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

(1)

Degree of Strong Approximation Bogdan Szal

vol. 10, iss. 4, art. 111, 2009

Title Page

Contents

JJ II

J I

Page1of 17 Go Back Full Screen

Close

ON THE DEGREE OF STRONG APPROXIMATION OF CONTINUOUS FUNCTIONS BY SPECIAL

MATRIX

BOGDAN SZAL

Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra

65-516 Zielona Góra ul. Szafrana 4a, Poland

EMail:B.Szal@wmie.uz.zgora.pl

Received: 13 May, 2009

Accepted: 20 October, 2009

Communicated by: I. Gavrea

2000 AMS Sub. Class.: 40F04, 41A25, 42A10.

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

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

(2)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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:= (ank) (k, n= 0,1, ...)be a lower triangular infinite matrix of real numbers satisfying 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, ...) 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.

(4)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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 sequence c := (cn) 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 constantKsuch that

0≤K(α_n)≤K

holds for all n, where K(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

|a_nk −a_nk+1| ≤Ka_nm (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)

(5)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

hold ifα_n:= (a_nk)^∞_k=0belongs toRBV SorM 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 functionHsuch 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 following 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 S6=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^m for m = 1,2,3, ..., belongs to the class M RBV S but it does not belong to the classRBV S.

(6)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

In the present paper we will generalize the mentioned result of L. Leindler [3] to the classM RBV S and extend it to strong summability with a mediate functionH defined by the following 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+).

ByK₁, K₂, . . .we shall denote either an absolute constant or a constant depending on the indicated parameters, not necessarily the same in each occurrence.

(7)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Page7of 17 Go Back Full Screen

Close

2. Main Results

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) kT_n(f, r)k=O

{a_n0H(r;a_n0)}¹^r . 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 ∈C2π

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 Theorem1.1follows from Corollary2.2.

(8)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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 Theorem2.1the next corollary.

Corollary 2.3. Under the conditions (1.2) and (1.7) we have, forf ∈C_2πandr >0,

kT_n(f, r)k=











O({a_n0}^α) if αr <1, O

n ln

π an0

an0

oα

if αr= 1, O

{a_n0}¹^r

if αr >1.

(9)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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

,

where En(f) denotes the best approximation of the function f by trigonometric polynomials of order at mostn.

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

(3.2) kσn,m−fk ≤2n+ 1

m+ 1En(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)

(10)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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

ani|Si(f;x)−f(x)|^r+

n

X

k=2^m+5

ank|Sk(f;x)−f(x)|^r. Applying the Abel transformation and (3.2) to the first sum we obtain

{Tn(f, r;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)

i

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

(a_nk−a_n,k+1)

k

X

l=2^m−1

|S_l(f;x)−f(x)|^r

+a_nn

n

X

k=2^m+2

|S_k(f;x)−f(x)|^r

(11)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

≤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|

2^k+1+3

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

|S_i(f;x)−f(x)|^r





+

n−1

X

k=2^m+2

|a_nk−a_n,k+1|

2^m+2+3

X

l=2^m+2

|S_l(f;x)−f(x)|^r

+a_nn

2^m+2+4

X

k=2^m+2

|S_k(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

|a_nk−a_n,k+1|+a_nn

!) .

(12)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

Using (1.7) we get {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)



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

|a_ni−a_ni+1| ≤K 1 2^k−1 + 2

2^k+2

X

i=2^k−1+1

a_ni 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.

(13)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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.

(14)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

4. Proof of Theorem 2.1

Using Lemma3.2we have

|T_n(f, r;x)| ≤K₁ ( _n

X

k=0

a_nkE_k^r(f) )_r¹ (4.1)

≤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

|a_nk−a_nk+1| ≤

∞

X

k=0

|a_nk−a_nk+1| ≤Ka_n0, 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

(15)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

≤K₃a_n0 Z n+1

1

ω^r f;π

t

dt

=πK3an0

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

a_nkω^r

f; π k+ 1

+

n

X

k=h

1 (K+1)an0

i

−1

a_nkω^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

(16)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

≤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;an0

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 Lemma3.1 we can see that (2.2) holds. This completes our proof.

(17)

vol. 10, iss. 4, art. 111, 2009

Title Page Contents

JJ II

J I

Close

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, Bu- dapest (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).