https://doi.org/10.33044/revuma.v60n1a08
DIRECT THEOREMS OF TRIGONOMETRIC APPROXIMATION FOR VARIABLE EXPONENT LEBESGUE SPACES
RAMAZAN AKG ¨UN
Abstract. Jackson type direct theorems are considered in variable exponent Lebesgue spaces Lp(x)with exponent p(x) satisfying 1 ≤ ess inf
x∈[0,2π]p(x), ess supx∈[0,2π]p(x) < ∞, and the Dini–Lipschitz condition. Jackson type di-rect inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages
Zvf (·) :=1 v
Z v
0
f (· + t) dt
in these spaces. We give the main properties of the modulus of smoothness Ωr(f, v)p(·):= k(I − Zv)rf kp(·) (r ∈ N)
in Lp(x), where I is the identity operator. An equivalence of the modulus of smoothness and Peetre’s K-functional is established.
1. Introduction
The main purpose of this work is to obtain a Jackson type direct theorem for functions in generalized Lebesgue spaces Lp(·) with variable exponent p (x) : [0, 2π) → [1, ∞), satisfying the Dini–Lipschitz condition and
1 ≤ p := ess infx∈[0,2π)p (x) ≤ ess supx∈[0,2π)p (x) =: P < ∞.
The main difficulty related to Lp(·) theory is that the spaces Lp(·) are, in general, not translation invariant, see e.g. [10, Proposition 3.6.1]. This inadequacy and the structure of Lp(·) cause some additional problems. For example Young’s Convolu-tion inequality and Cavalieri’s equality do not hold in the spaces Lp(·). Maximal, Poinc´are, and Sobolev inequalities do not hold in a modular form in Lp(·) either. Interpolation is not so useful in Lp(·). Solutions of the p (·)-Laplace equation are not scalable.
To obtain a Jackson type inequality in Lp(·)we can not use the classical modulus of smoothness ωr(f, ·)p because the classical translation taf (x) := f (x + a) of
2010 Mathematics Subject Classification. Primary 42A10; Secondary 41A17.
Key words and phrases. Lebesgue spaces with variable exponent; K-functional; Steklov mean;
Best approximation.
This work was supported by Balikesir University Research Project “A modulus of smoothness in Lp(x)” in 2019.
f (·), where a ∈ R, may not be in the class Lp(·) even if f is in Lp(·), see e.g. [10, Proposition 3.6.1]. So the classical modulus of smoothness ωrmay not be suitable for functions f ∈ Lp(·). Instead of using the classical translation operator f 7→ t
af , a ∈ R, we will consider the one sided Steklov average
Zvf (x) := 1 v Z v 0 tuf (x) du, v > 0; Z0f := f
in Lp(·) [22, 23, 2, 24]. Under some condition on p (x) Sharapudinov [22, 23] ob-tained that the family of operators {Zvf }0<v≤1 is uniformly bounded on Lp(·) (see Theorem 1.1 below).
Let T := [0, 2π) and let E be the class of Lebesgue measurable functions p (x) : T → [1, ∞) such that 1 ≤ p ≤ P < ∞. The variable exponent p (x) is said to satisfy the Dini–Lipschitz property on T ([22]) if there exists a (Dini–Lipschitz) constant A > 0, depending only on p (x), such that
|p (x) − p (y)| ln 2π |x − y|−1≤ A < ∞, (1.1) for all x, y ∈ T with x 6= y. We will denote by P the class of those exponents
p ∈ E that satisfy the Dini–Lipschitz property (1.1) on T. We define the variable
exponent Lebesgue space Lp(·)as the collection of 2π-periodic Lebesgue measurable functions f : T → R having the norm
kf kp(·):= inf ( α > 0 : Z T f (x) α p(x) dx ≤ 1 ) < ∞, (1.2)
where p ∈ E . The space Lp(·)is a Banach space. If p ∈ P, p0(x) := p (x) / (p (x) − 1)
for p (x) > 1, and p0(x) := ∞ for p (x) = 1, then H¨older’s inequality Z T |f (x) g (x)| dx ≤ 1 + 1 p− 1 P kf kp(·)kgkp0(·)= H kf kp(·)kgkp0(·)
holds when f ∈ Lp(·)and g ∈ Lp0(·). We know that
kf kp(·)≤ (2π + 1) kf kq(·)= D kf kq(·) (1.3) for p ∈ E and 1 ≤ p (x) ≤ q (x) ≤ ¯q := ess supx∈Tq (x) < ∞ a.e. on T.
Theorem 1.1. If p ∈ P, f ∈ Lp(·), and 0 < v ≤ 1, then
kZvf kp(·)≤ kZvkLp(·)→Lp(·)kf kp(·), (1.4) where kf kp(·) is the Luxemburg norm (1.2) of f and
kZvkLp(·)→Lp(·) ≤ B := 136
25 A
(2π + 1)ln 2A +P. (1.5)
Inequality (1.4) was obtained in [22, 23]; (1.5) can be proved similarly.
After this result one can define modulus of smoothness. For p ∈ P, f ∈ Lp(·), and 0 < v ≤ 1, we set
For r = 1 see [24]. Note that modulus of smoothness is the main notion in the approximation theory and it is used in many of the approximation results such as Jackson type direct theorems, Salem–Stechkin type inverse theorems, Marchaud type inequalities, etc. First of all the set of trigonometric polynomials is a dense subset ([23, Theorems 6.1 and 6.2]) of Lp(·)when p ∈ P. This allows us to consider approximation problems in Lp(·). Jackson type inequalities in Lp(·) were investi-gated by several mathematicians. For example in [14] Israfilov and Testici obtained the following estimate:
If f ∈ Lp(·), p ∈ P, then En(f )p(·):= inf t∈Πn kf − tkp(·)= O (1) sup 0≤h≤1/n 1 h Z h 0 (I − tu)rf (·) du p(·) (1.7)
holds for n ∈ N, with constant depending only on p, where Πn is the class of
trigonometric polynomials of degree not greater than n.
For p ∈ ˆP := {p ∈ P : 1 < ess infx∈Tp (x)} see also [13]. In [2] the author proved that If f ∈ Lp(·), r > 0, p ∈ ˆP, then En(f )p(·)= O (1) sup 0≤hi, t≤1/n (I − Φh1) · · · I − Φhbrc (I − Φt) {r} f p(·)
holds for n ∈ N, with constant depending only on p, r, where
Φhf (x) := 1 h Z x+h/2 x−h/2 f (t) dt,
brc := max {n ∈ N : n ≤ x}, {r} := r−brc, I is the identity operator, and (I − Φt){r} is the fractional binomial series expansion of I − Φt.
Here, and in what follows, A = O (1) B means that A/B is less than or equal to some constant, depending on essential parameters only.
In [24, 28] Sharapudinov and Volosivets considered the following type of modulus of smoothness of order r ∈ N:
sup
0≤vi≤δ
k(I − Zv1) · · · (I − Zvr) f kp(·) in the right hand side of (1.7):
If f ∈ Lp(·), p ∈ P, then ([24, r = 1], [28, r ≥ 1]) En(f )p(·)= O (1) sup
0≤vi≤δ
k(I − Zv1) · · · (I − Zvr) f kp(·) (1.8) holds for n, r ∈ N, with constant depending only on p, r.
In the present work we will consider a more natural and smaller modulus of smoothness:
for f ∈ Lp(·)and p ∈ P. Our result, given below, refines the Jackson type estimates (1.7) and (1.8).
Theorem 1.2. We suppose that p ∈ P, f ∈ Lp(·)
, and r ∈ N. In this case we have inequalities En(f )p(·)≤ CΩr f, 1 n p(·) , (1.9) n Y s=1 Es(f )p(·) !1/n ≤C 2 (2e [1 + B])r B Ωr f, 1 n p(·) , (1.10) where C = 2Bhmaxn216BHPr−1 j=0B j , 2 [1 + 108BH + 432BH ln 2]oi r .
For the constant exponent case p (x) = p, 1 ≤ p ≤ ∞, the Lp modulus of smoothness (1.6) was considered by Trigub in 1968 [27] for r = 1, 1 ≤ p ≤ ∞, and by Ditzian and Ivanov [11] for r ∈ N, 1 ≤ p ≤ ∞. In the case r = 1, p ∈ P, a Jackson type inequality was obtained in [24] by Sharapudinov (see (1.8) and [28]). For the constant exponent case p (x) = p, in Lpthe inequality (1.10) was obtained by Natanson and Timan (see [21]) for r ∈ N, p = ∞.
Let X be a Banach space with norm k·kX. By Xrwe denote the class of functions f ∈ X such that f(r−1)is absolutely continuous and f(r)∈ X. When r ∈ N, p ∈ P,
and X = Lp(·) we will denote Wr p(·):= X
r.
The following second type Jackson inequality holds:
Theorem 1.3. Suppose that r, k, n ∈ N, p ∈ P, and f ∈ Wp(·)r . Then the following inequality holds: nrEn(f )p(·)≤ C B 1 + D (2π)1p−1 r Ωk f(r), 1 n p(·) .
One of the results of this paper is the following theorem, which contains an equivalence of Ωrand Peetre’s K-functional.
Theorem 1.4. If r ∈ N, p ∈ P, f ∈ Lp(·), then the equivalence k(I − Zv)rf kp(·)≈ inf kf − gkp(·)+ vr g (r) p(·): g ∈ W r p(·) (1.11) holds for v > 0.
If A (t) = O (B (t)) and B (t) = O (A (t)), we will write A (t) ≈ B (t). Corollary 1.5. If k ∈ N, p ∈ P, f ∈ Lp(·), then Ωk(f, λv)p(·)= O (1) (1 + bλc) k Ωk(f, v)p(·), v, λ > 0, and Ωk(f, v)p(·) vk = O (1) Ωk(f, δ)p(·) δk , 0 < δ ≤ v,
The rest of the present work is organized as follows. In Section 2 we give upper estimates for the operator norm of Steklov and Jackson operators. Section 3 contains the main properties of the modulus of smoothness Ωr and some integral operators. In Section 4 we give a transference result. In Section 5 we give the proof of the equivalence of the modulus of smoothness Ωrand Peetre’s K-functional Kr. Finally, Section 6 contains the proof of the refined Jackson inequality and other proofs.
In what follows, letters c, p, A, B, C, . . . will stand for certain positive constants and these will not change in different places.
2. Upper estimates for the operator norm of Steklov and Jackson operators
Theorem 2.1 ([23]). If p ∈ P, ϑ > 0, 1 ≤ λ < ∞, |τ | ≤ πλ−ϑ, then the family of operators {Sλ,τ}1≤λ<∞, defined by
Sλ,τf (x) = λ
Z x+τ +1/(2λ)
x+τ −1/(2λ)
f (u) du,
is uniformly bounded (in λ and τ ) in Lp(·):
kSλ,τf kp(·)≤ kSλ,τkLp(·)→Lp(·)kf kp(·) (2.1) with
kSλ,τkLp(·)→Lp(·) ≤ B. (2.2)
We point out that (2.1) was obtained in [23] and (2.2) can be obtained in the same way.
Let p ∈ P, f ∈ Lp(·), 0 < v ≤ 1. We note that Zvf (·) = 1 v Z v 0 tuf (·) du = S1 v, v 2f (·) . Let n ∈ N and Dnf (x) := 1 π Z T t−uf (x)J2,bn 2c+1(u) du ∈ Πn
be the Jackson operator (polynomial) where J2,n is the Jackson kernel
J2,n(x) := 1 κ2,n sin(nx/2) sin(x/2) 4 , κ2,n:= 1 π Z π −π sin(nt/2) sin(t/2) 4 dt. It is known ([12, p. 147]) that 3 2√2n 3 ≤ κ2,n≤ 5 2√2n 3.
The Jackson kernel J2,nsatisfies the relations 1 π R TJ2,n(u) du = 1, |J2,n(u)| ≤ 2 √ 2 3 π 4, n−3/4 ≤ u ≤ π,
maxu∈T|J2,n(u)| ≤ π2−1 4 n, 1 π Rπ 0 uJ2,n(u) du ≤ 1 2n. (2.3)
The next theorem was obtained in [24].
Theorem 2.2. If p ∈ P and f ∈ Lp(·), then the sequence of Jackson operators {Dnf }1≤n<∞ is uniformly bounded (in n) in Lp(·):
kDnf kLp(·) ≤ kDnkLp(·)→Lp(·)kf kLp(·), where kDnkLp(·)→Lp(·) ≤ E := π 4 32 ln 23P4A 68 25 4A3P" 45π5D 16 +H 15π5 D 16 P 3 #1P + 2 √ 2D 3π−4 (2π)1P. 3. Modulus of smoothness By Theorem 1.1 we have (I − Zv)kf p(·)≤ (1 + B) k kf kp(·).
For k ∈ N we define the modulus of smoothness of f ∈ Lp(·), p ∈ P, as Ωk(f, v)p(·):= (I − Zv) k f p(·), v > 0. From [28, (3.2) and Corollary 2] we have
Lemma 3.1. Let p ∈ P, k ∈ N, and 0 ≤ v ≤ 1. Then (I − Zv) k f p(·)≤ B · 2 −1v (I − Zv) k−1 f0 p(·), f ∈ W 1 p(·), and (I − Zv) k f p(·)≤ B k · 2−kvk f (k) p(·), f ∈ W k p(·) hold. 4. Transference result
Here we state a variant of the transference result obtained in [5, Theorem 14]. Let p ∈ P, f ∈ Lp(·), G ∈ Lp0(·), and kGkp0(·)= 1. Note that the dual of Lp(·) is Lp0(·). For any ε > 0 there exists an h
1≤ 1 such that
(see [5, (18)]). Everywhere in this work, this h1 will be fixed. We define Ff(u) := hΦh1f (· + u) , Gi = R TΦh1f (x + u) |G (x)| dx, for −h1≤ u ≤ h1, Ff(h1) for h1< u ≤ π, Ff(−h1) for π ≤ u < −h1. (4.1) Lemma 4.1 ([5, Lemma 13]). If p ∈ P, f ∈ Lp(·), then the function F
f(u) defined in (4.1) is uniformly continuous on [−h1, h1] =: Ih1.
Theorem 4.2. Let p ∈ P and f, g ∈ Lp(·). If kFgkC[I
h1] ≤ c kFfkC[Ih1] , then
kgkp(·)≤ 3cBH kfkp(·). We define Peetre’s K-functional
Kr(f, v, X)X:= inf g∈Xr n kf − gkX+ vr g(r) X o , v > 0, and Kr(f, v, p (·)) := Kr f, v, Lp(·) Lp(·) for r ∈ N, p ∈ P, v > 0, and f ∈ L p(·).
Proposition 4.3 ([5, Corollary 20]). If 0 < h ≤ v ≤ 1 and f ∈ C [T], the class of
functions continuous on T, then
k(I − Zh) f kC[T]≤ 72 k(I − Zv) f kC[T]. (4.2)
Lemma 4.4. Let 0 < h ≤ v ≤ 1, p ∈ P, and f ∈ Lp(·). Then 1
3k(I − Zh) f kp(·)≤ 72BH k(I − Zv) f kp(·). (4.3) 5. Approximation by Jackson operators
Proposition 5.1 ([24]). For every n ∈ N, p ∈ P, f ∈ W1
p(·), the inequality En(f )p(·)≤ kf − Dnf kp(·)≤ B n1 f (1) p(·) holds. Theorem 5.2. If p ∈ P, f ∈ Lp(·) , and n ∈ N, then kf − Dnf kp(·)≤ B [216H + 2 [1 + 108BH + 432BH ln 2] + 216EH] Ω1 f,1 n p(·) . (5.1) To find an equivalence between the modulus of smoothness Ωr(f, δ)p(·) and the K-functional Kr(f, δ, p (·))p(·) we consider the operator (see [20, 24]) defined for f ∈ Lp(·), p ∈ P, as (Rvf ) (x) := 2 v Z v v/2 1 h Z h 0 f (x + t) dt ! dh, x ∈ T, v > 0.
Note that for 0 < v ≤ 1, p ∈ P we know that kRvf kp(·)≤ B kfkp(·), hence f − Rvf ∈ Lp(·)for f ∈ Lp(·). We set Rrvf := (Rvf )
r . Lemma 5.3. Let 0 < v, p ∈ P and f ∈ Lp(·). Then
k(I − Rv) f kp(·)≤ 216HB k(I − Zv) f kp(·).
Proof. If f ∈ Lp(·), we can use the generalized Minkowski integral inequality and Lemma 4.4 to obtain k(I − Rv) f kp(·)= 2 v Z v v/2 1 h Z h 0 (f (x + t) − f (x)) dt ! dh p(·) = 2 v Z v v/2 (Zhf (x) − f (x)) dh p(·) ≤ 2 v Z v v/2 kZhf − f kp(·)dh ≤ 216HB kZvf − f kp(·) 2 v Z v v/2 dh = 216HB k(I − Zv) f kp(·). Remark 5.4. Note that the function Rvf is absolutely continuous ([24]) and differentiable a.e. on T.
Lemma 5.5. Let 0 < v ≤ 1, p ∈ P, and f ∈ W1
p(·). Then d dxRvf (x) = Rv d dxf (x) and d dxZvf (x) = Zv d dxf (x) , a.e. on T. (5.2)
Proof. The proof is the same as the proof of Lemma 24 of [5].
Lemma 5.6. Let 0 < v ≤ 1, p ∈ P, and f ∈ Lp(·) be given. Then
v d dxRvf (x) p(·) ≤ 2 [1 + 108BH + 432BH ln 2] k(I − Zv) f kp(·). (5.3) Proof. For f ∈ Lp(·) by [24, p. 426] we know that
κ := v 2 d dxRvf (x) p(·) ≤ Zvf (x) − f (x) − 1 2 Zv/2f (x) − f (x) p(·) + Z v v/2 1 h Zhf (x) − f (x) − v 2h Zh/2f (x) − f (x) dh p(·) and hence κ ≤ k(I − Zv) f kp(·)+ 1 2 I − Zv/2 f p(·) + Z v v/2 1 h |Zhf (x) − f (x)| + v 2h Zh/2f (x) − f (x) dh p(·) .
One can find by Lemma 4.4 and the generalized Minkowski integral inequality: κ ≤ k(I − Zv) f kp(·)+ 1 2 I − Zv/2 f p(·) + Z v v/2 1 h|Zhf (x) − f (x)| dh p(·) + Z v v/2 1 h Zh/2f (x) − f (x) dh p(·) ≤ (1 + 108BH) k(I − Zv) f kp(·)+ Z v v/2 1 h h kZh− f kp(·)+ Zh/2− f p(·) i dh ≤ (1 + 108BH) k(I − Zv) f kp(·)+ 144BH kZvf − f kp(·) Z v v/2 1 hdh ≤ (1 + 108BH + 432BH ln 2) k(I − Zv) f kp(·)
Lemma 5.7. Let 0 < v ≤ 1, r − 1 ∈ N, p ∈ P, and f ∈ Lp(·) be given. Then dr dxrR r vf (x) = d dxRv dr−1 dxr−1R r−1 v f (x) , x ∈ T.
Proof. The proof is the same as the proof of Lemma 26 of [5].
Proof of Theorem 1.4. Let r = 1, p ∈ P, and f ∈ Lp(·). Since K1(f, v, p (·))p(·)≤ kf − Rvf kp(·)+ v d dxRvf (x) p(·) ,
from Lemma 5.3 and (5.3) we find
K1(f, v, p (·))p(·)≤ 2 (1 + 108BH + 432BH ln 2) k(I − Zv) f kp(·). From Lemma 3.1, for g ∈ Wp(·)1 ,
Ω1(f, v)p(·)≤ Ω1(f − g, v)p(·)+ Ω1(g, v)p(·)≤ (1 + B) kf − gkp(·)+ B 2v kg
0k
p(·), and taking infimum on g ∈ Wp(·)1 in the last inequality we get
Ω1(f, v)p(·)≤ (1 + B) K1(f, v, p (·))p(·)
and the equivalence of k(I − Zv) f kp(·) with K1(f, v, p (·))p(·) is established. Now we will consider the case r > 1. For r = 2, 3, . . . we consider the operator (see [2]) Ar v:= I − (I − Rrv) r =Xr−1 j=0(−1) r−j+1r j Rr(r−j)v . From the identity I − Rr
v= (I − Rv)Pr−1j=0Rjv we find k(I − Rr v) gkp(·)≤ r−1 X j=0 Bj k(I − Rv) gkp(·) ≤ 216BH r−1 X j=0 Bj k(I − Zv) gkp(·)
when 0 < v ≤ 1, p ∈ P, and g ∈ Lp(·). Since kf − Ar vf kp(·) = k(I − R r v) r f kp(·), a recursive procedure gives
kf − Ar vf kp,ω= k(I − R r v) r f kp(·)≤ · · · ≤ 216BH r−1 X j=0 Bj r k(I − Zv) r f kp(·).
On the other hand, using Lemmas 5.7 and 5.3 recursively,
vr dr dxrR r vf p(·) = vr−1v d dxRv dr−1 dxr−1R r−1 v f p(·) ≤ 2 (1 + 108BH + 432BH ln 2) vr−1 (I − Zv) dr−1 dxr−1R r−1 v f p(·) ≤ . . . ≤ [2 (1 + 108BH + 432BH ln 2)]r−1v d dxRv(I − Zv) r−1 f p(·) ≤ [2 (1 + 108BH + 432BH ln 2)]rk(I − Zv)rf kp(·). Thus Kr(f, v, p (·))p(·)≤ kf − A r vf kp(·)+ v r dr dxrA r vf (x) p(·) ≤ 2 max 216BH r−1 X j=0 Bj, 2 (1 + 108BH + 432BH ln 2) r k(I − Zv) r f kp(·).
For the reverse of the last inequality, when g ∈ Wp(·)r , from Lemma 3.1: Ωr(f, v)p(·)≤ (1 + B) r kf − gkp(·)+ Ωr(g, v)p(·) ≤ (1 + B)rkf − gkp(·)+ 2−rBrvr g (r) p(·); (5.4) taking infimum on g ∈ Wr p(·) in (5.4) we get Ωr(f, v)p(·)≤ (1 + B) r Kr(f, v, p (·))p(·), and hence Ωr(f, v)p(·)≈ Kr(f, v, p (·))p(·). (5.5)
6. Proofs of the results 6.1. Proof of the transference result.
Proof of Theorem 4.2. Let p ∈ P and f ∈ Lp(·). In this case, kFgkC[I h1] ≤ c kFfkC[Ih1] = c Z T Φh1f (x + u) |G (x)| dx C[I h1] = c max u∈Ih1 Z T Φh1f (x + u) |G (x)| dx ≤ cH max u∈Ih1kΦh1f (· + u)kp(·)kGkp0(·)≤ cHB kfkp(·), by Theorem 2.1.
On the other hand, for any ε, η > 0 and appropriately chosen G ∈ Lp0(·) with
hg, Gi =R
Tg (x) G (x) dx ≥
1
3kgkp(·)− ε, kGkp0(·)= 1, one can find
kFgkC[I h1] ≥ |Fg(0)| ≥ Z T Φh1g (x) |G (x)| dx > Z T g (x) |G (x)| dx − η Z T |G (x)| dx ≥ 1 3kgkp(·)− ε − ηH k1kp(·). Since ε, η > 0 are arbitrary we have
kFgkC[I h1] ≥ 1 3kgkp(·), ε, η → 0 +, and hence 1 3kgkp(·)≤ kFgkC[Ih1] ≤ cHB kf kp(·).
This gives the required result.
Proof of Lemma 4.4. Let 0 < h ≤ v ≤ 1, p ∈ P, and f ∈ Lp(·). In this case, by (4.2) and Lemma 4.1, max u∈Ih1 F(I−Zh)f(u) = max u∈Ih1 Z T Φh1(I − Zh) f (x + u) |G (x)| dx = max u∈Ih1 (I − Zh) Z T Φh1f (x + u) |G (x)| dx = max
u∈Ih1|(I − Zh) Ff(u)|
(4.2)
≤ 72 max
u∈Ih1|(I − Zv) Ff(u)| = 72 max u∈Ih1 (I − Zv) Z T Φh1f (x + u) |G(x)| dx ≤ 72 max u∈Ih1 Z T Φh1(I − Zv) f (x + u) |G(x)| dx = 72 max u∈Ih1 F(I−Zv)f(u)
≤ 72H max
u∈Ih1kΦh1(I − Zv) f (x + u)kp(·)kGkp0(·) ≤ 72HB k(I − Zv) f kp(·).
On the other hand, for any ε, η > 0 and appropriately chosen G ∈ Lp0(·) with
hf, Gi ≥ 1
3kf kp(·)− ε and kGkp0(·)= 1, one can get
max
u∈Ih1|(I − Zh) Ff(u)| ≥ |(I − Zh) Ff(0)| ≥ (I − Zh) Z T Φh1f (x) |G (x)| dx = Z T Φh1(I − Zh) f (x) |G (x)| dx > Z T (I − Zh) f (x) |G (x)| dx − ηH k1kp(·) ≥1 3k(I − Zh) f kp(·)− ε − ηH k1kp(·). Since ε, η > 0 are arbitrary we obtain
max
u∈Ih1|(I − Zh) Ff(u)| ≥ 1
3k(I − Zh) f kp(·) and hence
1
3k(I − Zh) f kp(·)≤ maxu∈Ih1|(I − Zh) Ff(u)| ≤ 72HB k(I − Zv) f kp(·). 6.2. Proof of Jackson’s inequalities.
Proof of Theorem 1.2. Let n, r ∈ N. For g ∈ Wp(·)r we have by Theorem 1.3 that En(f )p(·)≤ En(f − g)p(·)+ En(g)p(·)≤ kf − gkp(·)+ B nr g (r) p(·). Taking infimum in the last inequality with respect to g ∈ Wp(·)r , one can get, by (5.5), En(f )p(·)≤ BKr(f, 1/n, p (·))p(·) ≤ 2B max 216BH r−1 X j=0 Bj, 2 [1 + 108BH + 432BH ln 2] r Ωr f,1 n p(·) = CΩr f,1 n p(·) .
At this stage, we will use the method given by Natanson and Timan [21] to obtain (1.10). By Corollary 1.5 we have, for ρ ≤ n,
Ωr f,1 ρ p(·) ≤ [1 + B]rC B 1 +n ρ r Ωr f,1 n p(·)
and hence n Y ρ=1 Ωr f,1 ρ p(·) ≤ n Y ρ=1 [1 + B]rC B 1 +n ρ r Ωr f,1 n p(·) = [1 + B]rC B Ωr f, 1 n p(·) !n n Y ρ=1 1 +n ρ r .
Using Stirling’s formula we have n Y ρ=1 1 + n ρ r ≤ 2rer and consequently n Y ρ=1 Ωr f,1 ρ p(·) !1/n ≤ (2e [1 + B])rC B Ωr f, 1 n p(·) .
From Theorem 1.2 and the property En(f )p(·)↓ as n ↑, we find n Y ρ=1 Eρ(f )p(·) !1/n ≤ n Y ρ=1 CΩr f,1 ρ p(·) !1/n ≤ (2e [1 + B])rC2B−1Ωr f,1 n p(·)
and the result (1.10) follows.
Proof of Theorem 1.3. Suppose that Θn ∈ Πn, En(f0)p(·) = kf0− Θnkp(·), and β/2 is the constant term of Θn, namely,
β = 1 π Z T Θn(t) dt = 1 π Z T (Θn(t) − f0(t)) dt. We get |β/2| ≤ 1 2πkf 0− Θ nk1≤ D 2πkf 0− Θ nkp(·)= D 2πEn(f 0) p(·). Furthermore, kf0− (Θ n− β/2)kp(·)≤ En(f0)p(·)+ kβ/2kp(·) ≤ En(f0)p(·)+ D (2π) 1 p−1E n(f0)p(·) =1 + D (2π)p1−1 En(f0)p(·). We set un∈ Πn so that u0n= Θn− β/2. Then
En(f )p(·)= En(f − un)p(·)≤ B nkf 0− (Θ n− β/2)kp(·) ≤ B1 + D (2π)p1−1 1 nEn(f 0) p(·).
The last inequality gives En(f )p(·)≤ h B 1 + D (2π)p1−1 ir 1 nrEn f(r) p(·). Using Theorem 1.2 we obtain
nrEn(f )p(·)≤ C h B 1 + D (2π)1p−1 ir Ωr fr,1 n p(·) .
Proof of Theorem 5.2. From Proposition 5.1, Lemma 5.3, and Theorem 2.2 we get
kf − Dnf kp(·)= f − R1/nf + R1/nf − DnR1/nf + DnR1/nf − Dnf p(·) ≤ f − R1/nf p(·)+ R1/nf − DnR1/nf p(·)+ Dn(R1/nf − f ) p(·) ≤ 216BHΩ1 f, 1 n p(·) +B n (R1/nf )0 p(·)+ E R1/nf − f p(·) ≤ B [216H + 2 (1 + 108BH + 432BH ln 2) + 216EH] Ω1 f,1 n p(·) and (5.1) follows. Acknowledgements
The author is indebted to the referees for their valuable suggestions and com-ments.
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Ramazan Akg¨unB
Balıkesir University, Faculty of Arts and Sciences, Department of Mathematics, C¸ a˘gi¸s Yerle¸skesi, 10145, Balıkesir, T¨urkiye.
rakgun@balikesir.edu.tr
Received: December 7, 2017 Accepted: September 11, 2018
