepsilon i=0
i\in \{ 0,...,j - 1\}
1\sum
\rho l=0
l\in \{ 0,...,j - 1\}
\int
I| j| (\varepsilon )\times I| j| (\rho )
(f(.+ x) - f(.))\scrK \kappa
j (x)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,j - 1\}
1\sum
\rho l=0
l\in \{ 0,...,j - 1\}
| \scrK \kappa
j (\varepsilon , \rho )| \times
\times
\left( \int
G2
\left( \int
I| j| (\varepsilon )\times I| j| (\rho )
| f(y + x) - f(y)| d\mu (x)
\right)
p
d\mu (y)
\right)
1/p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,j - 1\}
1\sum
\rho l=0
l\in \{ 0,...,j - 1\}
| \scrK \kappa
j (\varepsilon , \rho )|
\int
I| j| (\varepsilon )\times I| j| (\rho )
\left( \int
G2
| f(y + x) - f(y)| pd\mu (y)
\right) 1/p
d\mu (x) \leq
\leq c
1\sum
\varepsilon i=0
i\in \{ 0,...,j - 1\}
1\sum
\rho l=0
l\in \{ 0,...,j - 1\}
| \scrK \kappa
j (\varepsilon , \rho )| \omega p(2
- | j| , f)
\int
I| j| (\varepsilon )\times I| j| (\rho )
d\mu (x) \leq
\leq c\| \scrK \kappa
j \| 1\omega p(2
- | j| , f) \leq c\omega p(2
- | j| , f).
This yields that
An,9 \leq qn - 2A - 1+1(2
A - 1 - 1)E2A - 1 - 1 \leq cqn - 2A - 1+12
A - 1\omega p(2
- (A - 2), f).
If qk \uparrow , then
An,9 \leq cqn - 2A - 22A - 2\omega p(2
- (A - 2), f).
If qk \downarrow , then
An,9 \leq cqn - 2A - 12A - 1\omega p(2
- A+1, f).
Moreover, we get
An,8 \leq c
2A - 1 - 2\sum
j=1
| qn - j - qn - j - 1| jEj \leq
A - 2\sum
j=0
2j+1 - 1\sum
l=2j
| qn - l - qn - l - 1| lEl \leq
\leq c
A - 2\sum
j=0
\omega p(2
- j , f)
2j+1 - 1\sum
l=2j
| qn - l - qn - l - 1| l.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
100 K. NAGY
If qk \uparrow , then
2j+1 - 1\sum
l=2j
| qn - l - qn - l - 1| l \leq
2j - 1\sum
l=0
qn - 2j - l \leq 2jqn - 2j ,
An,8 \leq c
A - 2\sum
j=0
qn - 2j2
j\omega p(2
- j , f).
If qk \downarrow , then
2j+1 - 1\sum
l=2j
| qn - l - qn - l - 1| l \leq 2j+1qn - 2j+1 ,
An,8 \leq c
A - 2\sum
j=0
qn - 2j+12j+1\omega p(2
- j - 1, f).
At last, we discuss the expressions An,6, An,7, An,12. Now, we investigate An,12 and the other
two term can be treated analogously. But, we will write some words about it later.
First, let qk \downarrow . We note that \scrL w
n - 2A
(\tau A(x
1), \tau A(x
2)) is constant on the sets IA(\varepsilon ) \times IA(\rho ) for
any \varepsilon , \rho \in G. This and the generalized Minkowski inequality give
FA
n - 2A :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))rA(x
1)rA(x
2)\scrL w
n - 2A(\tau A(x
1), \tau A(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
=
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
\int
IA(\varepsilon )\times IA(\rho )
(f(.+ x) - f(.)) \times
\times rA(x
1)rA(x
2)\scrL w
n - 2A(\tau A(x
1), \tau A(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
| \scrL w
n - 2A(\tau A(\varepsilon ), \tau A(\rho ))| \times
\times
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
IA(\varepsilon )\times IA(\rho )
(f(.+ x) - f(.))rA(x
1)rA(x
2)d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
\leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 101
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
| \scrL w
n - 2A(\tau A(\varepsilon ), \tau A(\rho ))| \times
\times
\left( \int
G2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA(\varepsilon )\times IA(\rho )
(f(y + x) - f(y))rA(x
1)rA(x
2)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p
d\mu (y)
\right)
1/p
.
In the way of estimation (5) we easily get\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
IA(\varepsilon )\times IA(\rho )
(f(y + x) - f(y))rA(x
1)rA(x
2)d\mu (x)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\int
IA+1(\varepsilon )\times IA+1(\rho )
\Delta Af(x, y)d\mu (x), (9)
where
\Delta Af(x, y) := | f(x+ y) - f(x+ y + e2A) - f(x+ y + e1A) + f(x+ y + e1A + e2A)| .
Inequality (9), condition (4) and Lemma 5 imply that
FA
n - 2A \leq
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
| \scrL w
n - 2A(\tau A(\varepsilon ), \tau A(\rho ))| \times
\times
\left( \int
G2
\left( \int
IA+1(\varepsilon )\times IA+1(\rho )
\Delta Af(x, y)d\mu (x)
\right)
p
d\mu (y)
\right)
1/p
\leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
| \scrL w
n - 2A(\tau A(\varepsilon ), \tau A(\rho ))| \times
\times
\int
IA+1(\varepsilon )\times IA+1(\rho )
\left( \int
G2
(\Delta Af(x, y))
pd\mu (y)
\right) 1/p
d\mu (x) \leq
\leq
1\sum
\varepsilon i=0
i\in \{ 0,...,A - 1\}
1\sum
\rho j=0
j\in \{ 0,...,A - 1\}
\int
IA+1(\varepsilon )\times IA+1(\rho )
| \scrL w
n - 2A(\tau A(\varepsilon ), \tau A(\rho ))| d\mu (x)\omega
p
1,2(2
- A, 2 - A, f) \leq
\leq \| \scrL w
n - 2A \circ (\tau A \times \tau A)\| 1\omega p
1,2(2
- A, 2 - A, f) \leq
\leq \| \scrL w
n - 2A\| 1\omega
p
1,2(2
- A, 2 - A, f) \leq c\omega p
1,2(2
- A, 2 - A, f). (10)
We note that we used that the transformation \tau A is measure-preserving [26]. From the above written
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
102 K. NAGY
An,12 \leq cQn - 2A\omega
p
1,2(2
- A, 2 - A, f) \leq cQn - 2A\omega p(2
- A, f).
Now, we discuss the expression An,12 for sequence qk \uparrow . By Abel’s transformation we write
Qn - 2A\scrL w
n - 2A =
n - 2A - 2\sum
j=1
(qn - 2A - j - qn - 2A - j - 1)j\scrK w
j + q1(n - 2A - 1)\scrK w
n - 2A - 1.
Let us set for | j| \leq A
\~FA
j :=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))rA(x
1)rA(x
2)\scrK w
j (\tau A(x
1), \tau A(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
.
The method of the discussion of FA
j (see inequality (10)) and Lemma 5 give immediately
\~FA
j \leq c\omega p
1,2(2
- A, 2 - A, f)\| \scrK w
j \circ (\tau A \times \tau A)\| 1 \leq c\omega p
1,2(2
- A, 2 - A, f)\| \scrK w
j \| 1 \leq c\omega p(2
- A, f)
and
An,12 \leq c\omega p(2
- A, f)
\bigl(
Qn - 2A + q1(n - 2A - 1)
\bigr)
.
(For more details see An,10, An,11.)
Let us define \~\~FA - 1
j (for any | j| \leq A - 1) by
\~\~FA - 1
j :=
:=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
G2
(f(.+ x) - f(.))rA - 1(x
1 + x2)\omega 2A - 1 - 1(x
1 + x2)\scrK w
j (\tau A - 1(x
1), \tau A - 1(x
2))d\mu (x)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p
.
The method presented for discussion FA
j (see inequality (10)) and Lemma 5 give
\~\~FA - 1
j \leq c\| \scrK w
j \circ (\tau A - 1 \times \tau A - 1)\| 1\omega p
1,2(2
- A+1, 2 - A+1, f) \leq
\leq c\| \scrK w
j \| 1\omega
p
1,2(2
- A+1, 2 - A+1, f) \leq c\omega p
1,2(2
- A+1, 2 - A+1, f).
Thus,
An,7 \leq cqn - 2A - 12A - 1\omega p
1,2(2
- A+1, 2 - A+1, f) \leq cqn - 2A - 12A - 1\omega p(2
- A+1, f)
and
An,6 \leq c
2A - 1 - 1\sum
j=1
| qn - 2A+j - qn - 2A+j+1| j\omega
p
1,2(2
- A+1, 2 - A+1, f).
If qk \uparrow , then by (7)
An,6 \leq c2A - 1qn - 2A - 1\omega p(2
- A+1, f).
If qk \downarrow , then by (8)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 103
An,6 \leq c(Qn - 2A - 1 - Qn - 2A+1)\omega p(2
- A+1, f) \leq c(Qn - 2A - 1 - Qn - 2A+1)\omega p(2
- A, f).
Summarising our results on An,i, i = 1, . . . , 12, we complete the proof of our main theorem.
Now, we discuss the following cases:
(A) The nondecreasing \{ qk\} , in sign qk \uparrow , satisfies the condition
nqn - 1
Qn
= O(1). (11)
In particular (11) is true if
qk \asymp k\beta or (\mathrm{l}\mathrm{o}\mathrm{g} k)\beta for some \beta > 0.
(B) The nonincreasing \{ qk\} , in sign qk \downarrow , satisfies
(\mathrm{B}i) qk \asymp k - \beta for some 0 < \beta < 1, or
(\mathrm{B}ii) qk \asymp (\mathrm{l}\mathrm{o}\mathrm{g} k) - \beta for some 0 < \beta .
(We note that the condition (4) is satisfied in these cases.) For more details see [17, 19].
The one-dimensional analogue of the following theorem was proven for Walsh – Paley system by
Móricz and Siddiqi in [17] for Walsh – Kaczmarz system by the author [18]. We mention that as
special case (set qk := 1 for all k) we get Marcinkiewicz means of Walsh – Kaczmarz – Fourier series.
More generally, when qk := A\beta
k :=
\bigl(
\beta +k
k
\bigr)
for k \geq 1, \beta \not = - 1, - 2, . . . , we have the (C, \beta ) mean of
Marcinkiewicz type discussed by Goginava [13] with respect to the double Walsh – Paley system and
by the author [22] with respect to double Walsh – Kaczmarz system.
Theorem 2. Let f \in \mathrm{L}\mathrm{i}\mathrm{p} (\alpha , p) for some \alpha > 0 and 1 \leq p \leq \infty .
Let \{ qk : k \geq 1\} be a sequence of nonnegative numbers such that in case qk \uparrow the condition (11)
is satisfied, while in case qk \downarrow the condition (\mathrm{B}i) or (\mathrm{B}ii) is satisfied, then
\| \bft \kappa n(f) - f\| p =
\left\{
O(n - \alpha ), if 0 < \alpha < 1,
O(n - 1 \mathrm{l}\mathrm{o}\mathrm{g} n), if \alpha = 1,
O(n - 1), if \alpha > 1.
Proof. Let f \in Lip(\alpha , p) for some \alpha > 0 and 1 \leq p \leq \infty .
First, let qk \uparrow , which satisfies the condition (11). From Theorem 1 by the method of Móricz and
Siddiqi [17] Theorem 2 can be proven.
Second, let qk \downarrow , which satisfies the condition (\mathrm{B}i), that is,
qk \asymp k - \beta for some 0 < \beta < 1, then Qn \asymp n1 - \beta .
From Theorem 1 it follows that
\| \bft \kappa n(f) - f\| p \leq
c
Qn
| n| - 1\sum
l=0
qn - 2l2
l2 - l\alpha +O(2 - | n| \alpha ).
For 0 \leq l \leq | n| - 1 we have 2| n| - 1 \leq n - 2l and qn - 2l \leq c2 - \beta (| n| - 1). Thus,
\| \bft \kappa n(f) - f\| p \leq
c
n1 - \beta
| n| - 1\sum
l=0
2 - \beta | n| 2l(1 - \alpha ) +O(2 - | n| \alpha ) \leq c
n
| n| - 1\sum
l=0
2l(1 - \alpha ) +O(2 - | n| \alpha ) =
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
104 K. NAGY
=
\left\{
O
\Biggl(
2| n| (1 - \alpha )
n
\Biggr)
, if 0 < \alpha < 1,
O
\biggl(
| n|
n
\biggr)
, if \alpha = 1,
O
\biggl(
1
n
\biggr)
, if \alpha > 1.
Let the condition (\mathrm{B}ii) be satisfied, that is,
qk \asymp (\mathrm{l}\mathrm{o}\mathrm{g} k) - \beta for some 0 < \beta , then Qn \asymp n(\mathrm{l}\mathrm{o}\mathrm{g} n) - \beta .
From now the proof goes along the same lines as that of case (\mathrm{B}i).
References
1. Agaev G. H., Vilenkin N. Ja., Dzhafarli G. M., Rubinstein A. I. Multiplicative systems of functions and harmonic
analysis on 0-dimensional groups (in Russian). – Baku: ELM, 1981.
2. Blahota I., Gát G. Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups // Anal. Theory
Appl. – 2008. – 24, № 1. – P. 1 – 17.
3. Fridli S., Manchanda P., Siddiqi A. H. Approximation by Walsh – Nörlund means // Acta Sci. Math. (Szeged). –
2008. – 74. – P. 593 – 608.
4. Gát G. On (C, 1) summability of integrable functions with respect to the Walsh – Kaczmarz system // Stud. Math. –
1998. – 130, № 2. – P. 135 – 148.
5. Gát G., Goginava U. Uniform and L-convergence of logarithmic means of Walsh – Fourier series // Acta Math.
Sinica. Eng. Ser. – 2006. – 22, № 2. – P. 497 – 506.
6. Gát G., Goginava U. Uniform and L-convergence of logarithmic means of cubical partial sums of double Walsh –
Fourier series // E. J. Approxim. – 2004. – 10, № 3. – P. 391 – 412.
7. Gát G., Goginava U., Nagy K. Marcinkiewicz – Fejér means of double Fourier series with respect to the Walsh –
Kaczmarz system // Stud. sci. math. hung. – 2009. – 46, № 3. – P. 399 – 421.
8. Glukhov V. A. On the summability of multiple Fourier series with respect to multiplicative systems (in Russian) //
Mat. Zametki. – 1986. – 39. – P. 665 – 673.
9. Goginava U., Tkebuchava G. Convergence of subsequences of partial sums and logarithmic means of Walsh – Fourier
series // Acta Sci. Math. (Szeged). – 2006. – 72. – P. 159 – 177.
10. Goginava U. On the approximation properties of Cesàro means of negative order of Walsh – Fourier series // J.
Approxim. Theory. – 2002. – 115. – P. 9 – 20.
11. Goginava U. Almost everywhere convergence of (C,\alpha )-means of cubical partial sums of d-dimensional Walsh –
Fourier series // J. Approxim. Theory. – 2006. – 141, № 1. – P. 8 – 28.
12. Goginava U. On the uniform convergence and L-convergence of double Fourier series with respect to the Walsh –
Kaczmarz system // Georg. Math. J. – 2003. – 10. – P. 223 – 235.
13. Goginava U. Approximation properties of (C,\alpha ) means of double Walsh – Fourier series // Anal. Theory Appl. –
2004. – 20, № 1. – P. 77 – 98.
14. Jastrebova M. A. On approximation of functions satisfying the Lipschitz condition by arithmetic means of their
Walsh – Fourier series (in Russian) // Math. Sb. – 1966. – 71. – P. 214 – 226.
15. Móricz F., Rhoades B. E. Approximation by Nörlund means of double Fourier series for Lipschitz functions // J.
Approxim. Theory. – 1987. – 50. – P. 341 – 358.
16. Móricz F., Schipp F. On the integrability and L1 convergence of Walsh series with coefficients of bounded variation
// J. Math. Anal. and Appl. – 1990. – 146, № 1. – P. 99 – 109.
17. Móricz F., Siddiqi A. Approximation by Nörlund means of Walsh – Fourier series // J. Approxim. Theory. – 1992. –
70, № 3. – P. 375 – 389.
18. Nagy K. Approximation by Nörlund means of Walsh – Kaczmarz – Fourier series // Georg. Math. J. – 2011. – 18. –
P. 147 – 162.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
APPROXIMATION BY NÖRLUND MEANS OF QUADRATICAL PARTIAL SUMS OF DOUBLE . . . 105
19. Nagy K. Approximation by Nörlund means of quadratical partial sums of double Walsh – Fourier series // Anal. Math. –
2010. – 36, № 4. – P. 299 – 319.
20. Nagy K. On the two-dimensional Marcinkiewicz means with respect to Walsh – Kaczmarz system // J. Approxim.
Theory. – 2006. – 142, № 2. – P. 138 – 165.
21. Nagy K. Approximation by Cesàro means of negative order of Walsh – Kaczmarz – Fourier series // E. J. Approxim. –
2010. – 16, № 3. – P. 193 – 207.
22. Nagy K. Approximation by Cesàro means of negative order of double Walsh – Kaczmarz – Fourier series // Tohoku
Math. J. – 2012. – 64, № 3. – P. 317 – 331.
23. Schipp F., Wade W. R., Simon P., Pál J. Walsh series. An introduction to dyadic harmonic analysis. – Bristol; New
York: Adam Hilger, 1990.
24. Schipp F. Pointwise convergence of expansions with respect to certain product systems // Anal. Math. – 1976. – 2. –
P. 63 – 75.
25. Simon P. (C, \alpha ) summability of Walsh – Kaczmarz – Fourier series // J. Approxim. Theory. – 2004. – 127. – P. 39 – 60.
26. Skvortsov V. A. On Fourier series with respect to the Walsh – Kaczmarz system // Anal. Math. – 1981. – 7. –
P. 141 – 150.
27. Skvortsov V. A. Certain estimates of approximation of functions by Cesàro means of Walsh – Fourier series (in Russian)
// Mat. Zametki. – 1981. – 29. – P. 539 – 547.
28. \u Sneider A. A. On series with respect to the Walsh functions with monotone coefficients // Izv. Akad. Nauk SSSR.
Ser. Mat. – 1948. – 12. – P. 179 – 192.
29. Persson L. E., Tephnadze G., Wall P. Maximal operators of Vilenkin – Nörlund means // J. Fourier Anal. and Appl. –
2015. – 21, № 1. – P. 76 – 94.
30. Tephnadze G. On the maximal operator of Walsh – Kaczmarz – Nörlund means // Acta Math. Acad. Pead.
Nyı́regyháziensis. – 2015. – 31, № 2. – P. 259 – 271.
31. Yano Sh. On Walsh series // Tohoku Math. J. – 1951. – 3. – P. 223 – 242.
32. Yano Sh. On approximation by Walsh functions // Proc. Amer. Math. Soc. – 1951. – 2. – P. 962 – 967.
33. Young W. S. On the a.e. convergence of Walsh – Kaczmarz – Fourier series // Proc. Amer. Math. Soc. – 1974. – 44. –
P. 353 – 358.
Received 23.04.13,
after revision — 05.06.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 1
|
| id | umjimathkievua-article-1824 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:13:20Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/86/8ec9b2f42ff8ce462e9e6dc90625dd86.pdf |
| spelling | umjimathkievua-article-18242019-12-05T09:28:56Z Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series Наближення середнiми Норлунда квадратичних часткових сум подвiйних рядiв Уолша–Качмаржа–Фур’є Nagy, K. Нагі, К. We discuss the Norlund means of quadratic partial sums of the Walsh – Kaczmarz – Fourier series of a function in $L_p$. We investigate the rate of approximation by this means, in particular, in $\text{Lip}(\alpha , p)$, where $\alpha > 0$ and $1 \leq p \leq \infty$. For $p = \infty$, by $L_p$, we mean $C$, i.e., the collection of continuous functions. Our main theorem states that the approximation behavior of this two-dimensional Walsh – Kaczmarz –Norlund means is as good as the approximation behavior of the one-dimensional Walsh– and Walsh – Kaczmarz –Norlund means. Earlier results for one-dimensional N¨orlund means of the Walsh – Fourier series was given by M´oricz and Siddiqi [J. Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] and Fridli, Manchanda and Siddiqi [Acta Sci. Math. (Szeged). – 2008. – 74. – P. 593 – 608], for one-dimensional Walsh – Kaczmarz –N¨orlund means by the author [Georg. Math. J. –2011. – 18. – P. 147 – 162] and for two-dimensional trigonometric system by M´oricz and Rhoades [J. Approxim. Theory. – 1987. – 50. – P. 341 – 358]. Розглядаються середнi Норлунда для квадратичних часткових сум рядiв Уолша – Качмаржа – Фур’є функцiї з простору $L_p$. Вивчено швидкiсть наближення цими середнiми, зокрема, в $\text{Lip}(\alpha , p)$, де $\alpha > 0$ та $1 \leq p \leq \infty$. Для $p = \infty$ пiд $L_p$ ми розумiємо $C$, тобто набiр всiх неперервних функцiй. Основна теорема у цiй статтi стверджує, що апроксимацiйна поведiнка таких двовимiрних середнiх Уолша – Качмаржа – Норлунда так само гарна, як i апроксимацiйна поведiнка одновимiрних середнiх Уолша та Уолша – Качмаржа – Норлунда. Ранiше результати для одновимiрних середнiх Норлунда рядiв Уолша – Фур’є були отриманi Морiчем та Сiддiкi [J. Approxim. Theory. – 1992. – 70, № 3. – P. 375 – 389] та Фрiдлi, Манчанда i Сiддiкi [Acta Sci. Math. (Szeged). –2008. – 74. – P. 593 – 608]. Для одновимiрних середнiх Уолша – Качмаржа – Норлунда вiдповiднi результати були отриманi автором [Georg. Math. J. – 2011. – 18. – P. 147 – 162]. Випадок двовимiрних тригонометричних систем було розглянуто Морiчем i Роадсом [J. Approxim Theory. – 1987. – 50. – P. 341 – 358]. Institute of Mathematics, NAS of Ukraine 2016-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1824 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 1 (2016); 87-105 Український математичний журнал; Том 68 № 1 (2016); 87-105 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1824/806 Copyright (c) 2016 Nagy K. |
| spellingShingle | Nagy, K. Нагі, К. Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title | Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title_alt | Наближення середнiми Норлунда квадратичних часткових сум подвiйних рядiв Уолша–Качмаржа–Фур’є |
| title_full | Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title_fullStr | Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title_full_unstemmed | Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title_short | Approximation by Norlund means of quadratical partial sums of double Walsh - Kaczmarz - Fourier series |
| title_sort | approximation by norlund means of quadratical partial sums of double walsh - kaczmarz - fourier series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1824 |
| work_keys_str_mv | AT nagyk approximationbynorlundmeansofquadraticalpartialsumsofdoublewalshkaczmarzfourierseries AT nagík approximationbynorlundmeansofquadraticalpartialsumsofdoublewalshkaczmarzfourierseries AT nagyk nabližennâseredniminorlundakvadratičnihčastkovihsumpodvijnihrâdivuolšakačmaržafurê AT nagík nabližennâseredniminorlundakvadratičnihčastkovihsumpodvijnihrâdivuolšakačmaržafurê |