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Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 19 (2018), No. 1, pp. 303–317 DOI: 10.18514/MMN.2018.2347

CONVERGENCE OF CES ´ARO MEANS WITH VARYING PARAMETERS OF WALSH-FOURIER SERIES

ANAS AHMAD ABU JOUDEH AND GY ¨ORGY G ´AT Received 06 June, 2017

Abstract. In 2007 Akhobadze [1] (see also [2]) introduced the notion of Ces`aro means of Fourier series with variable parameters. In the present paper we prove the almost everywhere conver- gence of the the Ces`aro.C; ˛n/means of integrable functionsn˛nf !f, whereN˛;K3n! 1forf 2L1.I /, whereI is the Walsh group for every sequence˛Dn/,0 < ˛n< 1. This theorem for constant sequences˛that is,˛˛1was proved by Fine [3].

2010Mathematics Subject Classification: 42C10

1. INTRODUCTION AND MAIN RESULTS

We follow the standard notions of dyadic analysis introduced by the mathem- aticians F. Schipp, P. Simon, W. R. Wade (see e.g. [9]) and others. Denote by NWD f0; 1; :::g;P WDNn f0g, the set of natural numbers, the set of positive integers andIWDŒ0; 1/the unit interval. Denote by.B/D jBjthe Lebesgue measure of the setB.BI /. Denote byLp.I /the usual Lebesgue spaces andk:kpthe correspond- ing norms.1p 1/. Set

JWD p

2n;pC1 2n

Wp; n2N

the set of dyadic intervals and for given x 2I and let In.x/ denote the interval In.x/2 J of length 2 n which contains x .n2N/. Also use the notion In WD In.0/ .n2N/. Let

xD

1

X

nD0

xn2 .nC1/

be the dyadic expansion of x2I, where xnD0 or1and if x is a dyadic rational number.x2 f2pn Wp; n2Ng/we choose the expansion which terminates in0’s. The

Research is supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K111651 and by project EFOP-3.6.1-16-2016-00022 supported by the European Union, co-financed by the European Social Fund.

c 2018 Miskolc University Press

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notion of the Hardy space H.I / is introduced in the following way [9]. A func- tiona2L1.I /is called an atom, if eitheraD1or a has the following properties:

suppaIa;kak1 jIaj 1;R

IaD0, for someIa 2J. We say that the functionf belongs toH, iff can be represented asf DP1

iD0iai;whereai’s are atoms and for the coefficients.i/the inequalityP1

iD0jij<1is true. It is known thatH is a Banach space with respect to the norm

kfkH WDinf

1

X

iD0

jij;

where the infimum is taken over all decompositionsf DP1

iD0iai 2H. Set the definition of thenth (n2N) Walsh-Paley function at pointx2I as:

!n.x/WD

1

Y

jD0

. 1/xjnj; whereN3nDP1

nD0nj2j.nj 2 f0; 1g.j 2N//. It is known (see [8] or [10]) that the system.!n; n2N/is the character system of.I;C/, where the group operation Cis the so-called dyadic or logical addition onI. That is, for anyx; y2I

xCyWD

1

X

nD0

jxn ynj2 .nC1/: Denote by

f .n/O WD Z

I

f !nd ; DnWD

n 1

X

kD0

!k; Kn1WD 1 nC1

n

X

kD0

Dk

the Fourier coefficients, the Dirichlet and the Fej´er or.C; 1/kernels, respectively. It is also known that the Fej´er or.C; 1/means off is

n1f .y/WD 1 nC1

n

X

kD0

Skf .y/D Z

I

f .x/Kn1.yCx/d .x/

D 1 nC1

n

X

kD0

Z

I

f .x/Dk.yCx/d .x/; .n2N; y2I /:

It is known [9] that forn2N; x2I it holds D2n.x/D

(2n;ifx2In

0 ;ifx…In

and also that

Dn.x/D!n.x/

1

X

kD1

D2k.x/nk. 1/xk;

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wherenDP1

iD1ni2i,niD f0; 1g.i 2N/.

Denote by Kn˛n the kernel of the summability method .C; ˛n/ and call it the .C; ˛n/kernel or the Ces`aro kernel for˛n2Rn f 1; 2; : : :g

Kn˛nD 1 A˛nn

n

X

kD0

A˛n kn 1Dk; where

A˛knD.˛nC1/.˛nC2/:::.˛nCn/

kŠ :

It is known [12] thatA˛nnDPn

kD0A˛kn 1; A˛kn A˛knC1D ˛nA

˛n k

kC1 :The.C; ˛n/ Ces`aro means of integrable functionf is

n˛nf .y/WD 1 A˛nn

n

X

kD0

A˛n kn 1Skf .y/D Z

I

f .x/Kn˛n.yCx/d .x/:

In [3] Fine proved the almost everywhere convergence n˛nf !f for all in- tegrable function f with constant sequence ˛n1> 0. On the rate of conver- gence of Ces`aro means in this constant case see the paper of Fridli [4]. For the two-dimensional situation see the paper of Goginava [7].

Comment1. With respect to other locally constant orthonormal sysytems for in- stance it was a question of Taibleson [8] open for a long time, that does the Fej´er- Lebesgue theorem, that is the a.e. convergencen1f !f hold for all integrable functionf with respect to the character system of the group of2-adic integers. In 1997 G´at answered [1] this question in the affirmative. Zheng and G´at generalized this result [9,2] for more general orthonormal systems.

Set two variable functionP .n; ˛/WDP1

iD0ni2i ˛ forn2N; ˛2R. For instance P .n; 1/Dn. Also set for sequences˛ D.˛n/ and positive reals K the subset of natural numbers

N˛;KWD

n2NW P .n; ˛n/ n˛n K

:

We can easily remark that for a sequence ˛ such that 1 > ˛n˛0> 0 we have N˛;K DNfor someK depending only on˛0. We also remark that2n2N˛;K for every˛D.˛n/,0 < ˛n< 1andK1.

In this paperC denotes an absolute constant andCK another one which may de- pend only onK. The introduction of.C; ˛n/means due to Akhobadze investigated [1] the behavior of theL1-norm convergence of n˛nf !f for the trigonometric system. In this paper it is also supposed that1 > ˛n> 0for alln.

The main aim of this paper is to prove :

Theorem 1. Suppose that1 > ˛n> 0. Letf 2L1.I /. Then we have the almost everywhere convergencen˛nf !f provided thatN˛;K 3n! 1.

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The method we use to prove Theorem 1 is to investigate the maximal operator ˛f WDsupn2N˛;Kjn˛nfj. We also prove that this operator is a kind of type.H; L/

and of type.Lp; Lp/for all1 < p 1. That is,

Theorem 2. Suppose that1 > ˛n> 0. Letjfj 2H.I /. Then we have k˛fk1CKkjfjkH:

Moreover, the operator˛is of type.Lp; Lp/for all1 < p 1. That is, k˛fkpCK;pkfkp

for all1 < p 1.

For the sequence˛nD1Theorem2is due to Fujii [5]. For the more general but constant sequence˛n1see Weisz [11].

Basically, in order to prove Theorem1we verify that the maximal operator ˛f .˛D.˛n//is of weak type.L1; L1/. The way we get this, the investigation of kernel functions, and its maximal function on the unit intervalI by making a hole around zero and some quasi locality issues (for the notion of quasi-locality see [9]).

To have the proof of Theorem2is the standard way. We need several Lemmas in the next section.

2. PROOFS

Lemma 1. Forj; n2N; j < 2nwe have

D2n j.x/DD2n.x/ !2n 1.x/Dj.x/:

Proof.

D2n.x/D

2n 1

X

kD0

!k.x/D

2n j 1

X

kD0

!k.x/C

2n 1

X

kD2n j

!k.x/DD2n jC

2n 1

X

kD2n j

!k.x/:

We have to prove :

2n 1

X

kD2n j

!k.x/D!2n 1.x/Dj.x/:

Fork < j; kDkn 12n 1C:::Ck121Ck0we have

!2n 1.x/!k

D!2n 1C:::C21C20.x/!kn 12n 1C:::Ck0.x/

D!.1Ckn 1.mod 2//2n 1C:::C.1Ck0.mod 2//20.x/

D!.1 kn 1.mod 2//2n 1C:::C.1 k0.mod 2//20.x/

D!2n 1C2n 2C:::C20 .kn 12n 1C:::Ck0/.x/D!2n 1 k.x/:

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Thus,

!2n 1.x/Dj.x/D!2n 1.x/

j 1

X

kD0

!k.x/D

j 1

X

kD0

!2n 1 k.x/D

2n 1

X

kD2n j

!k.x/:

This completes the proof of Lemma1.

Introduce the following notations: fora; n; j 2Nletn.j /WDPj 1

iD0ni2i, that is, n.0/D0; n.1/Dn0 and for2Bn < 2BC1, letjnj WDB,nDn.BC1/. Moreover, introduce the following functions and operators forn2Nand1 > ˛n> 0

Tn˛a WD 1 A˛na

2jnj 1

X

jD0

A˛n ja 1Dj;

TQn˛a WD 1 A˛naD2B

2B 1

X

jD0

A˛na 1

.B/Cj

C.1 ˛a/

2B 1

X

jD0

A˛na 1

.B/Cj

jC1 n.B/CjC1

ˇ ˇ ˇKj1

ˇ ˇ

ˇCA˛na 12Bˇ ˇK21B 1

ˇ ˇ; tn˛af .y/WD

Z

I

f .x/Tn˛a.yCx/d .x/;

Q

tn˛af .y/WD Z

I

f .x/TQn˛a.yCx/d .x/:

Now, we need to prove the next Lemma which means that maximal operator supn;ajQtn˛ajis quasi-local. This lemma together with the next one are the most im- portant tools in the proof of the main results of this paper.

Lemma 2. Let1 > ˛a> 0,f 2L1.I /such that suppf Ik.u/,R

Ik.u/f d D0 for some dyadic intervalIk.u/. Then we have

Z

InIk.u/

sup

n;a2NjQtn˛afjd Ckfk1: Moreover,ˇ

ˇTn˛aˇ

ˇ QTn˛a.

Proof. It is easy to have that forn < 2k and x2Ik.u/we haveTQn˛a.yCx/D TQn˛a.yCu/and

Z

Ik.u/

f .x/TQn˛a.yCx/d .x/

D QTn˛a.yCu/

Z

Ik.u/

f .x/d .x/D0:

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Therefore, Z

InIk.u/

sup

n;a2N

Q

tn˛af d D Z

InIk.u/

sup

n2k;a2N

Q

tn˛af d : Recall thatBD jnj. Then

A˛naTn˛a D

2B 1

X

jD0

A˛2aB 1

Cn.B/ jDj

D

2B 1

X

jD0

A˛na 1

.B/CjD2B j

By Lemma1we have A˛naTn˛a

DD2B

2B 1

X

jD0

A˛na 1

.B/Cj !2B 1

2B 1

X

jD0

A˛na 1

.B/CjDj: It is easy to have that A1˛a

n D2B.´/P2B 1 jD0 A˛na 1

.B/Cj D0, for any´2InIk. This holds becauseD2B.´/D0forBD jnj kand´2InIk. By the help of the Abel transform we get:

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ˇ ˇ ˇ ˇ ˇ ˇ

2B 1

X

jD0

A˛na 1

.B/CjDj

ˇ ˇ ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ ˇ ˇ

2B 1

X

jD0

.A˛na 1

.B/Cj A˛na 1

.B/CjC1/

j

X

iD0

DiCA˛na 1

.B/C2B 2B 1

X

iD0

Di

ˇ ˇ ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ ˇ ˇ

.1 ˛a/

2B 1

X

jD0

A˛na 1

.B/Cj

jC1

n.B/CjC1Kj1CA˛na 12BK21B 1

ˇ ˇ ˇ ˇ ˇ ˇ

D ˇ ˇ ˇ ˇ ˇ

.1 ˛a/

2k 1

X

jD0

A˛na 1

.B/Cj

jC1

n.B/CjC1Kj1C.1 ˛a/

2B 1

X

jD2k

A˛na 1

.B/Cj

jC1 n.B/CjC1Kj1 CA˛na 12BK21B 1

ˇ ˇ ˇ ˇ ˇ

.1 ˛a/

2k 1

X

jD0

A˛na 1

.B/Cj

jC1 n.B/CjC1

ˇ ˇ ˇKj1

ˇ ˇ ˇ

C.1 ˛a/

2B 1

X

jD2k

A˛na 1

.B/Cj

jC1 n.B/CjC1

ˇ ˇ ˇKj1

ˇ ˇ

ˇCA˛na 12Bˇ ˇK21B 1

ˇ ˇ

DWICIICIII:

These equalities above immediately proves inequalityˇ ˇTn˛aˇ

ˇ QTn˛a.

Since for any j < 2k we have that the Fej´er kernel Kj1.yCx/ depends (with respect tox) only on coordinatesx0; : : : ; xk 1, thenR

Ik.u/f .x/jKj1.yCx/jd .x/D jKj1.yCu/jR

Ik.u/f .x/d .x/D0givesR

Ik.u/f .x/I.yCx/d .x/D0.

On the other hand, IID.1 ˛a/

2B 1

X

jD2k

A˛na 1

.B/Cj

jC1

n.B/CjC1jKj1j sup

j2k

jKj1j.1 ˛a/

n

X

jD0

Aj˛a 1DA˛na.1 ˛a/ sup

j2k

jKj1j: This by Lemma 3 in [6] gives

Z

InIk

sup

n2k;a2N

1 A˛na

IId Z

InIk

sup

j2k

jKj1jd C:

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The situation withIII is similar. Namely, A˛na 1n

A˛na D ˛an

aCn/ ˛a< 1:

So, just as in the case ofII we apply Lemma 3 in [6]

Z

InIk

sup

n2k;a2N

1 A˛na

IIId Z

InIk

sup

n2k;a2N

jK21jnj 1jd C:

Therefore, substituting´DxCy2InIk (sincex2Ik.u/andy2InIk.u/) Z

InIk.u/

sup

n2k;a2N

Q tn˛af d

D Z

InIk.u/

sup

n2k;a2N

ˇ ˇ ˇ ˇ Z

Ik.u/

f .x/TQn˛a.yCx/d .x/

ˇ ˇ ˇ ˇ

d .y/

Z

InIk.u/

Z

Ik.u/jf .x/j sup

n2k;a2N

1

A˛na .II.yCx/CIII.yCx// d .x/

D Z

Ik.u/jf .x/j Z

InIk

sup

n2k;a2N

1

A˛na .II.´/CIII.´// d .´/d .x/

C Z

Ik.u/jf .x/jd .x/:

This completes the proof of Lemma2.

A straightforward corollary of this lemma is:

Corollary 1. Let1 > ˛a> 0. Then we havekTn˛ak1 k QTn˛ak1C,ktn˛afk1; kQtn˛afk1Ckfk1andktn˛agk1;kQtn˛agk1Ckgk1for all natural numbersa; n, whereC is some absolute constant andf 2L1; g2L1. That is, operatorstQn˛a; tn˛a

is of type.L1; L1/and.L1; L1/(uniformly inn).

Proof. The proof is a straightforward consequence of Lemma2and an easy es- timation below. LetBD jnj. Then

A˛naTQn˛a

1 kD2Bk1 2B 1

X

jD0

A˛na 1

.B/Cj

C.1 ˛a/

2B 1

X

jD0

A˛na 1

.B/Cj

jC1

n.B/CjC1kKj1k1CA˛na 12BkK21B 1k1: Then bykD2Bk1D1;kKj1k1C we complete the proof of Corollary1.

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Lemma 3. Letn; N be any natural numbers and0 < ˛ < 1. Then we have A˛n

A˛N 2 nC1

N ˛

: Proof. By definition we have

A˛n A˛N D

1 ˛

nC1C˛

1 ˛

NC˛

1 ˛

nC2

1 ˛

NC1

: It is well-known that

1 ˛

i.nC1/C1

1 ˛

.iC1/.nC1/

1 ˛

.iC1/.nC1/

nC1

e 1iC˛1 for everyn2N. This gives

1 ˛

nC2

1 ˛

NC1

e 1˛PbnNC1c

iD2 1 i

e 1˛loge j N

nC1

k 1Cc

2 e 1˛loge

N nC1

D2 nC1

N ˛

;

where c 0:5772 is the Euler-Mascheroni constant. This completes the proof of

Lemma3.

Recall that the two variable functionP .n; ˛/DP1

iD0ni2i ˛forn2N; ˛2Rand K2Rdetermines the set of natural numbers

N˛;K D

n2NW P .n; ˛n/ n˛n K

:

LetnD2hsC C2h0, wherehs> > h00are integers. That is,jnj Dhs. Let n.j /WD2hj C C2h0. This meansnDn.s/. Define the following kernel function and operators

KQn˛nWD QTn˛.s/n C

s

X

lD0

A˛nn.l 1/

A˛nn.s/ D2hlCA˛nn.l 1/

A˛nn.s/ TQn˛ln1

!

and

Q

n˛nf WDf QKn˛n; Q˛f WD sup

n2N˛;K

jf QKn˛j:

In the sequel we prove that maximal operatorQ˛f is quasi-local. This is the very base of the proof of the main results of this paper. That is, Theorem1and Theorem 2.

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Lemma 4. Let1 > ˛n> 0,f 2L1.I /such that suppf Ik.u/,R

Ik.u/f d D0 for some dyadic intervalIk.u/. Then we have

Z

InIk.u/Q˛f d CKkfk1; where constantCKcan depend only onK.

Proof. Recall thatnD2hsC C2h0, wherehs> > h00are integers. That is,jnj Dhs. Letn.j /WD2hjC C2h0. This meansnDn.s/. Use also the notation

Q Kn˛.s/n

D QTn˛.s/n C

s

X

lD0

A˛nn.l 1/

A˛nn.s/ D2hl CA˛nn.l 1/

A˛nn.s/ TQn˛ln1

!

DWG1CG2CG3:

Sincen.l 1/< 2h.l 1/C1, then by Lemma3we have A˛nn.l 1/

A˛nn.s/ 2 n.l 1/C1 n.s/

!˛n

22˛n.hl 1C1/

2˛nhs C2hl 1˛n n˛n : That is, by the above written we also have

Z

InIk.u/

sup

n2N

ˇ ˇ ˇ ˇ Z

Ik.u/

f .x/G2.yCx/d .x/

ˇ ˇ ˇ ˇ

d .y/

Z

InIk.u/

sup

n2N s

X

lD0

2hl 1˛n n˛n

ˇ ˇ ˇ ˇ Z

Ik.u/

f .x/D2h.yCx/d .x/

ˇ ˇ ˇ ˇ

d .y/D0

sincef D2hD0forhkbecause of theAk measurability ofD2h andR

f D0.

Besides, forh > k D2h.yCx/D0(yCx…Ik).

As a result of these estimations above and by the help of Lemma2, that is the quasi-locality of operatortQ˛Dsupn;a2NjQtn˛ajwe conclude

Z

InIk.u/

sup

n2N

ˇ ˇ ˇ ˇ Z

Ik.u/

f .x/.G1.yCx/CG3.yCx//d .x/

ˇ ˇ ˇ ˇ

d .y/

CK Z

InIk.u/

sup

n;a2N

ˇ ˇ ˇ ˇ Z

Ik.u/

f .x/TQn˛a.yCx/d .x/

ˇ ˇ ˇ ˇ

d .y/

CKkfk1:

This completes the proof of Lemma4.

Lemma 5. The operatorQ˛is of type.L1; L1/ .Q˛f WDsupn2N˛;Kˇ ˇQn˛n

ˇ/:

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Proof. By the help of the method of Lemma4and by Corollary1we have

KQn˛n 1D

KQn˛.s/n

1

TQn˛.s/n

1C

s

X

lD0

A˛nn.l 1/

A˛nn.s/ kD2hlk1CA˛nn.l 1/

A˛nn.s/ k QTn˛ln1k1

!

CCC

s

X

lD0

A˛nn.l 1/

A˛nn.s/ CK

becausen2N˛;K. HenceQ˛ is of type.L1; L1/(with constantCK). This com-

pletes the proof of Lemma5.

Now, we can prove the main tool in order to have Theorem1for operator˛f WD supn2N˛;Kˇ

ˇn˛nfˇ ˇ:

Lemma 6. The operatorsQ˛and˛are of weak type.L1; L1/.

Proof. First, we prove Lemma6for operatorQ˛. We apply the Calderon-Zygmund decomposition lemma [9]. That is, letf 2L1andkfk1< ı. Then there is a decom- position:

f Df0C

1

X

jD1

fj

such that kf0k1C ı ;kf0k1Ckfk1andIj DIkj.uj/are disjoint dyadic in- tervals for which

suppfj Ij ; Z

Ij

fjd D0 ; jFj Ckf1k ı

.uj 2I ; kj 2N ; j 2P /, whereF D [1iD1Ij. By the-sublinearity of the max- imal operator with an appropriate constantCK we have

.Q˛f > 2CKı/.Q˛f0> CKı/C.Q˛.

1

X

iD1

fi/ > CKı/WDICII :

Since by Lemma5k Q˛f0k1CKkf0k1CKıthen we haveID0. So, .Q˛.

1

X

iD1

fi/ > CKı/ jFj C.FN\ f Q˛.

1

X

iD1

fi/ > CKıg/

CKkfk1 ı CCK

ı

1

X

iD1

Z

InIjQ˛fjd DWCKkfk1 ı CCK

ı

1

X

iD1

IIIj;

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where

IIIj WD Z

InIjQ˛fjd

Z

InIkj.uj/

sup

n2N˛;K

ˇ ˇ ˇ ˇ ˇ Z

Ikj.uj/

fj.x/KQn˛n.yCx/d .x/

ˇ ˇ ˇ ˇ ˇ

d .y/:

The forthcoming estimation ofIIIj is given by the help Lemma4 IIIj CKkfjk1:

That is, operatorQ˛is of weak type.L1; L1/. Next, we prove the estimation jKn˛nj QKn˛n: (1) To prove (1) recall again thatnD2hsC C2h0, wherehs> > h00are integers.

SincenD2hsCn.s 1/, then we have

2hsCn.s 1/

X

jD2hs

A˛nn.s 11/

C2hs jDj D

n.s 1/

X

kD0

A˛nn.s 11/ kD2hsCk

DD2hs

n.s 1/

X

kD0

A˛nn.s 11/ kC!2hs

n.s 1/

X

kD0

A˛nn.s 11/ kDk DD2hsA˛nn.s 1/C!2hsA˛nn.s 1/Kn˛.sn 1/:

So, by the help of the equalities above we get Kn˛.s/n DTn˛.s/n CA˛nn.s 1/

A˛nn.s/

D2hs C!2hsKn˛.sn 1/

:

Apply this last formula recursively and Lemma 2. (Note that n. 1/ D0; T0˛n D K0˛nD0; A˛0nD1.)

jKn˛nj D jKn˛.s/n j jTn˛.s/nj C

s

X

lD0

0

@

s

Y

jDl

A˛nn.j 1/

A˛nn.j / D2hl C

s

Y

jDl

A˛nn.j 1/

A˛nn.j / jTn˛ln1j 1 A

D jTn˛.s/nj C

s

X

lD0

A˛nn.l 1/

A˛nn.s/ D2hl CA˛nn.l 1/

A˛nn.s/ jTn˛ln1j

!

QKn˛.s/n D QKn˛n:

This completes the proof of inequality (1). This inequality gives that the operator˛ is also of weak type.L1; L1/since

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.˛f > 2CKı/.Q˛jfj> 2CKı/CKkjfjk1

ı DCKkfk1 ı :

This completes the proof of Lemma6.

Proof of Theorem1. LetP 2Pbe a polynomial whereP .x/DP2k 1

iD0 ci!i. Then for all natural number n2k, n2N˛;K we have that SnP P. Consequently, the statementn˛nP !P holds everywhere (of course not only for restrictedn2 N˛;K). Now, let; ı > 0; f 2L1. LetP2Pbe a polynomial such thatkf Pk1< ı.

Then . lim

n2N˛;K

jn˛nf fj> / . lim

n2N˛;K

jn˛n.f P /j>

3/C. lim

n2N˛;K

jn˛nP Pj>

3/ C. lim

n2N˛;K

jP fj>

3/ . sup

n2N˛;K

jn˛n.f P /j>

3/C0C3

kP fk1CKkP fk13 CK

ı because˛is of weak type.L1; L1/(with any fixedK > 0). This holds for allı > 0.

That is, for an arbitrary > 0we have . lim

n2N˛;K

jn˛nf fj> /D0 and consequently we also have

. lim

n2N˛;K

jn˛nf fj> 0/D0:

This finally gives

n2limN˛;K

jn˛nf fj D0 a:e;

n˛nf !f a:e: .n2N˛;K/:

This completes the proof of Theorem1.

Proof of Theorem2. Inequality (1), Lemma5and Lemma6by the interpolation theorem of Marcinkiewicz [9] give that the operator˛ is of type.Lp; Lp/for all 1 < p 1. In the sequel we prove that operatorQ˛f Dsupn2N˛;Kjf QKn˛jis of type.H; L/.

Letabe an atom (a¤1can be supposed ), suppaIk.x/ ;kak12kfor some k2N andx2I. Then,n < 2k,n2N˛;K impliesa QKn˛D0becauseKQn˛ isAk measurable forn < 2k andR

Ik.x/a.t /d .t /D0. That is, Q

˛aD sup

N˛;K3n2k

j Qn˛nfj:

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By the help Lemma4we have Z

InIk.x/Q˛a d D Z

InIk.x/

sup

N˛;K3n2k

ˇ ˇ ˇ ˇ Z

Ik.x/

a.y/KQn˛n.´Cy/d .y/

ˇ ˇ ˇ ˇ

d .´/

CK Z

Ik.x/ja.y/jd .y/

CKkak1

CK:

Since the operatorQ˛is of type.L2; L2/(i.ek Q˛fk2CKkfk2for allf 2L2.I /), then we have

k Q˛ak1D Z

InIk.x/Q˛aC Z

Ik.x/Q˛a CKC jIk.x/j12k Q˛ak2

CKCCK2 2kkak2 CKCCK2 2k2k2 CK:

That isk Q˛ak1CKand consequently the-sublinearity ofQ˛gives k Q˛fk1

1

X

iD0

jijk Q˛aik1

CK 1

X

iD0

jij CKkfkH for allP1

iD0iai 2H. That is, the operatorQ˛is of type.H; L/. This by inequality (1) and then byk˛fk1 k Q˛jfjk1CKkjfjkH completes the proof of Theorem

2.

REFERENCES

[1] T. Akhobadze, “On the convergence of generalized Ces´aro means of trigonometric Fourier series.

I.”Acta Math. Hung., vol. 115, no. 1-2, pp. 59–78, 2007, doi: https://doi.org/10.1007/s10474- 007-5214-7.

[2] T. Akhobadze, “On the convergence of generalized Ces´aro means of trigonometric Fourier series.”

Bulletin of TICMI, vol. 18, no. 1, pp. 75–84, 2014.

[3] N. Fine, “Ces`aro summability of Walsh-Fourier series,”Proc. Nat. Acad. Sci. USA, vol. 41, pp.

588–591, 1955, doi:https://doi.org/10.1073/pnas.41.8.588.

[4] S. Fridli, “On the rate of convergence of ces`aro means of walsh-fourier series,” J. of Approx.

Theory, vol. 76, no. 1, pp. 31–53, 1994, doi:https://doi.org/10.1006/jath.1994.1003.

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[5] N. Fujii, “A maximal inequality forh1functions on the generalized walsh-paley group,”Proc.

Amer. Math. Soc., vol. 77, pp. 111–116, 1979, doi: https://doi.org/10.1090/S0002-9939-1979- 0539641-9.

[6] G. G´at, “On (c,1) summability for vilenkin-like systems,”Stud. Math., vol. 144, no. 2, pp. 101–

120, 2001, doi:10.4064/sm144-2-1.

[7] U. Goginava, “Approximation properties of (c, ˛) means of double walsh-fourier series,”

Analysis in Theory and Applications, vol. 20, no. 1, pp. 77–98, 2004, doi: ht- tps://doi.org/10.1007/BF02835261.

[8] E. Hewitt and K. Ross,Abstract Harmonic Analysis I. Heidelberg: Springer-Verlag, 1963.

[9] F. Schipp, W. Wade, P. Simon, and J. P´al,Walsh series,”An Introduction to dyadic harmonic analysis”. Bristol and New York: Adam Hilger, 1990.

[10] M. Taibleson,Fourier Analysis on Local Fields. Princeton, N.J.: Princeton Univ. Press., 1975.

[11] F. Weisz, “.C; ˛/summability of Walsh-Fourier series,”Analysis Mathematica, vol. 27, pp. 141–

155, 2001.

[12] A. Zygmund,Trigonometric Series. Cambridge: University Press, 1959.

Authors’ addresses

Anas Ahmad Abu Joudeh

Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary

E-mail address:anas.abujoudeh@mailbox.unideb.edu.hu, mr anas judeh@yahoo.com

Gy¨orgy G´at

Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary E-mail address:gat.gyorgy@science.unideb.hu

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