Vol. 44, No. 2, 2014, 161-172
APPROXIMATION BY LINEAR SUMMABILITY MEANS IN ORL˙ICZ SPACES
Sadulla Z. Jafarov1
Abstract. In the present work we estimate of deviations of periodic functions from linear operators constructed on basis of its Fourier series in terms of the best approximation of these functions in Orlicz space.
Specifically, we study the problem of the effect of metric of space on order of change of deviations.
AMS Mathematics Subject Classification(2010): 41A10, 41A17, 41A25, 42 A10, 46E30
Key words and phrases:Orlicz space, best approximation, trigonometric polynomials, Fej´er mean, Zygmund mean, Abel-Poisson mean.
1. Introduction and the main results
We suppose that [1] Φ is the class of strictly increasing functions Φ : [0,∞)−→ [0,∞) satisfying ϕ(∞) := lim
x→∞ϕ(x) = ∞. Let Y[p, q], −∞ < p≤ q <∞be the class of even functionsϕ∈Φ satisfying the following conditions
1. ϕ(t)/tp is non- decreasing as|t|increases;
2. ϕ(t)/tq is non- increasing as |t|increases.
Letp < q. The class of functions ϕ belonging toY[p+ϵ, q−δ] for some small numbers ϵ, δ > 0 we will denote by Y⟨p, q⟩. If 1 < p≤q, the class of functions M belonging to the classY⟨p, q⟩will be denoted by Φ∗p.
We use c, c1, c2, ... to denote constants (which may, in general, differ in different relations) depending only on numbers that are not important for the question of our interest.
Let T denote the interval [0,2π].We suppose that M ∈Φ∗p, p >1 and we putϕM(u) =M(u)/u.Note that 1< p < q <∞,thenϕM(u)→ ∞asu→ ∞. Let
ΦM(x) =
∫x 0
ϕM(u)du.
For some positive real constant c let LM(T) denote the set of all Lebesgue measurable functionsf :T→Rfor which
∫
T
ΦM(c|f(x)|)dx <∞.
1Department of Mathematics, Faculty of Art and Science, Pamukkale University, 20017 Denizli, Turkey; Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan , 9, B. Vaxabzade St., Baku, Az-1141, Azerbaijan, e-mail: [email protected]
LM(T) is called anOrlicz spaceand is a Banach function space with the norm
∥f∥LM(T):= inf
λ >0 :
∫
T
ΦM
(|f(x)| λ
) dx≤1
.
Every function inLM(T) is integrable onT [22, p. 50], i.e. LM(T)⊂L1(T).
Detailed information on properties Orlicz spaces can be found in [5, 16, 22].
Generally, the approximation problems in Orlicz spaces have been investigated, when M is a convex and quasiconvex Young function. According to [6] the conditionM ∈Φ∗p, p >1,need not imply M to be convex. Therefore, when M ∈Φ∗p, p >1 it is important to study the approximation of the functions in Orlicz spaces.
Definition 1.1. LetX be a normed space. X is said to beq−concave if for an arbitrary system of functions{ϕi(x)}ni=1,0≤ϕi∈X, the following inequality
holds: { n
∑
i=1
∥ϕi∥qX
}1q
≤c1
( n
∑
i=1
ϕqi )1q
X
,
X is said to be p−convex if for an arbitrary system of functions {ϕi(x)}ni=1, 0≤ϕi∈X, the following inequality holds:
{ n
∑
i=1
∥ϕi∥pX }1p
≥c2
( n
∑
i=1
ϕpi )p1
X
.
Let
(1.1) a0
2 +
∑∞ k=1
Ak(x;f), Ak(x;f) :=ak(f) coskx+bk(f) sinkx
be the Fourier series of the function f ∈ L1(T), where ak(f) and bk(f) are Fourier coefficients of the functionf.Thenthpartial sum of the series (1.1) is defined as:
Sn(x; f) =a0
2 +
∑n k=1
Ak(x;f).
We consider the sequence of the functions {λk(r)} defined in the setE of the number line, satisfying the conditions that
λ0(r) = 1, lim
r−→r0
λν(r) = 1 for an arbitrary fixed ν = 0,1,2, ...
For an arbitrary r∈E and for every functionf ∈LM(T) the series (1.2) U(f; x;λ) = a0
2 +
∑∞ k=1
λk(r)Ak(x; f)
converges in the spaceLM(T).
For each linear operatorUr(f;x;λ) we set
Rr(f; λ)M :=∥f−Ur(f; x;λ)∥LM(T). If we substitute the following
(1.3) λν(r) =
{ 1−r+1ν , 0≤ν≤r, 0, ν > r. ,
(1.4) λν(r) =
{
1−(r+1)νk k, 0≤ν≤r, 0, ν > r. , where k≥1,
(1.5) λν(r) =rν, (ν = 0,1,2, ...) (0≤r≤1)
into (1.2) we obtainFej´er means, Zygmund means of order kand Abel-Poisson means of the series (1.1) respectively.
We denote byEn(f)M the best approximation of f ∈LM(T) by trigono- metric polynomials of degree not exceeding n,i.e.,
En(f)M = inf{∥f −Tn ∥LM(T):Tn ∈Πn}
where Πn denotes the class of trigonometric polynomials of degree at mostn.
The approximation problems by trigonometric polynomials in Orlicz spaces were investigated by several authors (see, for example, [1, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 21, 23, 29]). In the present paper we investigate the problems of estimating the deviation of the functions from the linear operators constructed on the basis of its Fourier series in terms of the best approximation of these functions in Orlicz spaces. Obtained results show that the estimates ofRr(f; λ)
M depends on both the rate of decrease of the sequence {En(f)M} and in some cases the metric of the considered space. This is valid for the upper and lower estimates of the quantity Rr(f; λ)
M. The similar problems of the approximation theory in the different spaces were investigated in [2, 3, 18, 19, 20, 24, 25, 26, 27, 28].
Our main results are the following.
Theorem 1.2. Let{λν(r)} be an arbitrary triangular matrix(r= 0,1,2,3, ...;
λ0(r) = 1; λν(r) = 0, ν > r). Let M ∈Φ∗p, p > 1 and f ∈LM(T), then the following inequality holds:
Rr(f; λ)M ≤ c3{(1 +Kr)Er(f)M+
m∑−1 ν=0
δ(2ν+1;r)E2ν−1(f)M
+δ(r; r)E2m(f)M}, (1.6)
where 2m≤r <2m+1,c3 is a constant not depending on r,
Kr= 2 π
∫π 0
1
2+
∑r ν=1
λν(r) cosνθ dθ, (1.7) δ(µ;r) =
∫π 0
1−λµ(r)
2 +
µ−1
∑
ν=1
{1−λµ−ν(r)}cosνθ
dθ, µ≤r.
Corollary 1.3. Suppose that the conditions of Theorem 1.2 are satisfied.
1. Let λν(r), ν = 0,1,2, ... be a system of numbers defined by relations (1.3). Then the following inequality holds:
(1.8) Rr(f; λ)M ≤ c4
r+ 1
∑r ν=0
Eν(f)M.
2. Let λν(r), ν = 0,1,2, .. be a system of numbers defined by relations (1.4).
Then the following inequality holds:
(1.9) Rr(f; λ)M ≤ c5
(r+ 1)k
∑r ν=0
(ν+ 1)k−1Eν(f)M,
wherec5 is a positive constant depending onk.
Theorem 1.4. Let M ∈Φ∗p,1< p≤q,γ= max{2, q−δ} andf ∈LM(T), then for the system of numbers defined by (1.4)the following inequality holds:
Rr(f; λ)M ≥ c6 (r+ 1)k
{ r
∑
ν=1
νkγ−1Eνγ(f)M }1γ
,
whereδis some small positive number andc6 is a constant depending onpand k.
Theorem 1.5. Let M ∈Φ∗p, 1< p≤q, γ= max{2, q−δ}and f ∈LM(T), then for the system of numbers defined by (1.5)the following inequality holds:
Rr(f; λ)M ≥c7(1−r) {∞
∑
ν=0
rν(ν+ 1)γ−1Eνγ(f)M }γ1
,
whereδ is some small positive number andc7 is a constant depending on p.
2. Proofs of theorems
We need the following [1] theorems:
Theorem 2.1. Let a sequence λk satisfy the conditions
(2.1) |λk| ≤A,
2∑j−1 k=2j−1
|λk−λk+1| ≤A
where A >0 does not depend onkand j.Suppose that satisfied the conditions of Theorem 1.2 For given f ∈LM(T)there exists a functionF ∈LM(T)such that the series
λ0a0
2 +
∑∞ k=0
λk(akcoskx+bksinkx) is Fourier series for F and
(2.2) ∥F ∥LM(T)≤c8A∥f ∥LM(T), where c8>0does not depend on f ∈LM(T).
Theorem 2.2. Under the conditions of Theorem 1.2 there exist constantsc9>
0 andc10>0 such that
(2.3) c10∥f ∥LM(T)≤∥
∑∞ j=0
2∑j−1 k=2j−1
Ak(x, f)
2
1 2
∥LM(T)≤c9∥f ∥LM(T).
for all f ∈LM(T).
Proof of Theorem 1.2. We consider the trigonometric polynomial Tr(x) =
∑r ν=o
(ανcosνx+βνsinνx).
The following inequality holds:
Rr(f; λ)M
=
f(x)−
∑r ν=0
λν(r)Aν(x; f)
LM(T)
≤ ∥f(x)−Tr(x)∥LM(T)+
Tr(x)−
∑r ν=0
λν(r)(ανcosνx+βνsinνx) LM(T)
+
∑r ν=0
λν(r)Aν(x; f)−
∑r ν=0
(ανcosνx+βνsinνx)λν(r)
LM(T)
= ∥f(x)−Tr(x)∥LM(T)+Rr(Tr;λ)M
+ 1
π
∫ 2π 0
{f(x+θ)−Tr(x+θ}
1 2 +
∑r ν=1
λν(r) cosνθ
dθ LM(T)
.
Therefore, we obtain the following inequality
(2.4) Rr(f, λ)M ≤ ∥f(x)−Tr(x)∥LM(T)(1 +Kr) +Rr(Tr;λ)M, where
Kr= 2 π
∫ π 0
1
2+
∑r ν=1
λν(r) cosνθ dθ.
According to [27] the following identity holds:
(2.5)
∑n ν=1
{1−λν(r)}(ανcosνx+βνsinνx) = 2 π
∫
Tn(x+θ) cosnθBn(r, θ)dθ,
whereλ0(r) = 1 and
Tn(x) =
∑n ν=o
(ανcosνx+βνsinνx).
Bn(r, θ) = 1−λn(r)
2 +
n∑−1 ν=0
(1−λn−ν(r)) cosνθ.
Let f ∈ LM(T) and let Tn ∈Π (n= 0,12, ...) be the polynomial of best approximation tof i. e.
En(f)M =∥f(x)−Tn(x)∥LM(T). We set
(2.6) ρk(ν;r;x) = 1 π
∫ 2π 0
Tk(x+θ)
∑ν µ=1
{1−λµ(r)}cosµθ, (0≤k≤ν≤r),
It is clear that
Rr(Tr;λ)M.=∥ρr(r;r;x)∥LM(T),
ρ0(2;r;x) = 0, ρk(ν;r;x) = 0, ρk(k;r;x) = 0, (ν > k).
We suppose that the number m∈ N satisfies condition 2m ≤r < 2m+1. We have
Rr(Tr;λ)
M ≤ ∥ρ2(2;r;x)−ρ0(2;r;x)∥LM(T)
+
m∑−1 µ=1
ρ2µ+1(2µ+1;r;x)−ρ2µ(2µ+1;r;x)
LM(T)
+∥ρr(r;r;x)−ρ2m(r;r;x)∥LM(T). (2.7)
By (2.5) and (2.6) we get
ρ2µ+1(2µ+1;r;x)−ρ2µ(2µ+1;r;x)
LM(T)
= 1
π
∫2π 0
{T2µ+1(x+θ)−T2µ(x+θ)}
2∑m+1
j=1
{1−λj(r)}cosjθdθ LM(T)
=
2 π
∫ 2π 0
{T2µ+1(x+θ)−T2µ(x+θ)}cos 2µ+1θB2µ+1(r;θ) LM(T)
(2.8)
≤ c11δ(2µ+1;r)E2µ(f)M. By (2.7) and (2.8) we find
Rr(Tr;λ)M ≤ c12δ(2;r)E0(f)M +
m∑−1 µ=1
δ(2µ+1;r)E2µ(f)M
+δ(r;r)E2m(f)M. (2.9)
According to [27]Kr≤c12.The inequality (2.4) and (2.9) yield (1.6).
Proof of Corollary 1.3. If we put
λν(r) = 1− νk
(ν+ 1)k, (0≤ν ≤r) andλν(r) = 0, ν > r in the inequality (2.5) we have
∑n ν=1
νk(ανcosνx+βνsinνx)
= 2nk π
∫ 2π 0
Tn(x+θ) cosnθ
1 2 +
n−1
∑
ν=1
(1−ν
n)kcosνθ
dθ.
(2.10)
From (2.10) it is follows that
∑n ν=1
νk(ανcosνx+βνsinνx) LM(T)
≤c13nk∥Tn(x)∥LM(T).
If we put
λ2µ+1(r) = 1− 2(µ+1) (r+ 1)k
in (1.7) we have δ(2µ+1;r)
=
∫ π 0
1−λ2µ+1(r)
2 +
2∑µ+1
ν=1
{1−λ2µ+1−ν(r)}cosνθ
dθ
= 2(µ+1)k (r+ 1)k
∫ π 0
1 2+
2µ+1∑−1 ν=1
(1− ν
2µ+1)kcosνθ
dθ≤c14
2(µ+1)k (r+ 1)k. (2.11)
Then from (2.11) and (1.6) we obtain the inequalities (1.8) and (1.9) of Corol- lary 1.3.
Proof of Theorem 1.4. We suppose that the numberm∈N satisfies condition 2m≤n <2m+1. FromEn(f)M ↓0 we get
σγn,k = C (n+ 1)kγ
∑∞ ν=1
νkγ−1Eνγ(f)M
1 γ
≤ c15
(n+ 1)kγ
m+1∑
ν=0 2ν+1∑−1
µ=2ν
µkγ−1Enγ(f)M
1 γ
≤ c16
(n+ 1)kγ
m+1∑
ν=0
2νγkE2γν(f)M
1 γ
.
Using the estimate [1]
(2.12) ∥f(x)−Sn(x, f)∥LM(T)≤c17En(f)M and (2.3) we have
σn,kγ ≤ c18
(n+ 1)kγ
m+1∑
ν=0
2νγk
∑∞ µ=2ν
Aµ(x;f)
γ
LM(T)
1 γ
≤ c19
(n+ 1)kγ
m+1∑
ν=0
2νγk
(∞
∑
µ=ν
∆2µ+1 )12
γ
LM(T)
1 γ
.
By the Minkowski’s inequality we get
σn,kγ ≤c20
m+1∑
ν=0
( 22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1 )12
γ
LM(T)
1 γ
.
We suppose thatγ= 2.In this case we obtain 2≥(q−δ).Then we get
σ2n,k≤c21
m+1∑
ν=0
(
22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1 )12
2
LM(T)
1 2
.
Its clear that the normlp decreases withp↑. Then
σn,k2 ≤c22
m+1∑
ν=0
(
22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1 )12
q−δ
LM(T)
q−δ1
.
The spaceLM(T ) is of concavity (q−δ). Then we obtain
σ2n,k ≤ c23
m+1∑
ν=0
( 22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1
)(q−δ)/2
1/(q−δ) LM(T)
≤ c24
m+1∑
ν=0
2νk (n+ 1)k
∑∞ µ=ν
∆µ+1
LM(T)
.
Using Abel’s transformation and Minkowski’s inequality, we find that
σ2n,k ≤ c25
∑m
ν=0
2νk
(n+ 1)k∆ν+1+ 2(m+1)k (n+ 1)k
∑∞ µ=m+1
∆µ+1
LM(T)
≤ c26
∑m ν=0
2νk
(n+ 1)k∆ν+1
LM(T)
+c27
2(m+1)k (n+ 1)k
∑∞ µ=m+1
∆µ+1
LM(T)
. (2.13)
Taking the relations (2.3) and (2.12) into account we get
(2.14)
∑∞ µ=m+1
∆µ+1
LM(T)
≤c28
∑∞ µ=2m+1
Aµ(x;f) LM(T)
≤c29En(f)
M
.
Then from (2.13) and (2.14) we conclude that
σ2n,k≤c30
∑m ν=0
2νk
(n+ 1)k∆ν+1 LM(T)
+c31En(f)
M
.
Note that system of multipliers λµ = 2νk
µk(n+ 1)k (2ν ≤µ≤2ν+1−1, ν= 1,2, ..., 2m+1−1), λµ = 0 (µ≥2m+1)
satisfies the conditions (2.1). Therefore, by (2.2) we obtain
σn,k2 ≤c32
∑n µ=0
µk
(n+ 1)kAµ(x;f) LM(T)
+c33En(f)≤c34Rn(f;λ)M.
Letγ=q−δ.Then 2≤(q−δ). Using (q−δ) concavity ofLM(T) we get
σqn,k−δ ≤ c35
m+1∑
ν=0
(
22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1 )12
q−δ
LM(T)
q−δ1
≤ c36
m+1∑
ν=0
( 22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1
)(q−δ)/2
1/q−δ) LM(T)
≤ c37
(m+1
∑
ν=0
22νk (n+ 1)2k
∑∞ µ=ν
∆2µ+1 )12
LM(T)
.
Further, using the same Abel’s transformation and reasoning as in the case 2≥(q−δ) we have
σn,kq−δ ≤c38Rn(f;λ)M. Proof of Theorem 1.4 is completed.
Proof of Theorem 1.5 is similar to proof of Theorem 1.4.
Acknowledgement. The author thanks the referee for careful reading this article and useful comments.
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Received by the editors March 28, 2014
