Jackson-stechkin type inequality in weighted lorentz spaces

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Jackson-Stechkin type inequality in weighted Lorentz spaces

Article · October 2015 DOI: 10.7153/mia-18-100 CITATIONS 3 READS 78 2 authors: Ramazan Akgün Balikesir University 41PUBLICATIONS 303CITATIONS SEE PROFILE

Yunus Emre Yıldırır Balikesir University

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Inequalities & Applications

Volume 18, Number 4 (2015), 1283–1293 doi:10.7153/mia-18-100

JACKSON–STECHKIN TYPE INEQUALITY IN WEIGHTED LORENTZ SPACES

RAMAZANAKGUN AND¨ YUNUSEMREYILDIRIR

(Communicated by J. Peˇcari´c)

Abstract. In the present work we consider the modulus of smoothness, defined by means of the Steklov operator in weighted Lorentz spaces and prove the Jackson-Stechkin type direct theo-rem of trigonometric approximation. In the particular case we obtain a result on the constructive characterization of the generalized Lipschitz classes defined in these spaces. Simultaneous ap-proximation of functions is also considered.

1. Introduction and the main results

Jackson–Stechkin type inequalities in the normed space under consideration esti-mate the order of decrease of the best approximation of a function by a finite dimen-sional subspace in terms of some characteristic of its smoothness.

The starting point here is the classical theorem of Jackson ([8]) on the best uniform

approximation of a periodic function f by trigonometric polynomials of degree6 n :

For any 2π-periodic continuous function f, the following inequality holds

En( f ) 6 Cω f, 1 n+ 1 .

In this inequality, En( f ) denotes the best approximation of the function f by trigonometric polynomials of degree6 n , i.e.,

En( f ) := inf

Tn∈Tnx∈[0,2π]max | f (x) − Tn(x)| ,

where Tn is the class of trigonometric polynomials of degree6 n, and

ω( f ,δ) := sup

|h|6δ

max

x∈[0,2π]| f (x + h) − f (x)|

denotes the modulus of continuity of f.

In [15], Stechkin proved an analog of Jackson’s inequality for the Lebesgue spaces

Lp_{, 1 6 p 6 ∞. The elegant representation of the corresponding results in the Lebesgue} Mathematics subject classification(2010): 41A25, 41A27, 42A10.

Keywords and phrases: Modulus of smoothness, trigonometric polynomials, weighted Lorentz spaces, Muckenhoupt weight, direct and inverse theorem.

c

D l , Zagreb

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spaces Lp, 1 6 p 6 ∞, can be found in [3,16,17]. In weighted Lebesgue spaces with weights satisfying the Muckenhoupt’s condition Ap, 1 < p < ∞, the direct theorem of trigonometric approximation in the following form was proved in [7].

En( f )_Lp w.ωr f,1 n Lwp := sup 06hi61/n r

∏

i=1 I−σ_h i f Lpw , r∈ N,

where I is the identity operator on T := [−π,π) and σ_hf(x) := 1

2h x_Z+h

x−h

f(u)du, x ∈ T.

The shift operator σ_h and the modulus of smoothness ω_r( f , ·)_Lp

w are defined in

this way, because the weighted Lebesgue space Lpw is not, in general, invariant under the usual shift f(·) → f (· + h).

Some interesting results concerning to the best polynomial approximation in weigh-ted Lebesgue spaces were also proved in [4,5,11,19]. In weighted Lorentz spaces

some converse theorems were obtained in [10,18]. The detailed information on the

weighted polynomial approximation can be found in the books [6,12].

A measurable function w : T→ [0, ∞] is called a weight function if the preimage

w−1({0, ∞}) has Lebesgue measure zero. Let w be a weight function and fw∗(t) be a

decreasing rearrangement of f : T→ R with respect to the Borel measure

w(e) = Z e w(x)dx, i.e., f_w∗(t) = inf {τ_{> 0 : w(x ∈ T : | f (x)| >}τ_{) 6 t} .}

Let 1< p, q < ∞ and let Lwpq(T) be a weighted Lorentz space, i.e., the set of all measurable functions for which

k f k_Lpq w =  Z T ( f∗∗(t))qt q pdt t   1/q < ∞, where f∗∗(t) = 1 t t Z 0 f_w∗(u)du.

If p= q, Lwpq(T) is turn into the weighted Lebesgue space Lw(T).p

The weights w used in the paper are those which belong to the Muckenhoupt’s ([13]) class Ap(T), i.e., they satisfy the condition

sup 1 |I| Z I w(x)dx   1 |I| Z I w1−p′(x)dx   p−1 = CAp< ∞, p ′_:= p p− 1

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where the supremum is taken with respect to all the intervals I with length6 2π and |I| denotes the length of I. The constant CAp is called the Muckenhoupt constant of w .

The modulus of smoothness of a function f ∈ Lwpq(T) is given by Ω_r( f ,δ)_Lpq w := sup 06hi6δ,i=1,...,r r

∏

i=1 I−σ_h i f Lpqw , r∈ N.

Whenever w∈ Ap(T), 1 < p, q < ∞, the Hardy-Littlewood maximal function of

f ∈ Lwpq(T) belongs to Lwpq(T) ([2, Theorem 3]). Therefore the averageσhf belongs to Lpqw(T). Thus Ωr( f ,δ)_Lpq

w makes sense for every w∈ Ap(T).

By En( f )_Lpq

w we denote the best approximation of f ∈ L

pq

w (T) by trigonometric

polynomials of degree6 n, i.e.,

En( f )_Lpq

w = inf_T_n∈Tnk f − TnkLpqw .

Since Lwpq(T) ⊂ L1(T) when w ∈ Ap(T), 1 < p, q < ∞ (see [10, the proof of Prop. 3.3]), we can define the Fourier series of f∈ Lwpq(T)

f(x) ∽ a0( f )

2 +

∞

∑

k=1

(ak( f ) cos kx + bk( f ) sin kx) (1.1)

and the conjugate Fourier series ˜

f(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 .

The relation . is defined as “ A . B⇔ there exists a positive constant C, inde-pendent of essential parameters, such that A6 CB.”

In this work we prove the direct and simultaneous theorems of approximation theory in the weighted Lorentz spaces using the modulus of smoothness Ωr( f , ·)_Lpq

w .

Our new results are the following.

THEOREM1. Let w∈ Ap(T), 1 < p, q < ∞, r ∈ N. Then for every f ∈ Wr

pq,w, the inequality k f − Sn( f )k_Lpq w . 1 (n + 1)r f(r)− Sn( f(r)) Lpqw , n∈ N

holds with a positive constant depending only on r, p, q and the Muckenhoupt constant

CAp of w , where Sn( f ) denotes the n.th partial sum of the Fourier series (1.1) of f .

We define Wpqr,w:=

n

g∈ Lwpq: g(r)∈ Lwpq o

. Theorem1gives the following corol-lary.

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COROLLARY1. Let w∈ Ap(T), 1 < p, q < ∞, r, n ∈ N. Then for every f ∈ Wr pq,w, the inequalities En( f )_Lpq w . 1 (n + 1)rEn( f (r)₎ Lpqw, and En( f )_Lpq w . 1 (n + 1)r f(r) Lpqw ,

hold with some positive constants depending only on r, p, q and the Muckenhoupt

con-stant CAp of w.

THEOREM2. Let f ∈ Lwpq(T), w ∈ Ap(T), 1 < p, q < ∞. Then we have the

fol-lowing estimate En( f )_Lpq w . Ωr f,1 n Lpqw , r∈ N, (1.2) and k f − Sn( f )k_Lpq w . Ωr f,1 n Lwpq (1.3)

for n∈ N, with some constants depending only on r, p, q and the Muckenhoupt constant

CAp of w .

We note that the sharp inverse inequality to the Jackson-Stechkin type inequality was proved in [10]. In the sequel we use a weak version of inverse estimate:

Let f ∈ Lwpq(T), w ∈ Ap(T), 1 < p, q < ∞. Then Ω_r f,1 n Lwpq . 1 n2r n

∑

k=0 (k + 1)2r−1Ek( f )_Lpq w , r∈ N (1.4)

holds for n∈ N, with some constant depending only on r, p, q and the Muckenhoupt

constant CAp of w.

From Theorem2and (1.4), we obtain the following Marchaud type inequality.

COROLLARY2. Let f∈ Lpqw(T), w ∈ Ap(T), 1 < p, q < ∞. Then we have

Ω_r( f ,δ)_Lpq w .δ 2rZ 1 δ Ω_r₊₁( f , u)_Lpq w u2r du u , 0<δ < 1, for r∈ N.

From Theorem2and (1.4), we also obtain the following estimate.

THEOREM3. Let f∈ Lwpq(T), w ∈ Ap(T), 1 < p, q < ∞. If

En( f )_Lpq

w . n

−α_, _n_{∈ N}

for someα> 0, then, for a given r ∈ N, we have the estimations

Ω_r( f ,δ)_Lpq w =    δα , r>α/2; δ2r_log1 δ , r=α/2; δ2r _{, r}_<_α_/2.

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If we define the generalized Lipschitz class Lip α, Lwpq for α > 0 and k := [α/2] + 1, [x] := max {n ∈ Z : n 6 x} as Lip(α, Lwpq) := n f ∈ Lwpq: Ωk( f ,δ)Lwpq.δ α , δ> 0o,

then by virtue of Theorems3and2we obtain the following result which gives a

con-structive characterization of the Lipschitz classes Lip α, Lwpq

.

COROLLARY3. Let f∈ Lwpq(T), w ∈ Ap(T), 1 < p, q < ∞ andα> 0. The

fol-lowing assertions are equivalent.

(i) f ∈ Lip (α, Lwpq) (ii) En( f )Lwpq. n

−α_, _n_{∈ N.}

Jackson’s second type inequality is given in the following theorem.

THEOREM4. Let w∈ Ap(T), 1 < p, q < ∞ and r, k ∈ N. Then for every f ∈

Wr pq,w, the inequality En( f )_Lpq w . 1 (n + 1)rΩk f(r),1 n Lpqw , n∈ N

CAp of w.

Simultaneous approximation estimates are given in the next two theorems.

THEOREM5. Let w∈ Ap(T), 1 < p, q < ∞ and r, k, l ∈ N. Then for every f ∈

W_pqr _,w and06 k 6 r the inequality f(k)− S(k)n ( f ) Lwpq . 1 nr−kΩl f(r),1 n Lwpq , n∈ N

CAp of w.

THEOREM6. Let w∈ Ap(T), 1 < p, q < ∞ and r, k, n ∈ N. Then for every f ∈

Wr

pq,w and06 k 6 r the inequality

f(k)− Sn(k)( f ) Lwpq . 1 nr−kEn f(r) Lwpq

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2. Proofs of the main results

To prove Theorem2we need the following lemma. If A. B and B . A,

simulta-neously, we will write A≈ B.

For an f ∈ Lwpq(T) and r ∈ N the Peetre’s K -functional is defined as

K f,t; Lwpq,Wpqr ,w := inf g∈Wr pq,w k f − gk_Lpq w + t r g(r)(x) Lpqw for t> 0.

LEMMA1. Let f ∈ Lwpq(T ), r ∈ N, w ∈ Ap(T), 1 < p, q < ∞ and t, k > 0. Then

we have Ω_r( f ,t)_Lpq w ≈ K f ,t; L pq w ,Wpq2r,w (2.1) and Ω_r( f , kt)_Lpq w .(1 + [k]) 2r_Ω r( f ,t)Lpqw ,

with some constant depending only on p, q and the Muckenhoupt constant CAp of w.

Proof. If h∈ W2r

pq,w(T) , then from subadditivity of Ωr( f , ·)Lwpq and Ωr(h,t)Lwpq.

t2r h(2r) Lwpq (see Lemma 4.1 of [10]) Ω_r( f ,t)_Lpq w .k f − hkLwpq+ t 2r h(2r) Lpqw . Taking infimum on h we get Ωr( f ,t)_Lpq

w . K f,t; L pq w ,Wpq2r,w . We define (Lδf) (x) := 3δ−3 δ Z 0 u Z 0 t Z −t f(x + s) dsdtdu, x ∈ T. From [1, p.15] d2r dx2rL r δf= c δ2r(I −σδ) r , r∈ N. Because of estimates kLδfkLwpq. δ−3 3 δ Z 0 u Z 0 2tkσ_tfk_Lpq w dtdu.k f kLwpq

the operator Lδ is bounded in Lpqw. Defining Ar_δ := I − I − Lr δ r we obtain d2r dx2rA r δf Lwpq . d2r dx2rL r δf Lpqw = 1 δ2rk(I −σδ) r_k Lwpq . 1 δ2rΩr( f ,δ)Lwpq

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and hence Ar_δf∈ W2r

pq,w(T) . Since Lδ is bounded in Lwpqand I− Lrδ = (I − Lδ) ∑rj−1=0L

j δ we have I − Lr δ g Lwpq.k(I − Lδ) gkL pq w .δ −3 δ Z 0 u Z 0 2tk(I −σt) gk_Lpq w dtdu . sup 0<t6δ k(I −σt) gk_Lpq w for any g∈ Lwpq.

Applying this inequality r times in f − Ar_δf

Lpqw = I − Lr δ r f Lwpq we obtain f − Ar δf Lwpq . sup 0<t16δ (I −σ_t₁) I − Lrδ r−1 f Lwpq . sup 0<t1,t26δ (I −σ_t 1) (I −σt2) I − L r δ r−2 f Lwpq 6. . . 6 sup 0<ti6δ i=1,2,...,r r

∏

i=1 (I −σ_t i) f (x) Lwpq = Ωr( f ,δ)_Lpq w .

Using the equivalence (2.1) we have Ω_r( f , kt)_Lpq w . inf g∈W2r pq,w k f − gk_Lpq w + (kt) 2r g(2r)(x) Lwpq .(1 + [k])2r inf g∈W2r pq,w k f − gk_Lpq w + t 2r g(2r)(x) Lwpq .(1 + [k])2rΩ_r( f ,t)_Lpq w

and the lemma is proved.

Proof of Theorem1. We know that (see [9, Theorem 6.6.2], [10]) kSn( f )k_Lpq w .k f kLwpq, ˜f Lwpq.k f kL pq w , k f − Sn( f )k_Lpq w . En( f )Lwpq and Sn t, f(r)= S(r)n (t, f ) . Then (see inequality 6.15 of [14])

f(x) − Sn(x, f ) = ∞

∑

k=n+1 1 kr_π Z T f(r)(t) − Snt, f(r)cos k(x − t) −rπ 2 dt.

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When r= 2l f(x) − Sn(x, f ) = (−1)l ∞

∑

k=n+1 1 kr h ak f(r)− Snf(r) cos(kx) + + bkf(r)− Snf(r) sin(kx)i = (−1)l ∞

∑

k=n+1 Ak f(r)− Snf(r) kr = (−1)l ∞

∑

k=n+1 k−r− (k + 1)−rSk x, f(r)− Snf(r). Hence k f − Sn( f )k_Lpq w 6 ∞

∑

k=n+1 k−r− (k + 1)−r _Sk f(r)− Snf(r) Lwpq . ∞

∑

k=n+1 k−r− (k + 1)−r _f(r)− Snf(r) Lpqw . 1 (n + 1)r f(r)− Snf(r) Lwpq .

When r= 2l + 1 we have cos k (x − t) −π_r= sink(x − t)(−1)land

f(x) − Sn(x, f ) = (−1)l ∞

∑

k=n+1 1 kr h ak f(r)− Snf(r) sin(kx) − − bkf(r)− Snf(r) cos(kx)i = (−1)l ∞

∑

k=n+1 Ak g_f(r)_{− Sn}g_f(r) kr = (−1)l ∞

∑

k=n+1 k−r− (k + 1)−rSk x, gf(r)− Sngf(r) . Hence k f − Sn( f )k_Lpq w 6 ∞

∑

k=n+1 k−r− (k + 1)−r Skgf(r)− Sngf(r) Lwpq . ∞

∑

k=n+1 k−r− (k + 1)−r gf(r)− Sngf(r) Lpqw . 1 (n + 1)r gf(r)− Sngf(r) Lwpq . 1 (n + 1)r f(r)− Snf(r) Lwpq .

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and Theorem1is proved.

Proof of Theorem2. Let n∈ N and f ∈ Lwpq be fixed. We will use the operator

Ar_1/nf. From Corollary1and Lemma1

En( f )_Lpq w = En( f − A r 1/nf+ Ar1/nf)Lpqw 6 En( f − A r 1/nf)Lpqw + En(A r 1/nf)Lpqw . _{f − A}r_1/nf Lpqw + n−2r d2r dx2rA r 1/nf(x) Lwpq . Ωr f,1 n Lwpq . Hencek f − Sn( f )k_Lpq w . En( f )Lwpq. Ωr f, 1 n Lwpq.

Proof of Theorem3. Let f∈ Lwpq and

En( f )_Lpq

w . n

−α_, _n_{∈ N}

for someα> 0. We suppose that δ > 0 and n := [1/δ] . From (1.4) we get Ω_r( f ,δ)_Lpq w 6 Ωr f,1 n Lpqw . 1 n2r n

∑

j=0 ( j + 1)2r−1Ej( f )_Lpq w .δ2r E0( f )Lwpq+ n

∑

j=1 j2r−1Ej( f )_Lpq w ! .δ2r _E 0( f )Lwpq+ n

∑

j=1 j2r−1−α ! . If 2r>α_{, then we get Ωr}_{( f ,}δ₎ Lwpq.δ α_{. If 2r =}_α_{, then} n

∑

j=1 j2r−1−α= n

∑

j=1 j−16 1+ log(1/δ) and hence Ωr( f ,δ)_Lpq w .δ

2r_log(1/_δ_{) . If 2r <}_α_{, then the series} _∑n

j=0 j2r−1−α is con-vergent and Ω_r( f ,δ)_Lpq w .δ 2r _E 0( f )Lwpq+ n

∑

j=1 j2r−1−α ! .δ2r holds.

Proof of Theorem4. Using Corollary1and (1.2) we find

En( f )_Lpq w . 1 (n + 1)rEn f(r),1 n Lwpq . 1 (n + 1)rΩk f(r),1 n Lwpq .

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Proof of Theorem 5. For f ∈ Wr

pq,w we have f(k)∈ Wpqr−k,w. Using Corollary 1,

S(k)n ( f ) = Sn f(k) and (1.3) we find f(k)− Sn(k)( f ) Lwpq = _f(k)− Sn( f(k)) Lwpq . 1 nr−kΩl f(r),1 n Lwpq .

Proof of Theorem6. Let q,t∗

n∈ Tnand En f(k) Lpqw = _f(k)_{− q} Lpqw , En( f )_Lpq w = k f − t∗ nkLwpq. Then using (Sn( f , ·)) (k)_{= Sn}_f(k)_{, ·} f(k)− Sn(k)( f ) Lwpq 6 _f(k)− Snf(k), · Lwpq + S(k)n ( f ) − (tn∗)(k) Lpqw 6 _f(k)− q Lpqw + _{q − S}n f(k), · Lpqw + _(Sn( f , ·) − tn∗) (k) Lpqw . Enf(k) Lpqw + Sn q− f(k), · Lwpq + nkkSn( f , ·) − tn∗kLwpq . En f(k) Lpqw + nk_{kSn( f , ·) − Sn}_(t∗ n, ·)kLpqw . nk−rEn f(r) Lpqw + nk_E n( f )Lwpq. n k−r_E n f(r) Lpqw

and the proof of Theorem is completed.

Acknowledgements. Authors are indebted to referee for valuable and constructive suggestions which improve the presentation of the paper. The first author was supported by Balikesir University Scientific Research Unit with contract 2014.058.

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(Received May 16, 2014) Ramazan Akg¨un

Department of Mathematics, Faculty of Arts and Sciences Balikesir University 10145, Balikesir, Turkey e-mail:[email protected] Yunus Emre Yıldırır Department of Mathematics, Faculty of Education Balikesir University 10100, Balikesir, Turkey e-mail:[email protected]

Mathematical Inequalities & Applications

www.ele-math.com [email protected]

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