Research Article | Open Access

# Convergence in Measure of Logarithmic Means of Quadratical Partial Sums of Double Walsh-Kaczmarz-Fourier Series

**Academic Editor:**Anna Kaminska

#### Abstract

The main aim of this paper is to prove that the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series does not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of ββ, the set of the functions the logarithmic means of quadratical partial sums of the double Walsh-Kaczmarz series of which converge in measure is of first Baire category.

#### 1. Definitions and Notations

We denote by the Lebesgue space of functions that are measurable and finite almost everywhere on . is the Lebesgue measure of the set . The constants appearing in this paper are denoted by .

Let be the Orlicz space [1] generated by Youngβs function ; that is, is convex, continuous, even function such that and

This space is endowed with the norm In particular case, if , , then the corresponding space is denoted by .

We denote the set of nonnegative integers by . By a dyadic interval in we mean one of the form for some , . Given and , let denote the dyadic interval of length which contains the point .

Every point can be written in the following form: In that case when there are two different forms we choose the one for which . We use the notation

It is well-known that [2, 3] where .

Let be a function defined by The Rademacher functions are defined by

Let represent the Walsh functions, that is, , and if is a positive integer with , then The order of the is denoted by .

The Walsh-Kaczmarz functions are given by , and for For and , Skvorcov [4] defined a transformation by By the definition of , we have (see [4])

The Dirichlet kernels are defined by where for all or for all .

The th FejΓ©r means of the Walsh-(Kaczmarz-)Fourier series of function are given by where is the th partial sum of the Fourier series and is said to be the th Walsh-(Kaczmarz-)Fourier coefficient of the function .

The NΓΆrlund logarithmic (simply we say logarithmic) means and kernels of one-dimensional Walsh-(Kaczmarz-)Fourier series are defined as follows: where .

The Kronecker product () of two Walsh(-Kaczmarz) systems is said to be the two-dimensional Walsh(-Kaczmarz) system. That is,

If , then the number is said to be the th Walsh-(Kaczmarz-)Fourier coefficient of . The rectangular partial sums of double Fourier series with respect to both system are defined by

The logarithmic means and kernels of quadratical partial sums of double Walsh-(Kaczmarz)-Fourier series are given by

It is evident that where denotes the dyadic addition [3]. The th Marcinkiewicz kernels are given by

In 1948 Ε neΔder [5] showed that the inequality holds a.e. for the Walsh-Kaczmarz Dirichlet kernel. This inequality shows that the behavior of the Walsh-Kaczmarz system is worse than the behavior of the Walsh system in the Paley enumeration. This βspreadnessβ property of the kernel makes it easier to construct examples of divergent Fourier series [6]. On the other hand, Schipp [7] and Young [8] in 1974 proved that the Walsh-Kaczmarz system is a convergence system. Skvorcov in 1981 [4] showed that the FejΓ©r means with respect to the Walsh-Kaczmarz system converge uniformly to for any continuous functions . For any integrable functions, GΓ‘t [9] proved that the FejΓ©r means with respect to the Walsh-Kaczmarz system converge a.e. to the function. Recently, GΓ‘tβs result was generalized by Simon [10, 11]. The a.e. convergence of the Walsh-Kaczmarz-Marcinkiewicz means of integrable functions was discussed by the second author [12].

The partial sums of the Walsh-Fourier series of a function , converge in measure on [2]. The condition provides convergence in measure on of the rectangular partial sums of double Walsh-Fourier series [13]. The first example of a function from classes wider than with divergent in measure on was obtained in [14, 15]. Moreover, in [16] Tkebuchava proved that in each Orlicz space wider than the set of functions where quadratic Walsh-Fourier sums converge in measure on is of first Baire category (see Goginava [17] for Walsh-Kaczmarz-Fourier series). In the paper [18] it was showed that the NΓΆrlund logarithmic means of one-dimensional Walsh-Kaczmarz-Fourier series are of weak type (1,1), and this fact implies that converge in measure on for all functions and converge in measure on for all functions .

At last, we note that the Walsh-NΓΆrlund logarithmic means are closer to the partial sums than to the classical logarithmic means or the FejΓ©r means. Namely, it was proved that there exists a function in a certain class of functions and a set with positive measure, such that the Walsh-NΓΆrlund logarithmic means of the function diverge on the set [19].

For results with respect to logarithmic means of cubical and rectangular partial sums of two-dimensional Walsh-Fourier series, see [17, 19β22].

In the present paper we investigate convergence in measure of logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series and prove Theorem 2.1 that is, for any Orlicz space, which is not a subspace of , the set of the functions where logarithmic means converges in measure is of first Baire category. From this result it follows that (Corollary 2.2) in classes wider than there exists functions for which logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series diverge in measure.

Thus, in question of convergence in measure logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series differ from the Marcinkiewicz means and are like the usual quadratical partial sums of double Walsh-Fourier series.

#### 2. Main Results

The main results of this paper are presented in the following proposition.

Theorem 2.1. *Let be an Orlicz space, such that
**Then the set of the functions in the Orlicz space with logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series convergent in measure on is of first Baire category in .*

Corollary 2.2. *Let be a nondecreasing function satisfying for the condition
**Then there exists a function such that*

(a)*(b)**logarithmic means of quadratical partial sums of double Walsh-Kaczmarz series of diverge in measure on .*

#### 3. Auxiliary Results

It is well-known [2, 3] for the Dirichlet kernel function that for any . Then for these βs we also get where is a nonnegative integer. It is also well-known for the Walsh-Paley-Dirichlet kernel functions the following lower pointwise estimates holds. Let (). Then for any and , we have

Since this inequality plays a prominent role in the proofs of some divergence results concerning the partial sums of the Fourier series, then it seems that it would be useful to get a similar inequality also for the logarithmic kernels. In [20] the first author, GΓ‘t, and Tkebuchava proved the inequality for all , and . We have the exceptional set , such that it is βrare around zero.β For set (where denotes the lower integral part of ), and we take a βsmall partββ of the interval . This way the intervals are defined as follows: The exceptional set is given by

For the logarithmic kernels of quadratical partial sums of double Walsh-Kaczmarz-Fourier series, we prove an analogue result. To do this we need the following lemma of the first author, GΓ‘t and Tkebuchava [22].

Lemma 3.1 (see [22]). *Let , , and . Then one has
**
where and
*

Now, for the Walsh-Kaczmarz logarithmic kernels , we prove the following.

Lemma 3.2. *Let ,ββ for and . Then
*

*Proof. *Set . Let
for or . Now, we write that

First, we discuss . We use equalityββ(1.13)

If , then (see (1.14))
Moreover, by Lemma 3.1, we have

Now, we discuss by the help of β(1.13)
Since , for all , thus,

We use Abelβs transformation for ()
We write
Since , we have
For , we obtain that
Moreover,
Let . Then we have
that is, , .

Now, we introduce the following notation:
and (). A simple consideration gives
It is known [23] that if and , then
Using this inequality, we have that
Moreover,
Analogously,
Taking into account (3.11), we get
Therefore by (3.14)β(3.29), it follows that
with suitable constants . It is clear that for large enough we have
where is an absolute constant.

We apply the reasoning of [24] formulated as the following proposition in particular case.

Theorem A. *Let be a linear continuous operator, which commutes with family of translations , that is, for all for all . Let and . Then for any under condition, , there exist and , , such that
*

Theorem B. *Let be a sequence of linear continuous operators, acting from Orlicz space into the space . Suppose that there exist the sequence of functions from unit bull of space , and sequences of integers and increasing to infinity such that
**Then βthe set of functions from space , for which the sequence converges in measure to an a.e. finite functionβis of first Baire category in space .*

Theorem GGT. *Let be an Orlicz space, and let be measurable function with condition as . Then there exists Orlicz space , such that as , and for .*

The proof of Theorems B and GGT can be found in [20].

#### 4. Proof of the Theorem

*Proof of Theorem 2.1. *By Theorem B the proof of Theorem 2.1 will be complete if we show that there exist sequences of integers and increasing to infinity, and a sequence of functions from the unit bull of Orlicz space , such that for all

First, we prove that there exists such that

It is easy to show that
Hence, from Lemma 3.2, we can write
for .

Since
for and , from (4.4) we conclude that
Hence, (4.2) is proved.

From the condition of the theorem, we write [1]
Consequently, there exists a sequence of integers increasing to infinity, such that

From (4.2) we write
Then by the virtue of Theorem A, there exists and such that
where .

Set
where
Denote
It is easy to show that
Hence, (4.1) is proved.

Since
Moreover,
From (4.8) we can write

Hence, , and Theorem 2.1 is proved.

The validity of Corollary 2.2 follows immediately from Theorems 2.1 and GGT.

#### References

- M. A. Krasnosel'skii and Y. B. Rutickii,
*Convex Functions and Orlicz Space*, P. Noorhoff, Groningen, The Netherlands, 1961. - B. I. Golubov, A. V. Efimov, and V. A. Skvorcov,
*Series and Transformations of Walsh*, Nauka, Moscow, Russia, 1987. - F. Schipp, W. R. Wade, and P. Simon,
*Walsh Series, Introduction to Dyadic Harmonic Analysis*, Adam Hilger Ltd, Bristol, UK, 1990. - V. A. Skvorcov, βOn Fourier series with respect to the Walsh-Kaczmarz system,β
*Analysis Mathematica*, vol. 7, no. 2, pp. 141β150, 1981. View at: Publisher Site | Google Scholar - A. A. Ε neΔder, βOn series of Walsh functions with monotonic coefficients,β
*Izvestiya Akademii Nauk SSSR*, vol. 12, pp. 179β192, 1948. View at: Google Scholar - L. A. BalaΕ‘ov, βSeries with respect to the Walsh system with monotone coefficients,β
*SibirskiΔ MatematiΔeskiΔ Ε½urnal*, vol. 12, pp. 25β39, 1971 (Russian). View at: Google Scholar - F. Schipp, βPointwise convergence of expansions with respect to certain product systems,β
*Analysis Mathematica*, vol. 2, no. 1, pp. 65β76, 1976. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. S. Young, βOn the a.e. convergence of Walsh-Kaczmarz-Fourier series,β
*Proceedings of the American Mathematical Society*, vol. 44, pp. 353β358, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. GΓ‘t, βOn (
*C*, 1) summability of integrable functions with respect to the Walsh-Kaczmarz system,β*Studia Mathematica*, vol. 130, no. 2, pp. 135β148, 1998. View at: Google Scholar | Zentralblatt MATH - P. Simon, βOn the Cesaro summability with respect to the Walsh-Kaczmarz system,β
*Journal of Approximation Theory*, vol. 106, no. 2, pp. 249β261, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Simon, β$(C,\mathrm{\Xi \pm})$ summability of Walsh-Kaczmarz-Fourier series,β
*Journal of Approximation Theory*, vol. 127, no. 1, pp. 39β60, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Nagy, βOn the two-dimensional Marcinkiewicz means with respect to Walsh-Kaczmarz system,β
*Journal of Approximation Theory*, vol. 142, no. 2, pp. 138β165, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. V. Zhizhiashvili, βSome problems of multidimensional harmonic analysis,β TGU, Tbilisi, 1996. View at: Google Scholar
- R. Getsadze, βOn the divergence in measure of multiple Fourier seties,β
*Some Problems of Functions Theory*, vol. 4, pp. 84β117, 1988 (Russian). View at: Google Scholar - S. V. Konyagin, βOn a subsequence of Fourier-Walsh partial sums,β
*RossiΔskaya Akademiya Nauk*, vol. 54, no. 4, pp. 69β75, 1993. View at: Publisher Site | Google Scholar - G. Tkebuchava, βSubsequences of partial sums of multiple Fourier and Fourier-Walsh series,β
*Bulletin of the Georgian Academy of Sciences*, vol. 169, no. 2, pp. 252β253, 2004. View at: Google Scholar - U. Goginava, βConvergence in measure of partial sums of double Fourier series with respect to the Walsh-Kaczmarz system,β
*Journal of Mathematical Analysis and Approximation Theory*, vol. 2, no. 2, pp. 160β167, 2007. View at: Google Scholar | Zentralblatt MATH - U. Goginava and K. Nagy, βWeak type inequality for logarithmic means of Walsh-Kaczmarz-Fourier series,β
*Real Analysis Exchange*, vol. 35, no. 2, pp. 445β461, 2010. View at: Google Scholar | Zentralblatt MATH - G. GΓ‘t and U. Goginava, βOn the divergence of NΓΆrlund logarithmic means of Walsh-Fourier series,β
*Acta Mathematica Sinica*, vol. 25, no. 6, pp. 903β916, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. GΓ‘t, U. Goginava, and G. Tkebuchava, βConvergence in measure of logarithmic means of double Walsh-Fourier series,β
*Georgian Mathematical Journal*, vol. 12, no. 4, pp. 607β618, 2005. View at: Google Scholar | Zentralblatt MATH - G. GΓ‘t, U. Goginava, and G. Tkebuchava, βConvergence of logarithmic means of multiple Walsh-Fourier series,β
*Analysis in Theory and Applications*, vol. 21, no. 4, pp. 326β338, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - G. GΓ‘t, U. Goginava, and G. Tkebuchava, βConvergence in measure of logarithmic means of quadratical partial sums of double Walsh-Fourier series,β
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 1, pp. 535β549, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - K. Nagy, βSome convergence properties of the Walsh-Kaczmarz system with respect to the Marcinkiewicz means,β
*Rendiconti del Circolo Matematico di Palermo. Serie II*, no. 76, pp. 503β516, 2005. View at: Google Scholar | Zentralblatt MATH - A. M. Garsia,
*Topics in Almost Everywhere Convergence*, vol. 4, Markham Publishing Co., Chicago, Ill, USA, 1970.

#### Copyright

Copyright © 2012 Ushangi Goginava and Károly Nagy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.