(1)ON THE DEGREE OF STRONG APPROXIMATION OF CONTINUOUS FUNCTIONS BY SPECIAL MATRIX BOGDAN SZAL FACULTY OFMATHEMATICS, COMPUTERSCIENCE ANDECONOMETRICS UNIVERSITY OFZIELONAGÓRA 65-516 ZIELONAGÓRA,UL

ON THE DEGREE OF STRONG APPROXIMATION OF CONTINUOUS FUNCTIONS BY SPECIAL MATRIX

BOGDAN SZAL

FACULTY OFMATHEMATICS, COMPUTERSCIENCE ANDECONOMETRICS

UNIVERSITY OFZIELONAGÓRA

65-516 ZIELONAGÓRA,UL. SZAFRANA4A, POLAND

B.Szal@wmie.uz.zgora.pl

Received 13 May, 2009; accepted 20 October, 2009 Communicated by I. Gavrea

ABSTRACT. In the presented paper we will generalize the result of L. Leindler [3] to the class M RBV Sand extend it to the strong summability with a mediate function satisfying the standard conditions.

Key words and phrases: Strong approximation, matrix means, classes of number sequences.

2000 Mathematics Subject Classification. 40F04, 41A25, 42A10.

1. INTRODUCTION

Letf be a continuous and2π-periodic function and let

(1.1) f(x)∼ a₀

2 +

∞

X

n=1

(ancosnx+bnsinnx)

be its Fourier series. Denote byS_n(x) =S_n(f, x)then-th partial sum of (1.1) and byω(f, δ) the modulus of continuity off ∈C_2π. The usual supremum norm will be denoted byk·k.

LetA := (a_nk) (k, n = 0,1, ...)be a lower triangular infinite matrix of real numbers satisfy- ing the following conditions:

(1.2) a_nk ≥0 (0≤k ≤n), a_nk = 0, (k > n) and

n

X

k=0

a_nk = 1, wherek, n= 0,1,2, ....

Let theA−transformation of(S_n(f;x))be given by (1.3) t_n(f) := t_n(f;x) :=

n

X

k=0

a_nkS_k(f;x) (n = 0,1, ...)

132-09

and the strongA_r−transformation of(S_n(f;x))forr >0be given by T_n(f, r) := T_n(f, r;x) :=

( _n X

k=0

a_nk|S_k(f;x)−f(x)|^r )¹_r

(n= 0,1, ...). Now we define two classes of sequences.

A sequencec := (c_n) of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or brieflyc∈RBV S, if it has the property

(1.4)

∞

X

n=m

|c_n−c_n+1| ≤K(c)c_m

form = 0,1,2, ..., whereK(c)is a constant depending only onc(see [3]).

A null sequencec:= (c_n)of positive numbers is called of Mean Rest Bounded Variation, or brieflyc∈M RBV S, if it has the property

(1.5)

∞

X

n=2m

|c_n−c_n+1| ≤K(c) 1 m+ 1

2m

X

n=m

c_n form = 0,1,2, ...(see [5]).

Therefore we assume that the sequence(K(α_n))^∞_n=0 is bounded, that is, there exists a con- stantK such that

0≤K(α_n)≤K

holds for alln, whereK(α_n) denotes the sequence of constants appearing in the inequalities (1.4) or (1.5) for the sequenceα_n := (a_nk)^∞_k=0. Now we can give some conditions to be used later on. We assume that for alln

(1.6)

∞

X

k=m

|ank −ank+1| ≤Kanm (0≤m ≤n) and

(1.7)

∞

X

k=2m

|a_nk −a_nk+1| ≤K 1 m+ 1

2m

X

k=m

a_nk (0≤2m≤n) hold ifαn:= (ank)^∞_k=0 belongs toRBV S orM RBV S, respectively.

In [1] and [2] P. Chandra obtained some results on the degree of approximation for the means (1.3) with a mediate functionH such that:

(1.8)

Z π u

ω(f;t)

t² dt=O(H(u)) (u→0+), H(t)≥0 and

(1.9)

Z t 0

H(u)du=O(tH(t)) (t →O₊).

In [3], L. Leindler generalized this result to the classRBV S. Namely, he proved the follow- ing theorem:

Theorem 1.1. Let (1.2), (1.6), (1.8) and (1.9) hold. Then forf ∈C2π

kt_n(f)−fk=O(a_n0H(a_n0)).

It is clear that

(1.10) RBV S ⊆M RBV S.

In [7], we proved thatRBV S 6=M RBV S.Namely, we showed that the sequence d_n:=







1 ifn = 1,

1+m+(−1)ⁿm

(2^µm)²m ifµ_m ≤n < µ_m+1,

whereµ_m = 2^mform = 1,2,3, ..., belongs to the classM RBV Sbut it does not belong to the classRBV S.

In the present paper we will generalize the mentioned result of L. Leindler [3] to the class M RBV S and extend it to strong summability with a mediate function H defined by the fol- lowing conditions:

(1.11)

Z π u

ω^r(f;t)

t² dt=O(H(r;u)) (u→0₊), H(t)≥0andr >0, and

(1.12)

Z t 0

H(r;u)du=O(tH(r;t)) (t→O₊).

By K₁, K₂, . . . we shall denote either an absolute constant or a constant depending on the indicated parameters, not necessarily the same in each occurrence.

2. MAINRESULTS

Our main results are the following.

Theorem 2.1. Let (1.2), (1.7) and (1.11) hold. Then forf ∈C_2π andr >0

(2.1) kT_n(f, r)k=O

n

a_n0H r;π

n o¹_r

. If, in addition (1.12) holds, then

(2.2) kTn(f, r)k=O

{an0H(r;an0)}^1r . Using the inequality

kt_n(f)−fk ≤ kT_n(f,1)k, we can formulate the following corollary.

Corollary 2.2. Let (1.2), (1.7) and (1.11) hold. Then forf ∈C_2π kt_n(f)−fk=O

a_n0H 1;π

n

. If, in addition (1.12) holds, then

kt_n(f)−fk=O(a_n0H(1;a_n0)).

Remark 1. By the embedding relation (1.7) we can observe that Theorem 1.1 follows from Corollary 2.2.

For special cases, putting

H(r;t) =







t^rα−1 if αr < 1, ln^π_t if αr = 1, K₁ if αr > 1,

wherer >0and0< α≤1, we can derive from Theorem 2.1 the next corollary.

Corollary 2.3. Under the conditions (1.2) and (1.7) we have, forf ∈C_2π andr >0,

kT_n(f, r)k=















O({a_n0}^α) if αr < 1, On

ln

π an0

a_n0oα

if αr = 1, O

{a_n0}^1r

if αr > 1.

3. LEMMAS

To prove our main result we need the following lemmas.

Lemma 3.1 ([6]). If (1.11) and (1.12) hold, then forr >0 Z s

0

ω^r(f;t)

t dt =O(sH(r;s)) (s→0₊). Lemma 3.2. If (1.2) and (1.7) hold, then forf ∈C_2π andr > 0

(3.1) kT_n(f, r)k_C ≤O



 ( _n

X

k=0

a_nkE_k^r(f) )¹_r

,

whereE_n(f)denotes the best approximation of the functionf by trigonometric polynomials of order at mostn.

Proof. It is clear that (3.1) holds forn= 0,1, ...,5. Namely, by the well known inequality [8]

(3.2) kσ_n,m−fk ≤2n+ 1

m+ 1E_n(f) (0≤m≤n), where

σ_n,m(f;x) = 1 m+ 1

n

X

k=n−m

S_k(f;x), form = 0,we obtain

{T_n(f, r;x)}^r ≤12^r

n

X

k=0

a_nkE_k^r(f) and (3.1) is obviously valid, forn ≤5.

Letn ≥6and letm=m_nbe such that

2^m+1+ 4 ≤n <2^m+2+ 4.

Hence

{T_n(f, r;x)}^r ≤

3

X

k=0

a_nk|S_k(f;x)−f(x)|^r

+

m−1

X

k=1 2^k+1+4

X

i=2^k+2

a_ni|S_i(f;x)−f(x)|^r+

n

X

k=2^m+5

a_nk|S_k(f;x)−f(x)|^r.

Applying the Abel transformation and (3.2) to the first sum we obtain {T_n(f, r;x)}^r

≤8^r

3

X

k=0

a_nkE_k^r(f) +

m−1

X

k=1





2^k+1+3

X

i=2^k+2

(a_ni−a_n,i+1)

i

X

l=2^k+2

|S_l(f;x)−f(x)|^r

+a_n,2^k+1₊₄

2^k+1+4

X

i=2^k+2

|Si(f;x)−f(x)|^r





+

n−1

X

k=2^m+2

(ank−an,k+1)

k

X

l=2^m−1

|Sl(f;x)−f(x)|^r

+a_nn

n

X

k=2^m+2

|S_k(f;x)−f(x)|^r

≤8^r

3

X

k=0

ankE_k^r(f) +

m−1

X

k=1





2^k+1+3

X

i=2^k+2

|ani−an,i+1|

2^k+1+3

X

l=2^k+2

|Sl(f;x)−f(x)|^r

+a_n,2^k+1₊₄

2^k+1+4

X

i=2^k+2

|S_i(f;x)−f(x)|^r





+

n−1

X

k=2^m+2

|ank−an,k+1|

2^m+2+3

X

l=2^m+2

|Sl(f;x)−f(x)|^r

+ann

2^m+2+4

X

k=2^m+2

|Sk(f;x)−f(x)|^r.

Using the well-known Leindler’s inequality [4]

( 1 m+ 1

n

X

k=n−m

|S_k(f;x)−f(x)|^s )¹_s

≤K₁En−m(f)

for0≤m≤n,m=O(n)ands >0, we obtain

{T_n(f, r;x)}^r ≤8^r

3

X

k=0

a_nkE_k^r(f)

+K₂







m−1

X

k=1



 2^k+ 3

E₂^rk+2(f)





2^k+1+3

X

i=2^k+2

|a_ni−a_n,i+1|+a_n,2^k+1₊₄









3 (2^m+ 1)E₂^rm+2

n−1

X

k=2^m+2

|ank −an,k+1|+ann

!) .

Using (1.7) we get {Tn(f, r;x)}^r ≤8^r

3

X

k=0

ankE_k^r(f)

+K₂







m−1

X

k=1



 2^k+ 3

E₂^rk+2(f)



K 1 2^k−1+ 2

2^k+2

X

i=2^k−1+1

a_ni+a_n,2^k+1₊₄









3 (2^m+ 1)E₂^rm+2(f) K 1 2^m−1+ 2

2^m+2

X

i=2^m−1+1

a_ni+a_nn

!) . In view of (1.7), we also obtain for1≤k ≤m−1,

a_n,2^k+1₊₄ =

∞

X

i=2^k+1+4

(a_ni−a_ni+1)≤

∞

X

i=2^k+1+4

|a_ni−a_ni+1|

≤

∞

X

i=2^k+2

|ani−ani+1| ≤K 1 2^k−1+ 2

2^k+2

X

i=2^k−1+1

ani

and

a_nn =

∞

X

i=n

(a_ni−a_ni+1)≤

∞

X

i=n

|a_ni−a_ni+1|

≤

∞

X

i=2^m+2

|a_ni−a_ni+1| ≤K 1 2^m−1+ 2

2^m+2

X

i=2^m−1+1

a_ni.

Hence

{T_n(f, r;x)}^r ≤8^r

3

X

k=0

a_nkE_k^r(f)

+K3







m−1

X

k=1

E₂^rk+2(f)

2^k+2

X

i=2^k−1+1

ani+E₂^rm+2(f)

2^m+2

X

i=2^m−1+1

ani







≤8^r

3

X

k=0

a_nkE_k^r(f) + 2K₃

2^m+2

X

k=3

a_nkE_k^r(f)

≤K₄

n

X

k=0

a_nkE_k^r(f).

This ends our proof.

4. PROOF OFTHEOREM2.1 Using Lemma 3.2 we have

(4.1) |T_n(f, r;x)| ≤K₁ ( _n

X

k=0

a_nkE_k^r(f) )¹_r

≤K₂ ( _n

X

k=0

a_nkω^r

f; π k+ 1

)¹_r .

If (1.7) holds, then, for anym = 1,2, ..., n,

a_nm−a_n0 ≤ |a_nm−a_n0|=|a_n0−a_nm|=

m−1

X

k=0

(a_nk −a_nk+1)

≤

m−1

X

k=0

|ank−ank+1| ≤

∞

X

k=0

|ank−ank+1| ≤Kan0, whence

(4.2) a_nm ≤(K + 1)a_n0.

Therefore, by (1.2),

(4.3) (K+ 1) (n+ 1)a_n0 ≥

n

X

k=0

a_nk = 1.

First we prove (2.1). Using (4.2), we get

n

X

k=0

a_nkω^r

f; π k+ 1

≤(K+ 1)a_n0

n

X

k=0

ω^r

f; π k+ 1

≤K₃a_n0 Z n+1

1

ω^r f;π

t

dt

=πK₃a_n0 Z π

π n+1

ω^r(f;u) u² du and by (4.1), (1.11) we obtain that (2.1) holds.

Now, we prove (2.2). From (4.3) we obtain

n

X

k=0

a_nkω^r

f; π k+ 1

≤

h 1

(K+1)an0

i−1

X

k=0

ankω^r

f; π k+ 1

+

n

X

k=h

1 (K+1)an0

i

−1

ankω^r

f; π k+ 1

.

Again using (1.2), (4.2) and the monotonicity of the modulus of continuity, we get

n

X

k=0

a_nkω^r

f; π k+ 1

≤(K+ 1)a_n0

h 1

(K+1)an0

i

−1

X

k=0

ω^r

f; π k+ 1

+K₄ω^r(f;π(K+ 1)a_no)

n

X

k=

h 1

(K+1)an0

i

−1

a_nk

≤K₅a_n0

Z _(K+1)¹

an0

1

ω^r f;π

t

dt+K₄ω^r(f;π(K+ 1)a_no)

≤K₆

a_n0 Z π

an0

ω^r(f;u)

u² du+ω^r(f;a_n0)

. (4.4)

Moreover

ω^r(f;a_n0)≤4^rω^r f;a_n0

2 (4.5)

≤2·4^r Z an0

an0 2

ω^r(f;t) t dt

≤2·4^r Z an0

0

ω^r(f;t) t dt.

Thus collecting our partial results (4.1), (4.4), (4.5) and using (1.11) and Lemma 3.1 we can see

that (2.2) holds. This completes our proof.

REFERENCES

[1] P. CHANDRA, On the degree of approximation of a class of functions by means of Fourier series, Acta Math. Hungar., 52 (1988), 199–205.

[2] P. CHANDRA, A note on the degree of approximation of continuous function, Acta Math. Hungar., 62 (1993), 21–23.

[3] L. LEINDLER, On the degree of approximation of continuous functions, Acta Math. Hungar., 104 (1-2), (2004), 105–113.

[4] L. LEINDLER, Strong Approximation by Fourier Series, Akadèmiai Kiadò, Budapest (1985).

[5] L. LEINDLER, Integrability conditions pertaining to Orlicz space, J. Inequal. Pure and Appl. Math., 8(2) (2007), Art. 38.

[6] B. SZAL, On the rate of strong summability by matrix means in the generalized Hölder metric, J.

Inequal. Pure and Appl. Math., 9(1) (2008), Art. 28.

[7] B. SZAL, A note on the uniform convergence and boundedness a generalized class of sine series, Comment. Math., 48(1) (2008), 85–94.

[8] Ch. J. DE LA VALLÉE - POUSSIN, Leçons sur L’Approximation des Fonctions d’une Variable Réelle, Paris (1919).