255–270 DOI: 10.18514/MMN.2019.2702 LOGARITHMIC MEANS OF WALSH-FOURIER SERIES USHANGI GOGINAVA Received 11 October, 2018 Abstract

Vol. 20 (2019), No. 1, pp. 255–270 DOI: 10.18514/MMN.2019.2702

LOGARITHMIC MEANS OF WALSH-FOURIER SERIES

USHANGI GOGINAVA Received 11 October, 2018

Abstract. In this paper we discuss some convergence and divergence properties of subsequences of logarithmic means of Walsh-Fourier series. We give necessary and sufficient conditions for the convergence regarding logarithmic variation of numbers.

2010Mathematics Subject Classification: 42C10

Keywords: Walsh-Fourier series, N¨orlund Logarithmic means, almost everywhere converges, convergence in norm

1. WALSHFUNCTIONS

We shall denote the set of all non-negative integers byN, the set of all integers byZand the set of dyadic rational numbers in the unit intervalIWDŒ0; 1/byQ. In particular, each element ofQhas the form ₂^pn for somep; n2N; 0p2ⁿ.

Denote the dyadic expension ofn2Nandx2Iby nD

1

X

jD0

"j.n/ 2^j; "j.n/D0; 1

and

xD

1

X

jD0

xj

2^jC1; xj D0; 1:

In the case ofx2Qchose the expension which terminates in zeros. Define the dyadic additionCas

xCyD

1

X

kD0

jx_k y_kj2 ^.kC1/:

The setsIn.x/WD fy2IWy0Dx0; :::; yn 1Dxn 1gforx2I; InWDIn.0/ for 0 < n2NandI0.x/WDI are the dyadic intervals ofI. For0 < n2Ndenote by jnj WDmax˚

j2NWnj ¤0 ;that is,2^jnjn < 2^jnjC1:

The author supported by Shota Rustaveli National Science Foundation grant 217282 (Operators of Fourier analysis in some classical and new function spaces ).

c 2019 Miskolc University Press

The Rademacher system is defined by

n.x/WD. 1/^xn .x2I; n2N/ :

The Walsh-Paley system is defined as the sequence of the Walsh-Paley functions:

wn.x/WD

1

Y

kD0

._k.x//^nk D. 1/

jnj

P

kD0

nkxk

.x2I; n2N/ :

The Walsh-Dirichlet kernel is defined by Dn.x/D

n 1

X

kD0

w_k.x/ .n2N/ ; D0D0:

Recall that (see [20])

D2ⁿ.x/D

2ⁿ;ifx2In.0/

0; ifx2InIn.0/ : (1.1) As usual, denote byL1.I/the set of measurable functions defined onI, for which

kfk1WD Z

I

jf .t /jdt <1.

Letf 2L1.I/. The partial sums of the Walsh-Fourier series are defined as follows:

SM.x; f /WD

M 1

X

iD0

f .i / wb i.x/ ;

where the number

bf .i /D Z

I

f .t / wi.t / dt

is said to be the ith Walsh-Fourier coefficient of the functionf: Set En.x; f /D S2ⁿ.x; f / :The maximal function is defined by

E.x; f /Dsup

n2N

E_n.x;jfj/ :

The notiation a.b in the proofs stands for a < cb, where c is an absolute constant.

2. LOGARITHMIC MEANS

In the literature, there is the notion of Riesz’s logarithmic means of a Fourier series.

Then-th Riesz’s logarithmic means of the Fourier series of an integrable functionf is defined by

Rn.x; f /WD 1 ln

n

X

kD1

Sk.x; f /

k ;

wherelnWDPn

kD1.1=k/.

Riesz’s logarithmic means with respect to the trigonometric system was studied by a lot of authors. This means with respect to the Walsh and Vilenkin systems was discussed by Simon [21], Blahota, G´at [1], G´at [4], G´at, Goginava [8].

Letfq_kWk0gbe a sequence of nonnegative numbers. Then-th N¨orlund means for the Fourier series off is defined by

1 Qn

n 1

X

kD0

q_{n k}S_k.f /;

where

QnWD

n

X

kD1

q_k:

Ifqk Dk, then we get the N¨orlund logarithmic means tn.x; f /WD 1

ln n 1

X

kD0

Sk.x; f /

n k :

In this paper we call it logarithmic mean altough, it is a kind of ”reverse” Reisz’s logarithmic mean.

It is easy to see that

tn.x; f /D Z

I

f .t / Fn.xCt / dt;

where byFn.t /we denotenth logarithmic kernel, i. e.

Fn.t /WD 1 ln

n 1

X

kD0

Dk.t / n k :

and Fej´er kernel is defined by

Kn.t /WD 1 n

n

X

kD1

Dk.t / :

3. L1-ESTIMATION FOR LOGARITHMIC KERNEL

FornD P¹

jD0

"j.n/ 2^j; "j.n/D0; 1we define

n .k/WD

k

X

jD0

"j.n/ 2^j:

It is easy to see thatn .jnj/Dn. In this paper forL1-norm of logarithmic means we prove the following two sides estimation.

Theorem 1. Letn2N. Then

1 ln

n

X

jD1

Dn j

j ₁

1 jnj

jnj

X

kD1

j"_k.n/ "_kC1.n/jl_{n.k 1/}: Proof. We can write

n

X

jD1

Dn j.t / j

D

n.jnj 1/

X

jD1

Dn j.t /

j C

n

X

jDn.jnj 1/C1

Dn j.t / j

D

n.jnj 1/

X

jD1

D_"

jnj.n/2^jnjCn.jnj 1/ j.t /

j C

n

X

jDn.jnj 1/C1

Dn j.t /

j :

(3.1)

Since D_"

jnj.n/2^jnjCn.jnj 1/ j.t /D"_jnj.n/ D₂jnj.t /C w₂jnj.t /"_jnj.n/

D_{n.jnj 1/ j}.t / ; from (3.1) we have

n

X

jD1

Dn j.t / j

D"_jnj.n/ D₂jnj.t / ln.jnj 1/

C w₂jnj.t /"_jnj.n/

n.jnj 1/

X

jD1

Dn.jnj 1/ j.t / j

C"_jnj.n/

2^jnj 1

X

jD1

D₂jnj j.t / jCn .jnj 1/:

Iterating this equality we obtain

n

X

jD1

Dn j.t / j

D 0

@

jnj

X

jD2

"j.n/ D₂j.t / l_{n.j 1/}

1 A

jnj

Y

kDjC1

._k.t //^"k.n/

C 0

@

jnj

X

jD2

"j.n/

2^j 1

X

kD1

D₂j k.t / kCn .j 1/

1 A

jnj

Y

sDjC1

.s.t //^"s.n/

C 0

@

n.1/

X

jD1

Dn.1/ j.t / j

1 A

jnj

Y

kD2

._k.t //^"k.n/:

(3.2)

Since

"j.n/ D₂j.t /

j

Y

kD0

._k.t //^"k.n/D"j.n/ D₂j.t / j.t /

we have

0

@

jnj

X

jD2

"j.n/ D₂j.t / l_{n.j 1/}

1 A

jnj

Y

kDjC1

.k.t //^"k.n/

Dwn.t / 0

@

jnj

X

jD2

"j.n/ D₂j.t / l_{n.j 1/}

1 A

j

Y

kD0

._k.t //^"k.n/

Dwn.t / 0

@

jnj

X

jD2

"j.n/ D₂j.t / j.t / l_{n.j 1/}

1 A:

(3.3)

Combining (3.2) and (3.3) we conclude that

n

X

jD1

Dn j.t / j

Dwn.t / 0

@

jnj

X

jD2

"j.n/ D₂^j.t / j.t / ln.j 1/

1 A

C 0

@

jnj

X

jD2

"j.n/

2^j 1

X

kD1

D₂j k.t / kCn .j 1/

1 A

jnj

Y

sDjC1

.s.t //^"s.n/

C 0

@

n.1/ 1

X

jD1

D_{n.1/ j}.t / j

1 A

jnj

Y

kD2

._k.t //^"k.n/

DWH_n^.1/.t /CH_n^.2/.t /CH_n^.3/.t / :

(3.4)

Since (see [9])

D₂j k.t /DD₂j.t / w₂j 1.t / D_k.t / ; kD1; 2; :::; 2^j 1 forH_n^.2/.t /we can write

H_n^.2/.t /

D 0

@

jnj

X

jD2

"j.n/ D₂j.t /

2^j 1

X

kD1

1 kCn .j 1/

1 A

jnj

Y

sDjC1

.s.t //^"s.n/

0

@

jnj

X

jD2

"j.n/ w₂j 1.t /

2^j 1

X

kD1

Dk.t / kCn .j 1/

1 A

jnj

Y

sDjC1

.s.t //^"s.n/

DWH_n^.21/.t /CH_n^.22/.t / :

(3.5)

Since

H_n^.21/.t /D 0

@

jnj

X

jD2

"j.n/ D₂j.t / l_{n.j / 1} l_{n.j 1/}

1 A

jnj

Y

sDjC1

.s.t //^"s.n/

from (1.1) we get H_n^.21/

₁

jnj

X

jD2

"j.n/ l_{n.j /} l_{n.j 1/}

cjnj: (3.6)

Usin Abel’s transformation we obtain

2^j 1

X

kD1

Dk.t / kCn .j 1/

D

2^j 2

X

kD1

1

kCn .j 1/

1

kC1Cn .j 1/

kK_k.t /

C 2^j 1

2^j 1Cn .j 1/K₂j 1.t / : Since (see [20]) sup

n kKnk1<1forH_n^.22/.t /we can write

H_n^.22/

₁

.

jnj

X

jD2

"j.n/

2^j 1

X

kD1

1

kCn .j 1/

1

kC1Cn .j 1/

k

C

jnj

X

jD2

"j.n/ 2^j 1 2^j 1Cn .j 1/

.

jnj

X

jD2

"j.n/

2^j 1

X

kD1

1 .kCn .j 1//

C

jnj

X

jD2

"j.n/

2^j 1

X

kD1

n .j 1/

.kCn .j 1//²C

jnj

X

jD2

"j.n/ 2^j 1 2^j 1Cn .j 1/

.

jnj

X

jD2

"j.n/ l_{n.j /} l_{n.j 1/}

C jnj.jnj:

(3.7)

Combining (3.5)-(3.7) we conclude that H_n^.2/

₁cjnj: (3.8)

It is easy to see that

sup

n

H_n^.3/

₁c: (3.9)

First, we find upper estimation for Hn^.1/

₁. We can write H_n^.1/.t /Dwn.t /

0

@

jnj

X

jD2

"j.n/ l_{n.j 1/}.D₂jC1.t / D₂j.t //

1 A

Dwn.t / 0

@

jnj 1

X

jD2

"j.n/ l_{n.j 1/} "jC1.n/ l_{n.j /}

D₂jC1.t / 1 A Cwn.t / l_{n.jnj 1/}D₂jnjC1 wn.t / "2.n/ l_n.1/D₂².t / : Hence, from (1.1) we obtain

H_n^.1/

₁

jnj 1

X

jD2

ˇˇ"j.n/ l_{n.j 1/} "jC1.n/ l_{n.j /}ˇ ˇCcjnj

jnj 1

X

jD2

ˇˇ"j.n/ "jC1.n/ˇ ˇl_{n.j 1/}

C

jnj 1

X

jD2

"jC1.n/ l_{n.j /} l_{n.j 1/}

Ccjnj

jnj

X

jD2

ˇ

ˇ"j.n/ "jC1.n/ˇ

ˇln.j 1/Ccjnj:

Now, we find lower estimation for Hn^.1/

₁. Letai andbi; i D1; :::; sbe strictly increasing sequences, i. e.

0a1b1< a2b2< < asbs< asC1D 1 for which

"j.n/D

1; ai jbi

0; bi < j < aiC1 : (3.10) Then it is evident that

bjC1 < ajC1: (3.11)

Set

A_kWD 1

2^akC1; 1 2^ak

; B_kWD 1

2^bkC2; 1 2^bkC1

; kD1; :::; s:

Letx2A_k. Then we can write ˇ

ˇ

ˇH_n^.1/.t / ˇ ˇ ˇD

ˇ ˇ ˇ ˇ ˇ ˇ

jnj

X

jD2

"j.n/ .D₂jC1.t / D₂j.t // l_{n.j 1/}

ˇ ˇ ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ ˇ ˇ

k 1

X

iD1 bi

X

jDa_i

.D₂jC1.t / D₂j.t // l_{n.j 1/}

C

bk

X

jDa_k

.D₂jC1.t / D₂j.t // l_{n.j 1/}

ˇ ˇ ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ ˇ ˇ

k 1

X

iD1 bi

X

jDa_i

2^jln.j 1/ 2^akln.a_k 1/

ˇ ˇ ˇ ˇ ˇ ˇ :

From (3.11) we can write

k 1

X

iD1 b_i

X

jDai

2^jln.j 1/ln.b_{k 1} 1/

k 1

X

iD1

2^biC1 2^ai

l_{n.bk 1 1/}

k 1

X

iD1

2^biC1 2^{bi 1C1}

2^{bk 1C1}ln.b_{k 1} 1/

2^{bk 1C1}l_{n.ak 1/}: Consequently,

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ

ˇ2^akl_{n.ak 1/} 2^{bk 1C1}l_{n.ak 1/}2^{ak 1}l_{n.ak 1/}: Integrating onA_k we get

Z

A_k

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdt

Z

A_k

2^{ak 1}l_{n.ak 1/}dtD l_{n.ak 1/}

4 : (3.12)

On the intervalB_kwe have ˇ

ˇ

ˇH_n^.1/.t / ˇ ˇ ˇ D

ˇ ˇ ˇ ˇ ˇ ˇ

k

X

iD1 b_i

X

jDai

.D₂jC1.t / D₂j.t // l_{n.j 1/}

ˇ ˇ ˇ ˇ ˇ ˇ

D

k

X

iD1 b_i

X

jDai

2^jl_{n.j 1/}l_{.bk 1/}2^bk:

Hence,

Z

B_k

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdt

Z

B_k

l_{.bk 1/}2^bkdt Dl_{n.bk 1/}

4 : (3.13)

SinceAi; Bi; i D1; :::; sare pairwise disjoint from (3.12) and (3.13) we have Z

I

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdt

s

X

kD1

0 B

@ Z

A_k

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdtC

Z

B_k

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdt

1 C A

1 4

s

X

kD1

l_{n.ak 1/}Cl_{n.bk 1/}

:

(3.14)

Since (see (3.10))

n .a_k 1/Dn .a_k 2/

and

ˇˇ"j.n/ "jC1.n/ˇ ˇD

1; j Da_k 1orb_k; kD1; 2; :::; s 0otherwise

we conclude that Z

I

ˇ ˇ

ˇH_n^.1/.t / ˇ ˇ ˇdt 1

4

jnj

X

kD1

j"k.n/ "kC1.n/jln.k 1/:

Combining (3.4)-(3.10) and (3.14) we complete the proof of Theorem1.

4. ALMOST EVERYWHERE CONVERGENCE OF LOGARITHMIC MEANS

For a non-negative integernlet us denote VS.n/WD

1

X

iD0

j"i.n/ "iC1.n/j C"0.n/

and

VL.n/WD 1 jnj

jnj

X

kD1

j"_k.n/ "_kC1.n/jl_{n.k 1/}: It is known that ifnj < njC1,

sup

j

VS nj

<1; (4.1)

then a. e.Sn_j.f /!f. On the other hand, Konyagin [14] proved that the condition (4.1) is not necessary for a. e. convergence of subsequence of partial sums. Moreover, he gave negative answer to the question of Balashov and proved the validity of the following theorem.

Theorem K(Konyagin). SupposefnAgis an increasing sequence of natural num- bers,kAWDŒlog₂nAC1;and2^kA is a divider ofnAC1for allA. ThenSnA.f /!f a. e. for any functionf 2L1.I/.

For instance, a sequencefnAg; nAWD2^A2

A

P

iD0

4ⁱ;such that sup

nA

V .nA/D 1;satis- fies the hypotheses of the theorem.

Almost ewerywhere convergence offt₂A.f /WA1gwith respect to Walsh-Paley system was studied by author [11]. In particular, the following is proved

Theorem G. Letf 2L1.I/. Thent₂A.x; f /!f .x/asA! 1a. e. x2 I.

Nagy in [18] established a similar result for the Walsh-Kaczmarz system. How- ever, a divergence on the set with positive measure for the whole sequence ftn.f /Wn1g was proved by G´at and Goginava [7]. Memi´c [16] improved The- orem G and proved that the following is true.

Theorem M. Letf 2L1.I/and sup

A

1 jmAj

jmAj

X

kD1

"_k.m_A/ l_{mA.k 1/}<1: (4.2) Thentm_A.x; f /!f .x/asA! 1for a. e.x2I.

In this paper we are going to replace condition (4.2) with more weaker condition sup

A

1 jmAj

jmAj

X

kD1

j"_k.m_A/ "_kC1.m_A/jl_{mA.k 1/}<1: (4.3) It is easy to see that condition (4.2) imply condition (4.3), on other hand, for the sequence˚

2^A 1WA2N condition (4.2) does not holds and condition (4.3) holds.

So, we prove that the following is valid.

Theorem 2. Letf 2L1.I/and condition (4.3) is holds. ThentmA.x; f /!f .x/

asA! 1for a. e.x2I.

Proof. From (3.4) we have

f .lmAFmA/ .x/Df H_m^.1/_A.x/Cf H_m^.2/_A.x/Cf H_m^.3/_A.x/ : (4.4) It is easy to see that

sup

A

ˇ ˇ

ˇf H_m^.3/_A ˇ ˇ ˇ 1

.kfk1: (4.5)

From (3.5) we can write

f H_m^.2/_A.x/Df H_m^.21/_A .x/Cf H_m^.22/_A .x/Cf H_m^.22/_A .x/ : (4.6) Using (1.1) we have

ˇ ˇ

ˇf H_m^.21/_A .x/

ˇ ˇ ˇ

jm_Aj 1

X

jD1

.jfj D₂j.x// jmAjE.x; f / :

Hence

sup

A

ˇ ˇ

ˇf H_m^.21/_A .x/

ˇ ˇ ˇ

jmAj E.x; f / : (4.7)

We can write ˇ ˇ

ˇf H_m^.22/

A .x/ˇ ˇ ˇ

jmAj 1

X

jD1

"j.m_A/ l_{mA.j /} l_{mA.j 1/}

.jfj D₂j.x//

.jmAjE.x; f / ;

sup

A

ˇ ˇ

ˇf Hm^.22/_A .x/

ˇ ˇ ˇ

jmAj .E.x; f / : (4.8)

It is proved in [7]

sup

ˇ ˇ ˇ ˇ ˇ ˇ

8

<

: sup

A

ˇ ˇ

ˇf Hm^.23/_A

ˇ ˇ ˇ jmAj >

9

=

; ˇ ˇ ˇ ˇ ˇ ˇ

.kfk1: (4.9)

Since sup

jfE.f / > gj.kfk1from (4.7)- (4.9) we get

sup

ˇ ˇ ˇ ˇ ˇ ˇ

8

<

: sup

A

ˇ ˇ

ˇf Hm^.2/A

ˇ ˇ ˇ jmAj >

9

=

; ˇ ˇ ˇ ˇ ˇ ˇ

.kfk1: (4.10)

Now, we estimate ˇ ˇ

ˇf H_m^.1/_A.x/ˇ ˇ

ˇ. We have ˇ

ˇ

ˇf H_m^.1/_A.x/

ˇ ˇ ˇ.

jmAj 1

X

jD1

"j.mA/ lm_A.j 1/ "jC1.mA/ lm_A.j /

.jfj D₂^j.x//

Cl_{.jmAj 1/}

jfj D_2jmAjC1.x/

.E.x; f / 0

@l.jmAj 1/C

jmAj 1

X

jD1

"j.mA/ l_mA.j 1/ "jC1.mA/ l_mA.j / 1 A

.E.x; f / 0

@l_{.jmAj 1/}C

jmAj 1

X

jD1

ˇˇ"j.mA/ "jC1.mA/ˇ

ˇl_{mA.j 1/}

C

jm_Aj 1

X

jD1

"jC1.mA/ l_{mA.j /} l_{mA.j 1/}

1 A

.E.x; f / 0

@l_{.jmAj 1/}C

jmAj 1

X

jD1

ˇˇ"j.m_A/ "jC1.m_A/ˇ

ˇl_{mA.j 1/}

1 A:

From the condition of the Theorem we can write sup

A

ˇ ˇ

ˇf Hm^.1/A.x/

ˇ ˇ ˇ lmA

.E.x; f / VL.mA/

and

sup

ˇ ˇ ˇ ˇ ˇ ˇ

8

<

: sup

A

ˇ ˇ

ˇf H_m^.1/_A ˇ ˇ ˇ jm_Aj >

9

=

; ˇ ˇ ˇ ˇ ˇ ˇ

.kfk1: (4.11)

Combining (3.5), (4.5), (4.10) and (4.11) we conclude that sup

ˇ ˇ ˇ ˇ

sup

A

jf F_mAj jmAj >

ˇ ˇ ˇ ˇ

.kfk1:

By the well-known density argument we complete the proof of Theorem2.

5. UNIFORM ANDL-CONVERGENCE OF LOGARITHMIC MEANS

Denote byCw.I/the space of uniformly continuous functions onI, with the su- premum norm

kfkCwWDsup

x2Ijf .x/j .f 2Cw.I// :

LetXDX .I/be either the spaceL1.I/, or the space of uniformly continuous func- tions, that is,Cw.I/. The corresponding norm is denoted bykkX.

For Walsh-Fourier series Fine [2] has obtained a sufficient condition for the uni- form convergence which is in a complete analogy with the Dini-Lipshitz condition (see also [20]). Similar results are true for the space of integrable functionsL1.I/

[19]. Gulicev [13] has estimated the rate of uniform convergence of a Walsh-Fourier series using Lebesgue constant and modulus of continuity. Uniform convergence of Walsh-Fourier series of the functions of classes of generalized bounded variation was investigated by author [10]. This problem has been considered for Vilenkin group by Fridli [3] and G´at [5]. Lukomskii [15] considered uniform andL1-convergence of subsequence of partial sums of Walsh-Fourier series. In particular, he proved that the condition sup

A

VS.mA/ <1is necessary and sufficient condition for uniform and L1-convergence of subsequence of partial sums SmA.f / of Walsh-Fourier series.

In M´oricz and Siddiqi [17] investigated approximation properties of N¨orlund means

1 Qn

n 1

P

kD0

q_{n k}S_kf. The case when we have q_k WD1=k differs from the types dis- cussed by M´oricz and Siddiqi in [17]. His method is not applicable for logarithmic

means. In [6] it is proved that Theorem of M´oricz does not hold for L1; Cw and q_k WD1=k:

In [12] it is investigatedX-norm convergence of subsequence of logarithmic means of Walsh-Fourier series. In particular, the following are proved.

Theorem GT. a) Letf 2X .I/and

sup

A

log

mA 2^jmAjC1 logmA

1 l_m

A 2jmAj

mA 2jmAj 1

X

jD1

Dn j

j ₁

<1: (5.1) Then subsequence of N¨orlund logarithmic meanstmA.f / converges in the norm of spaceX .I/.

b) If the condition (5.1) does not holds then we can find a function from the space X .I/for which the convergence of logarithmic meansLmA.f /in the norm of space X .I/does not holds.

Since sup

A

log

mA 2^jmAjC1 logmA

1 l_m

A 2jmAj

mA 2jmAj 1

X

jD1

Dn j

j ₁

sup

A

log

mA 2^jmAjC1

logmA

1 ˇˇm_A 2^jmAjˇ

ˇ

ˇ ˇ

ˇmA 2jmAjˇ ˇ ˇ

X

kD1

j"_k.n/ "_kC1.n/jl_{n.k 1/}

sup

A

1 jmAj

jmAj

X

kD1

j"_k.n/ "_kC1.n/jl_{n.k 1/};

from Theorem GT and Theorem1we can prove necessary and sufficint condition for norm convergence of subsequence of N¨orlund logarithmic means

Theorem 3. Letf 2X .I/. Then the conditionsup

A

VL.mA/ <1is neccessary and sufficient for convergence subsequence of N¨orlund logarithmic means of Walsh- Fourier series in norm of spaceX .I/ :

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Author’s address

Ushangi Goginava

U. Goginava, I. Vekua Institute of Applied Mathematics and Faculty of Exact and Natural Sciences of I. Javakhishvili Tbilisi State University, Tbilisi 0186, 2 University str., Georgia

E-mail address:zazagoginava@gmail.com