# Pointwise approximation of modified conjugate functions by matrix operators of their Fourier series.

1. INTRODUCTION

Let Lp (1 [less than or equal to] p < [infinity]) be the class of all 2[pi]-periodic real-valued functions, integrable in the Lebesgue sense with the pth power over Q=[-[pi], [pi]] with the norm

[mathematical expression not reproducible]

Given a function of class Lp let us consider its conjugate trigonometric Fourier series

[mathematical expression not reproducible]

with the partial sums [S.sub.k] f. We know that if f [member of] [L.sup.1], then

[mathematical expression not reproducible]

where, for r [member of] N,

[mathematical expression not reproducible]

and [mathematical expression not reproducible] with [mathematical expression not reproducible] exist for almost all x (cf. [5, Theorem (3.1)IV])

Let A := ([a.sub.n,k]) be an infinite matrix of real numbers such that

[mathematical expression not reproducible]

We will use the notations [mathematical expression not reproducible] and

[mathematical expression not reproducible]

for the A-transformationof Sf.

In this paper, we will estimate the deviation [T.sub.n,A]f(x)-[f.sub.r](x,[epsilon]) by the function of modulus of continuity type, i.e. nondecreasing continuous function [omega] having the following properties: [omega](0) = 0, [mathematical expression not reproducible] for any [mathematical expression not reproducible] We will also consider functions from the following subclass [L.sub.p] [([omega]).sub.[beta]] of Lp:

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

It is clear that for [mathematical expression not reproducible] and it is easy to see that [mathematical expression not reproducible] is the classical integral modulus of continuity of f.

The above deviation was estimated with r = 1 in  and generalized in  as follows:

Theorem [1, Theorem 8, p. 95]. If f [MEMBER OF] [L.sup.p] [([member of]).sub.[beta]] with [beta] < 1 - 1 p where [omega] satisfies the conditions

[mathematical expression not reproducible] (1)

and

[mathematical expression not reproducible] (2)

with 0 < [gamma] < [beta] + 1/p, then

[mathematical expression not reproducible]

In our theorems we generalize the above results using [f.sub.r] (x, [epsilon]) with r [member of] N instead of [f.sub.1] (x, [epsilon]) = f(x, [epsilon]). In the paper [[SIGMA].sup.b.sub.k=a]= 0 when a >b.

2. STATEMENT OF THE RESULTS

First we will present the estimates of the quantity [mathematical expression not reproducible] Finally, we will formulate some remarks and corollaries.

Theorem 1. Let f [member of] [L.sup.p], 0 [less than or equal to] [beta] < 1 - 1 p and let a function of modulus of continuity type [omega] satisfy the conditions: for r [member of] N

[mathematical expression not reproducible] (3)

for a natural r[greater than or equal to]3

[mathematical expression not reproducible] (4)

where m [member of] { 1,... [1/2]} when r is anoddnatural number or m [member of][mathematical expression not reproducible] when r is an even natural number, and for r [member of] N

[mathematical expression not reproducible] (5)

with 0<[gamma]<[beta] + 1 p, where m [member of]{ 0,... [ 2 r ]} when r is anodd natural number or m [member of] { 0, ... [ 2 r ] -1 } whenr is an even natural number. Moreover, let [omega] satisfy, for a natural r [greater than or equal to] 2, the conditions:

[mathematical expression not reproducible] (6)

[mathematical expression not reproducible] (7)

with 0<[gamma]<[beta] + 1 p, where m [member of] {0,... [r/2] -1 }. If a matrix A is such that

[mathematical expression not reproducible] (8)

and

[mathematical expression not reproducible] (9)

are true for r [member of] N, then

[mathematical expression not reproducible]

Theorem 2. Let f [member of] [L.sup.p], 0 [less than or equal to] [beta] < 1 - 1 p and let a function of modulus of continuity type [omega] satisfy, for r [member of] N, theconditions:

[mathematical expression not reproducible] (10)

and(5)wth0<[gamma]<[beta]+ 1 p where m [member of]{ 0,... [r/2]} when r is an even natural number. Moreover, let [omega] satisfy for natural r [greater than or equal to] 2, the conditions (6) and (7) with 0<[gamma]<[beta]3 + 1 pwhere m [member of] { 0,... [ 2 r ] -1 }. If a matrix A is such that (8) and

[mathematical expression not reproducible] (11)

are true for r [member of] N then

[mathematical expression not reproducible]

We can observe that if f [member of] [L.sup.p][([omega]).sub.[beta]] and

[mathematical expression not reproducible] (12)

is fulfilled, then the conditions (3)-(7) and (10) always hold with [mathematical expression not reproducible] instead of [mathematical expression not reproducible]. Hence from Theorems 1 and 2 we can obtain the following corollary:

Corollary 1. Let f [member of] [L.sup.p] [([omega]).sub.[beta]] with 0 [less than or equal to] [beta] < 1 - 1 p, where [omega] satisfy condition (12). Ifa matrix A is such that (8) and (9) or (8) and (11) are true for r [member of] N, then

[mathematical expression not reproducible]

Corollary 2. We can observe that in the case r = 1, the conditions (3)-(7) in Theorem 1 reduce to (1) and (2). Thus we obtain the results from  and .

Remark 1. If we consider the following more natural conditions

[mathematical expression not reproducible]

[mathematical expression not reproducible]

for [gamma] [member of] (1/p, 1/p + [beta]) where [beta] > 0; instead of (5) and (7), and

[mathematical expression not reproducible]

[mathematical expression not reproducible]

instead of conditions (6) and (10), respectively, then our estimate takes the form

[mathematical expression not reproducible]

Remark 2. We note that our extra conditions (8), (9), and (11) for a lower triangular infinite matrix A always hold.

Corollary 3. Under the above remarks and the obvious inequality

[A.sub.n,r][less than or equal to]r[A.sub.n,1]for r [member of] N, (13)

our results also improve and generalize the mentioned result of Krasniqi .

Remark 3. We note that instead of [L.sup.p][([omega]).sub.[beta]] one can consider other subclasses of [L.sup.p] generated by any function of modulus continuity type, e.g. [[omega].sub.x] such that

[mathematical expression not reproducible]

Remark 4. We note that our condition (12) holds if we take [omega]([delta]) = [[delta].sup.[alpha]] with 0 < [alpha] < [beta] + 1 p.

3. AUXILIARY RESULTS

We begin this section with some notations from  and [5, Section 5 of Chapter II]. Let for r = 1,2,...

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

It is clear by  that

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Now we present a very useful property of the modulus of continuity.

Lemma 1 (). A function [omega] of a modulus of continuity type on the interval [0,2[pi]] satisfies the condition [mathematical expression not reproducible]

Next, we present the known estimates.

Lemma 2(). If 0 < |t| [less than or equal to] [pi],then

and for any real t we have

[mathematical expression not reproducible].

Lemma 3 ([3,4]). Let r [member of] N,l [member of] Z and ([a.sup.n]) [subset] C If t [not equal to] 2l [pi]/r, then for every m[greater than or equal to]n

[mathematical expression not reproducible]

4. PROOFS OF THEOREMS

4.1. Proof of Theorem 1

It is clear that for an odd r

[mathematical expression not reproducible].

and for an even r

[mathematical expression not reproducible].

Then

[mathematical expression not reproducible]

By Lemma 3,

[mathematical expression not reproducible]

Therefore,

[mathematical expression not reproducible]

Using the estimates [mathematical expression not reproducible], where m [member of] {0,..., [r/2]}, we obtain

[mathematical expression not reproducible]

Hence by (5)

[mathematical expression not reproducible]

Using Lemma 1 we get

[mathematical expression not reproducible]

for 0 <[gamma]<[beta] + 1/p. Therefore

[mathematical expression not reproducible]

Next,

[mathematical expression not reproducible]

Using the estimates [mathematical expression not reproducible] for [mathematical expression not reproducible], where m [member of] {0,..., [r/2]-1}, we get

[mathematical expression not reproducible]

Analogously as before, by (7)

[mathematical expression not reproducible]

for 0 < [gamma] < [beta] + 1/p.

Further, by Lemma 2

[mathematical expression not reproducible]

Hence, by condition (4) for 0 [less than or equal to] < 1 - 1/p

[mathematical expression not reproducible]

Finally, we note that applying condition (8) we have

[mathematical expression not reproducible]

whence

[mathematical expression not reproducible]

By Lemma 2, (9), (3), and (8) for 0 [less than or equal to] [beta] < 1 - 1/p

[mathematical expression not reproducible]

Analogously as in the estimate of [mathematical expression not reproducible] we get

[mathematical expression not reproducible]

Hence, by (6)

[mathematical expression not reproducible]

Thus our proof is complete. D

4.2. Proof of Theorem 2

As in the above proof

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Next, by Lemma 2, (11), (10) for 0 [less than or equal to] [beta] < 1 - 1/p and (8), we get

[mathematical expression not reproducible]

Finally, applying Lemma 2 and conditions (11), (6), and (8) we obtain

[mathematical expression not reproducible]

Collecting the partial estimates we get our statement.

5. CONCLUSIONS

We investigated pointwise approximation of modified conjugate functions by matrix operators of their Fourier series. In particular, we estimated the deviation [mathematical expression not reproducible] by the function of modulus of continuity type in the case when conjugate function [f.sub.r] depends on r. In the obtained results the measures of approximation depend on r-differences of the entries.

ACKNOWLEDGEMENTS

We appreciate helpful and constructive comments from the anonymous referee that improved the paper. The publication costs of this article were covered by the University of Zielona Gora and the Estonian Academy of Sciences.

REFERENCES

[1.] Krasniqi, Xh. Z. Slight extensions of some theorems on the rate of pointwise approximation of functions from some subclasses of [L.sup.p]. Acta Comment. Univ. Tartu. Math, 2013, 17, 89-101.

[2.] Lenski, W. and Szal, B. Approximation of functions belonging to the class [L.sup.p]([omega]) by linear operators. Acta Comment. Univ. Tartu. Math, 2009, 13,11-24.

[3.] Szal, B. A new class of numerical sequences and its applications to uniform convergence of sine series. Math. Nachr., 2011, 284, 1985-2002.

[4.] Szal, B. On L-convergence of trigonometric series. J. Math. Anal. Appl., 2011, 373, 449-463.

[5.] Zygmund, A. Trigonometric Series. Cambridge University Press, 2002.

Wlodzimierz Lenski ja Bogdan Szal

Modifitseeritud kaasfunktsioonide punktiviisi lahendamine nende Fourier' ridade maatriksoperaatoritega

Olgu [L.sup.p] (1 [less than or equal to] p < [infinity]) reaalvaartustega 2[pi]-perioodiliste loigul [-[pi],[pi]] Lebesgue'i mottes integreeruvate funktsioonide klassja Sf(x) funktsiooni f [member of] [L.sup.p] trigonomeetrilineFourier' kaasrida. Onteada,etfunktsiooni f [member of] [L.sup.1] kaasfunktsioon f. on esitatav teatud viisil defineeritud funktsioonide [f.sub.r](x, [epsilon]) (r on naturaalarv) kaudu piirvaartusena

[mathematical expression not reproducible].

Olgu A teatud omadustega mittenegatiivne regulaarne maatriks ja [T.sub.n,A]f(x) rea Sf A-teisendus. Artiklis on antud hinnang vahele [mathematical expression not reproducible] pidevuse mooduli tuupi funktsiooni abil.

Wlodzimierz Lenski (*) and Bogdan Szal

University of Zielona Gora, Faculty of Mathematics, Computer Science and Econometrics, 65-516 Zielona Gora, ul. Szafrana 4a, Poland; B.Szal@wmie.uz.zgora.pl

(*) Corresponding author, W.Lenski@wmie.uz.zgora.pl

Received 3 April 2017, revised 27 August 2017, accepted 18 September 2017, available online 1 February 2018
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