On trigonometric approximation in weighted lorentz spaces using norlund and riesz submethods

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Journal of Mathematical Analysis

ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 9 Issue 6 (2018), Pages 17-27.

ON TRIGONOMETRIC APPROXIMATION IN WEIGHTED

LORENTZ SPACES USING N ¨ORLUND AND RIESZ

SUBMETHODS

AHMET HAMDI AVSAR, YUNUS EMRE YILDIRIR

Abstract. In this study, we obtain the degree of approximation by the N¨orlund and Riesz submethods of the partial sums of the Fourier series of derivatives of functions in the weighted Lorentz spaces with Muckenhoupt weights. There-fore we generalize the theorems given in [22] to their sharper approximation versions with weaker conditions.

1. Introduction

Let T := [−π, π] . When ω : T → [0, ∞] is a non-negative measurable 2π-periodic function on (0, ∞) that is not identically zero, we say that ω is a weight.

Given a weight function ω and a measurable set e we put ω(e) =

Z

e

ω(x)dx. (1.1)

We define the decreasing rearrangement f_ω∗_{(t) of f : T → R with respect to the} Borel measure (1.1) by

f_ω∗(t) = inf {τ ≥ 0 : ω (x ∈ T : |f (x)| > τ ) ≤ t} and the maximal function of f is defined by

f∗∗(t) = 1 t t Z 0 f_ω∗(u)du.

The weighted Lorentz space Lps

ω(T) is defined [8, p.20], [3, p.219] as Lps_ω(T) =      f ∈ M(T) : kf k_ps,ω=   Z T (f∗∗(t))stspdt t   1/s < ∞      ,

where M(T) is the set of 2π periodic measurable functions on T, 1 < p, s < ∞. 2010 Mathematics Subject Classification. 41A10, 42A10.

Key words and phrases. Weighted Lorentz space, N¨orlund submethod, Riesz submethod, Fourier series, Muckenhoupt weight.

c

2018 Ilirias Research Institute, Prishtin¨e, Kosov¨e. Submitted August 11, 2018. Published October 10, 2018. Communicated by M.T. Garayev.

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Another thing to notice is that if p = s, then Lpsω(T) turns into the weighted

Lebesgue space Lpω(T) as [8, p. 20].

By En(f )Lpsω we denote the best approximation of f ∈ L

ps

ω(T) by trigonometric

polynomials of degree ≤ n, i.e.,

En(f )Lpsω = inf kf − Tkkps,ω,

where the infimum is taken with respect to all trigonometric polynomials of degree k ≤ n.

The weight functions ω used in the paper belong to the Muckenhoupt class Ap(T)

[18] which is defined by sup 1 |I| Z I ω(x)dx   1 |I| Z I ω1−p0(x)dx   p−1 < ∞, p0= p p − 1, 1 < p < ∞, where the supremum is taken with respect to all the intervals I with length ≤ 2π and |I| denotes the length of I.

The modulus of continuity of the function f ∈ Lps

ω(T) is defined [14] as Ω (f, δ)_Lps ω = sup |h|≤δ kAhf kps,ω, δ > 0, where (Ahf ) (x) := 1 h h Z 0 |f (x + t) − f (x)| dt.

In Lebesgue spaces Lp _{(1 < p < ∞), the traditional modulus of continuity is}

defined as

wp(f, δ) = sup 0<h≤δ

kf (· + h) − f (·)k_p, δ > 0, It is known that the modulus of continuity Ω(f, δ)_Lps

ω and traditional modulus of continuity wp(f, δ) are equivalent in [14].

Whenever ω ∈ Ap(T), 1 < p, s < ∞, the Hardy-Littlewood maximal function

of every f ∈ Lps

ω(T). The existence of the modulus Ω(f, δ)Lpsω follows from the boundedness of the Hardy-Littlewood maximal function in the space Lps

ω(T) [4,

Theorem 3]. That is the modulus of continuity Ω (f, δ)_Lps

ω is well defined for every ω ∈ Ap(T).

The modulus of continuity Ω (f, δ)_Lps

ω is non-decreasing, non-negative, continu-ous function such that

lim

δ→0Ω (f, δ)L

ps

ω = 0, Ω (f1+ f2, δ)Lpsω ≤ Ω (f1, δ)Lpsω + Ω (f2, δ)Lpsω . The modulus of continuity Ω (f, δ)_Lps

ω is defined in this way, since the space Lps

ω(T) is non-invariant, in general, under the usual shift f (x) → f (x + h) , (h > 0) .

For 0 < α ≤ 1, it is defined the Lipschitz class Lip (α, Lps ω) [9] as

Lip (α, Lps_ω) =f ∈ Lps

ω(T) : Ω (f, δ)Lpsω = O (δ

α_{) , δ > 0} and for r = 1, 2, ... the classes Wr

ps,ω , Wps,ωr,α as

W_ps,ωr = nf ∈ Lps_ω(T) : f(r−1) is absolutely continuous and f(r)∈ Lps ω o W_ps,ωr,α = nf ∈ W_ps,ωr : f(r)∈ Lip (α, Lps ω) o .

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Since Lpsω (T) ⊂ L1(T) when ω ∈ Ap(T), 1 < p, s < ∞ (see [8, the proof of

Prop. 3.3]), we can define the Fourier series of f ∈ Lpsω(T)

f (x) v a0₂(f )+

∞

X

k=1

(ak(f ) cos kx + bk(f ) sin kx) (1.2)

and the conjugate Fourier series ˜ f (x) v ∞ X k=1 (ak(f ) sin kx − bk(f ) cos kx) .

Here a0(f ) , ak(f ) , bk(f ) , k = 1, ..., are Fourier coefficients of f . Let Sn(x, f ),

(n = 0, 1, 2, ...) be the nth partial sums of the series (1.2) at the point x, that is, Sn(x, f ) := n X k=0 Ak(f )(x), where A0(f )(x) = a0 2 , Ak(f )(x) = ak(f ) cos kx + bk(f ) sin kx, k = 1, 2, .... Let (λn)∞_n=1be a strictly increasing sequence of positive integers. For a sequence

(xk) of the real or complex numbers, the Ces`aro submethod Cλ is defined by

(Cλx)n := 1 λn λn X k=1 xk, (n = 1, 2, ...).

Particularly, when λn = n we note that (Cλx)n is the classical Ces`aro method

(C, 1) of (xk) . Thus, the Ces`aro submethod Cλyields a subsequence of the Ces`aro

method (C, 1) . The detailed information about Ces`aro submethod Cλcan be found

in the papers [2, 19].

Let (pn) be a positive sequence of real numbers. We define N¨orlund and Riesz

submethods means as N_nλ(f ; x) := 1 Pλn λn X m=0 p_λn−mSm(f ; x) and Rλ_n(f ; x) := 1 Pλn λn X m=0 pmSm(f ; x) where Pλn= p0+ p1+ p2+ ... + pλn6= 0 (n > 0) and by convention, p−1= P−1= 0.

In the case pn= 1, n ≥ 0, λn = n, both of Nnλ(f )(x) and Rnλ(f )(x) are equal to

the Ces`aro mean

σn(f )(x) = 1 n + 1 n X m=0 Sm(f ; x).

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2. Historical Background

The basic properties of the Ces`aro submethod Cλ were investigated firstly by

Armitage and Maddox in [2] and Osikiewicz [19]. In these works, the relations between the classical Ces`aro method σn and Ces`aro submethod Cλ were obtained.

Deger et al. [6] obtained theorems of trigonometric approximation using trigono-metric polynomials obtained by Ces`aro submethod Cλ in Lebesgue spaces. Deger

and Kaya [5] investigated the degree of approximation of functions in Lebesgue spaces by trigonometric polynomials Nλ

n(f ; x) and Rλn(f ; x). In [17], Mittal and

Singh improved the results given by Deger et al. [6] by dropped monotonicity con-ditions on the elements of matrix rows. In [16], Mittal and Singh examined the approximation rate of functions using matrix submethods obtained by means of Cesàro Submethod in Lebesgue spaces. In [20], the results given by Deger and Kaya [5] were improved using more general summability methods. In [13], the some results obtained in variable exponent Lebesgue spaces were extended using a wider class of numerical sequences, a sharper degrees of approximation and Nörlund submethod Nnλ(f ; x) instead of Nörlund method Nn(f ; x). In [7], in the variable

ex-ponent Lebesgue spaces the results on the degree of approximation by the N¨orlund and the Riesz submethods of the partial sums Fourier series of functions were given. In [10], the approximation properties of the matrix method τn of partial sums of

Fourier series of functions in the weighted variable exponent Lebesgue spaces were investigated.

Lebesgue space may be generalized in different ways. One of the important generalizations of this space is Lorentz space. Lorentz space was firstly introduced by G. G. Lorentz in [15]. By means of the weight functions satisfying Muckenhoupt condition, the weighted Lorentz spaces were defined in [3, 8].

In weighted Lorentz spaces, some researchers obtained results about approxi-mation theory using different methods [1, 12, 21, 22]. But in these papers degree of approximation using Ces`aro submethod were not examined for derivatives of functions in the weighted Lorentz spaces.

In this paper, we generalize the results obtained by Mittal and Singh [16] to the weighted Lorentz spaces for derivatives of functions in these spaces. Also, we obtain the similars of the results in [22] using Ces`aro submethod under weaker conditions.

3. Auxiliary Results We need some helpful lemmas.

Lemma 3.1. Let 1 < p, s < ∞, ω ∈ Ap(T), r ∈ N. If f ∈ Lip(1, Lpsω), then f(r) is

absolutely continuous and f(r+1)∈ Lps ω(T).

Proof. We follow the method in [14, Th. 3]. If f(r) _{∈ L}ps

ω(T), then there exists

p0> 1 such that f(r)∈ Lp0 and

f (r) _L_p0 f (r) _Lps ω (3.1) [12, Prop. 3.3]. Using (3.1) and the equivalence of traditional modulus of continuity w(f, δ)Lp and Ω(f, δ)_Lps

ω , we get

w(f(r), δ)Lp0 Ω(f(r), δ)_Lps ω.

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Since Ω(f(r), δ)Lpsω = O(δ), the same estimate holds for w(f

(r)_{, δ)}

Lp0, too. From here, we obtain that f(r)_{is absolutely continuous in [−π, π] and for almost every x}

f(r)(x + t) − f(r)(x)

t → f

(r+1)_{(x) ,} _{(t → 0) .} _(3.2)

From (3.2) we obtain for almost every x 2 δ δ Z δ 2 f(r)_{(x + t) − f}(r)_(x) t dt → f (r+1)_(x) , (δ → 0+) .

Using Fatou Lemma, we get f (r+1) _Lps ω ≤ lim inf δ→0+ 2 δ δ Z δ 2 f(r)_{(x + t) − f}(r)_(x) t dt Lpsω ≤ lim sup4 δ δ→0+ 1 δ δ Z 0 f (r)_{(x + t) − f}(r)_(x) dt Lpsω ≤ lim sup 4 δ→0+ Ω(f(r)_{, δ)} δ < ∞.

This proves the lemma.

Lemma 3.2. [22]Let 1 < p, s < ∞, ω ∈ Ap(T), 0 < α ≤ 1, r ∈ N. Then, the

estimate f (r)_{− S} n f(r) _ps,ω= O n −α (3.3) holds for every f ∈ Wr,α

ps,ω and n = 1, 2, ....

Lemma 3.3. Let 1 < p, s < ∞, ω ∈ Ap(T), r ∈ N. Then, the estimate

Sn f(r)− σn(f(r)) _ps,ω= O n −1 (3.4) holds for every f ∈ Wr,1

ps,ω and n = 1, 2, ....

Proof. If f(r) _{has the Fourier series}

f(r)(x) ∼

∞

X

k=0

Ak(f(r))(x),

then the Fourier series of the conjugate function ˜f(r+1)_{(x) is}

˜ f(r+1)(x) ∼ ∞ X k=0 kAk(f(r))(x).

On the other hand, Sn f(r)(x) − σn(f(r)) (x) = n X k=1 k n + 1Ak(f (r)_)(x) = 1 n + 1Sn ˜_f(r+1)_{(x) .}

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Since the partial sums and the conjugate operator is uniform bounded in the space Lpsω(T) (see [12], [11, Th. 6.6.2], [23, Chap. VI]), we get from Lemma 3.1

Sn f(r)− σn(f(r)) _ps,ω = 1 n + 1 Sn ˜_f(r+1) _ps,ω≤ C 1 n + 1 ˜ f(r+1) _ps,ω ≤ C 1 n + 1 f (r+1) _ps,ω= O n −1 for n = 1, 2, ....

Lemma 3.4. Let (pn) be a non-increasing sequence of positive numbers. Then, λn X m=1 m−αpλn−m= O λ −α n Pλn for 0 < α < 1.

Proof. Due to (pn) is non-increasing sequence, we have λn X m=1 m−αpλn−m = r X m=1 m−αpλn−m+ λn X m=r+1 m−αpλn−m ≤ pλn−r r X m=1 m−α+ (r + 1)−α λn X m=0 pλn−m = O(λ1−α_n )pλn−r+ O(λ −α n )Pλn = O(λ−αn )Pλn

where r integer part of λn/2. The proof is completed.

4. Main Results

Theorem 4.1. Let 1 < p, s < ∞, ω ∈ Ap(T), 0 < α ≤ 1, r ∈ N and let (pn) ∞ 0 be

a monotonic sequence of positive numbers such that

(λn+ 1)pλn= O(Pλn). (4.1) If f ∈ Wr,α ps,ω then we have f (r) − Nnλ f (r) _ps,ω= O λ −α n .

Theorem 4.2. Let 1 < p, s < ∞, ω ∈ Ap(T), 0 < α ≤ 1, r ∈ N and let (pn)∞₀ be

a sequence of positive real numbers satisfying the relation

λn−1 X λm=0 Pλm λm+ 1 − Pλm+1 λm+ 2 = O _P λn λn+ 1 . (4.2)

If f ∈ W_ps,ωr,α then the estimate f (r)_{− R}λ n f (r) _ps,ω= O λ −α n holds.

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If we take pλn= A β−1 λn (β > 0) , where Aβ₀ = 1, Aβ_k =β (β + 1) ... (β + k) k! , k ≥ 1, we have Nnλ f (r)(x) = σ_λβ n(f (r) ) (x) = 1 Aβ_λ n λn X m=0 Aβ−1_λ n−mSm(x, f (r) ).

We can estimate the deviation of f ∈ W_ps,ωr,α using the Ces`aro submethods σβ_λ

n(f

(r)_{). We formulate this estimate in the following corollary.}

Corollary 4.3. Let 1 < p, s < ∞, ω ∈ Ap(T), 0 < α ≤ 1, r ∈ N. If f ∈ Wps,ωr,α, we have f (r) − σ_λβ n(f (r)₎ _ps,ω = O λ −α n .

Note that the our main submethod results are sharper than the results obtained using Ces`aro method, because λ−αn ≤ n−αfor 0 < α ≤ 1.

5. Proof of Main Results Proof of Theorem 4.1. Let 0 < α < 1. We can write

f(r)(x) = 1 Pλn λn X m=0 pλn−mf (r)_(x), then we get f(r)(x) − N_nλf (r)(x) = 1 Pλn λn X m=0 pλn−m h f(r)(x) − Sm f(r)(x)i. By considering Lemma 3.2, Lemma 3.4 and condition (4.1) we get

f (r)_{− N}λ n f (r) _ps,ω ≤ 1 Pλn λn X m=0 pλn−m f (r)_{− S} m f(r) _ps,ω = 1 Pλn λn X m=1 pλn−mO m −α₊ pλn Pλn f (r) − S0 f(r) _ps,ω = 1 Pλn O λ−α_n Pλn + O 1 λn = O λ−α_n . Let α = 1. It is easily seen that

N_nλf (r)(x) = 1 Pλn λn X m=0 pλn−mAm(f (r)_)(x).

Using Abel transform, Sn f(r)(x) − N_nλf (r)(x) = 1 Pλn λn X m=1 (Pλn− Pλn−m) Am(f (r)_)(x)

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= 1 Pλn λn X m=1 Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 m X k=1 kAk(f(r))(x) ! + 1 λn+ 1 λn X k=1 kAk(f(r))(x), and so Sn f(r)− Nnλ f (r) _ps,ω ≤ 1 Pλn λn X m=1 Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 × m X k=1 kAk(f(r)) ps,ω + 1 λn+ 1 λn X k=1 kAk(f(r)) ps,ω . Since Sn f(r)(x) − σn f(r)(x) = 1 λn+ 1 λn X k=1 kAk(f(r)) (x) ,

using Lemma 3.3 we obtain λn X k=1 kAk(f(r)) ps,ω = (λn+ 1) Sn f(r)− σn f(r) _ps,ω= O (1) . Therefore we have Sn f(r)− Nnλ f (r) _ps,ω ≤ 1 Pλn λn X m=1 Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 O (1) + O λ−1_n = O ₁ Pλn λn X m=1 Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 + O λ−1_n . (5.1) then it can be easily seen that

Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 = 1 m (m + 1) λn X k=λn−m+1 pk− mpλn−m ! . This equality implies that

Pλn− Pλn−m m

λn+1

m=1

is non-increasing whenever (pn) is non-decreasing and non-decreasing whenever (pn)

is non-increasing. This shows the following equality

λn X m=1 Pλn− Pλn−m m − Pλn− Pλn−(m+1) m + 1 = pλn− Pλn λn+ 1 = 1 λn+ 1 O (Pλn) . From this and the inequality (5.1), we have

Sn f(r)− Nλ n f (r) _ps,ω = O λ −1 n .

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Combining the last estimate with (3.3) we get f (r) − Nnλ f (r) _ps,ω= O λ −1 n . Proof of Theorem 4.2. Let 0 < α < 1. Using definition of Rλ

n f (r) (x), we can write f(r)(x) − R_nλf (r)(x) = 1 Pλn λn X m=0 pm h f(r)(x) − Sm f(r)(x)i, then using Lemma 3.2, we get

f (r)_{− R}λ n f (r) _ps,ω≤ 1 Pλn λn X m=0 pm f (r)_{− S} m f(r) _ps,ω = O ₁ Pλn λn X m=1 pmm−α+ p0 Pλn f (r)_{− S} 0 f(r) _ps,ω = O ₁ Pλn λn X m=1 pmm−α. (5.2)

Using Abel transform, we get

λn X m=1 pmm−α = λn−1 X m=1 Pm h m−α− (m + 1)−αi+ λ−αn Pλn ≤ λn−1 X m=1 m−α Pm m + 1+ λ −α n Pλn, and using condition (4.2) we can write

λn−1 X m=1 m−α Pm m + 1 = λn−1 X m=1 _P m m + 1− Pm+1 m + 2 m X k=1 k−α ! + Pλn λn+ 1 λn−1 X m=1 m−α = O λ−αn Pλn This implies λn X m=1 pmm−α= O λ−αn Pλn .

Combining last inequality and and the condition (5.2), we obtain that f (r)_{− R}λ n f(r) _ps,ω= O λ −α n .

Let α = 1. From Abel transform, Rλ_nf (r)(x) = 1 Pλn λn−1 X m=0 h Pm Sm f(r)(x) − Sm+1 f(r)(x)+ PλnSn f(r)(x)i = 1 Pλn λn−1 X m=0 Pm −Am+1 f(r)(x)+ Sn f(r)(x) ,

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and so Rλ_nf(r)(x) − Snf(r)(x) = − 1 Pλn λn−1 X m=0 PmAm+1 f(r)(x) . Using Abel transform again, we get

λn−1 X m=0 PmAm+1 f(r)(x) = λn−1 X m=0 Pm m + 1(m + 1) Am+1 f(r)(x) = λn−1 X m=0 _P m m + 1− Pm+1 m + 2 m X k=0 (k + 1) Ak+1 f(r)(x) ! + Pλn λn+ 1 λn−1 X k=0 (k + 1) Ak+1 f(r)(x) . By considering (3.3) and (4.2) we get

λn−1 X m=0 PmAm+1 f(r) _ps,ω ≤ λn−1 X m=0 Pm m + 1− Pm+1 m + 2 m X k=0 (k + 1) Ak+1 f(r) ps,ω + Pλn λn+ 1 λn−1 X k=0 (k + 1) Ak+1 f(r) ps,ω = λn−1 X m=0 Pm m + 1− Pm+1 m + 2 (m + 2) Sm+1 f(r)− σm+1 f(r) _ps,ω +Pλn Sn f(r)− σn f(r) _ps,ω = O(1) λn−1 X m=0 Pm m + 1− Pm+1 m + 2 + O Pλn λn . This yields R λ n f(r)− Sn f(r) _Lps w = 1 Pλn λn−1 X m=0 PmAm+1 f(r) _ps,ω = 1 Pλn O Pλn λn = O 1 λn . Combining this estimate with (3.3) , we get

f (r)_{− R}λ n f (r) _ps,ω= O λ −1 n . References

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Balikesir University, Department of Mathematics, Balikesir, Turkey E-mail address: ahmet.avsar@balikesir.edu.tr

Yunus Emre Yildirir

Balikesir University, Department of Mathematics, Balikesir, Turkey E-mail address: yildirir@balikesir.edu.tr