Convolutions and best approximations in variable exponent lebesgue spaces

IN VARIABLE EXPONENT LEBESGUE SPACES

DANIYAL M. ISRAFILOV and ELIFE YIRTICI

Communicated by Vasile Brˆınz˘anescu

In the variable exponent Lebesgue spaces a convolution is defined and its es-timations in the variable exponent Lebesgue spaces by the best approximation numbers are obtained.

AMS 2010 Subject Classification: 41A10, 42A10.

Key words: convolution, variable exponent Lebesgue space, best approximation.

1. INTRODUCTION AND MAIN RESULTS

Let p(·) : [0, 2π] → [1, ∞) be a Lebesgue measurable function. We define the modular functional

ρp(·)(f ) :=

Z 2π

0

|f (x)|p(x)dx

on the Lebesgue measurable functions f on [0, 2π]. By Lp(·)_2π we denote the class of 2π periodic Lebesgue measurable functions f , such that for a constant λ = λ (f ) > 0

ρ_p(·)(f /λ) < ∞. Equipped with the norm

kf k_p(·):= infλ > 0 : ρp(·)(f /λ) ≤ 1

the class Lp(·)_2π creates a Banach space.

The variable exponent Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p(·). Interest in the variable exponent Lebesgue spaces has increased since 1990s, because of their use in the different applications to problems in mechanic, especially in fluid dynamic for the modeling of electrorheological fluids and also in the study of image processing and some problems in physics (see, for example the monographs [4, 5, 14] and the references cited therein). Nowadays there are sufficiently wide investigations relating to the fundamen-MATH. REPORTS 18(68), 4 (2016), 497–508

tal problems of these spaces, in view of potential theory, maximal and singular integral operator theory and others. The sufficiently wide presentation of the corresponding results can be found in the monographs [4, 5, 14].

In the variable exponent Lebesgue spaces some fundamental problems of approximation theory were investigated also. Some necessary and sufficient conditions in term of the variable exponent for the basicity of the well known classical systems of functions were obtained [15, 16], the different modulus of smoothness were defined [7, 8, 17] and the direct and inverse theorems of ap-proximation theory in these spaces defined on the intervals of the real line and on the domains of the complex plane were proved [1–3, 7–9, 17]. The detailed information on these results and also on the general aspects of approximation theory in the variable exponent Lebesgue spaces can be found in the mono-graph [18].

Note that in the variable exponent Lebesgue space theory some convolu-tion operators were commonly used. This type of operators have some applica-tions also in the approximation theory, in particular for the construction of the approximation polynomials. Therefore, the estimation problem of convolution operators by using the best approximation numbers is an actual problem of ap-proximation theory. In this work, we investigate this problem in the variable exponent Lebesgue spaces. For the formulation of the main results obtained in this work we give some notations.

By β2π we denote the class of the exponents p (·), for which 1 < p− ≤

p+ < ∞, where p−:= essinf x∈[0,2π] p (x) , and p+:= essup x∈[0,2π] p (x) .

Definition 1. Let p(·) : [0, 2π] → [1, ∞) be a 2π periodic, measurable function. We say that p(·) is a log-Holder continuous function on [0, 2π] if (1.1) |p (x) − p (y)| ≤ c0

− log (|x − y|), x, y ∈ [0, 2π] with |x − y| ≤ 1/2 with some constant c0 > 0.

If p(·) ∈ β2πand satisfies the condition (1.1), then we say that p(·) ∈ bβ2π.

In the space Lp(·)_2π with p(·) ∈ bβ2π we define a mean value operator σh

(σhf ) (x, u) := 1 2h Z h −h f (x + tu) dt, 0 < h < π, x ∈ [0, π] , − ∞ < u < ∞, which is linear and bounded by [6], moreover k(σhf )kp(·)≤ c1kf kp(·) for some

constant c1 > 0.

For f ∈ Lp(·)_2π we define also the best approximation number En(f )_p(·):= inf

Tn

by trigonometric polynomials Tn(x) = n X k=0 ckeikx

of degree at most n and by T_n∗the best approximation trigonometric polynomial of degree ≤ n, such that

kf − T_n∗k_p(·)= En(f )_p(·).

Let f ∈ Lp(·)_2π . We define a convolution type operator Z ∞

−∞

(σhf ) (·, u) dσ (u)

with a bounded variation function σ (u) on the real line R and denote D (f, σ, h, p (·)) := Z ∞ −∞ (σhf ) (·, u) dσ (u) p(·) .

In this work, we estimate the quantity D (f, σ, h, p (·)) using the best approximation number En(f )p(·).

Our new results are following:

Theorem 1. If f ∈ Lp(·)2π with p(·) ∈ˆß2π, then

D (f, σ, h, p (·)) ≤ c

m

X

k=0

E₂k₋₁(f )_p(·)δ₂k_,h+ c_p(·)E₂m+1(f )_p(·)

for every m ∈ N, where δ₂k_,h : = 2k+1₋₁ X l=2k |ˆσ (lh) − ˆσ ((l + 1) h)| +   σˆ  2kh , ˆ σ (x) : = Z ∞ −∞ sin (ux) ux dσ (u) , 0 < h < π.

Theorem 2. Let f ∈ Lp(·)2π with p(·) ∈ˆß2πp(·) and let F (x) be a function

with bounded variation, i.e. kF (x)k ≤ c1,

2µ+1₋₁

X

θ=2µ

|F (θh) − F ((θ + 1) h)| ≤ c2, h ≤ 2−m−1.

If σ1 and σ2 are two functions satisfying the condition

ˆ σ1(x) = ˆσ2(x) F (x) , |x| < 1, then D (f, σ1, h, p(·)) = c h D (f, σ2, h, p(·)) + E2m+1(f )_p(·) i .

Using the usual shift f (x + t) in the definition of the convolution operator Theorems 1 and 2 in the Orlicz spaces were proved in [13]. In the weighted Orlicz spaces in term of the mean value operator (σhf ) (x, u), Theorems 1 and

2 were obtained in [10]. We also use this operator in the variable exponent spaces Lp(·)_2π , because Lp(·)_2π in general is non invariant with respect to the usual shift f (x + t).

Throughout this paper, the constant c denotes a generic constant, i.e. a constant whose values can change even between different occurrences in a chain of inequalities.

2. AUXILIARY RESULTS

As was proved in [15], if p (·) ∈ ˆß2π, then the systemeikx

k∈Z of

expo-nents greats a base in the space Lp(·)_2π , which is equivalent to the inequality kS_n(f )k_p(·)≤ c_p(·)kf k_p(·) (n = 0, 1, 2, ...) , where Sn(f ) (x) := n X k=−n ˆ fk(x) eikx

is the nth partial sum of the Fourier series

∞ X k=1 ˆ fkeikx ; fˆk(x) = 1 2π Z 2π 0 f (y) e−ikydy of f . This inequality implies the following

Lemma 1. If f ∈ Lp(·)2π with p(·) ∈ˆß2π, then

kf − Snf k_p(·)≤ cEn(f )_p(·). Proof. kf − Snf kp(·) ≤ kf − Tnkp(·)+ kSnf − Tnkp(·) ≤ E_n(f )_p(·)+ kSn(f − Tn)kp(·) ≤ E_n(f )_p(·)+ c k(f − Tn)kp(·) ≤ c1En(f )_p(·)

with some constant c1=c1(p)¿ 0. 

The proofs of the following two Theorems A and B can be found in [4, p. 27, Theorem 2.26] and [18, p. 39, Theorem 1.6.5], respectively.

Theorem A. Let p : [0, 2π] → [1, ∞) be a 2π periodic, measurable func-tion. If f ∈ Lp(·)_2π and g ∈ Lp_2π0(·), where 1/p (·) + 1/p0(·) = 1, then f g ∈ L1

2π and Z .2π 0 |f (x) g (x)| dx ≤ K_p(·)kf k_p(·)kgk_p0_(·) with K_p(·)= 1/p−− 1/p++ 1.

Theorem B. Let p (·) : [0, 2π] → [1, ∞) and let f : [0, 2π] × [0, 2π] → R be a measurable function and f (·, y) ∈ Lp(·)_2π for every y ∈ [0, 2π]. Then

2π Z 0 f (·, y) dy p(·) ≤ c 2π Z 0 kf (·, y)k_p(·)dy

with some positive constant c = c_p(·).

Theorem C ([6]). If f ∈ Lp(·)2π with p(·) ∈ˆß2π, then the maximal operator

M (f ) (x) := sup I x 1 |I| Z I |f (t)| dt is bounded in Lp(·)_2π and kM f k_p(·)≤ c kf k_p(·) with some constant c = cp(·)> 0.

Theorem CF ([4, p. 212]). Let = be a family of pairs (f, g) of non-negative, measurable function f and g defined on (0, 2π) such that for all ω ∈ Ap0(0, 2π) with some p0≥ 1 2π Z 0 fp0_{(x) ω (x) dx ≤ c} 2π Z 0 gp0_{(x) ω (x) dx, (f, g) ∈ =,}

where a constant c independent of (f, g). If p(·) ∈ˆß2π, then for every r,

1 < r < ∞, and sequence {(fi, gi)} ⊂ = X i f_ir !1/r _p(·) ≤ c_p(·) X i g_ir !1/r _p(·) .

Lemma 2 ([11]). Let f ∈ Lp(·)2π with p (·) ∈ˆß2π. If Al(x) = eilxfˆl and A2−1(x) = 0, l = 0, 1, 2, ..., then c1kf kp(·)≤   ∞ X j=0       2j−1 X l=2j−1 Al(x)       2  1/2 p(·) ≤ c₂kf k_p(·).

Lemma 3. Let {fn}∞1 be a sequence of functions fn∈Lp(·)2π with p (·) ∈ˆß2π

and let Sn,kn be the kth partial sum of fn with k = kn. Then

(2.1) ∞ X n=1 |Sn,kn(x)| 2 !1/2 p(·) ≤ c ∞ X n=1 |fn(x)|2 !1/2 p(·)

with some constant c > 0 independent of fn.p

Proof. The inequality (2.1) is a consequence of the extrapolation Theo-rem CF (for the case of r = 2, gi := fi and fi := |Si,ki(x)|, i = 1, 2, ..., )

and of the norm inequality, proved by Kurtz in [12] in the weighted Lebesgue spaces Lpω.

Let L∞_comp be the set of all bounded functions with compact support on [0, 2π]and let {fn}∞₁ be a sequence in L∞comp. If

f (x) := ∞ X n=1 |fn(x)|2 !1/2 and T f (x) := ∞ X n=1 |Sn,kn(x)| 2 !1/2 , then by [12] for a number p0 with 1 < p0 < ∞ there is a constant c such that

for all weight ω ∈ Ap0

kT f k_p

0,ω ≤ c kf kp0,ω.

Hence the conditions of the above cited extrapolation Theorem CF are fulfilled which implies the inequality (2.1) for the sequence in {fn}∞1 from L∞comp. Since

the set L∞_comp is dense in Lp(·)_2π the inequality (2.1) is also valid for all sequences {f_n}∞₁ from Lp(·)_2π . 

Theorem D. Let λ0, λ1, ... be a sequence of the numbers such that

(2.2) |λ_l| ≤ M ,

2l+1₋₁

X

υ=2l

|λυ− λυ+1| ≤ M (l = 0, 1, 2, ...) .

If p (·) ∈ˆß2π and aυ, bυ be the Fourier coefficients of f ∈ L p(·)

2π , then the series

a0λ0/2 + ∞

X

υ=1

is the Fourier series of function F ∈ Lp(·)_2π and kF k_p(·)≤ c kf k_p(·) with some constant c = cp(·)> 0 independent of f .

Proof. Let for s > 2µ−1, µ = 1, 2, ..., ∆µ,s : = s X υ=2µ−1 Aυ(x) , Aυ(x) := aυcos υx + bυsin υx ∆µ : = 2µ₋₁ X υ=2µ−1 Aυ(x) and ∆ 0 µ:= 2µ₋₁ X υ=2µ−1 λυAυ(x) .

As in [19, p. 347] we have the estimation   ∆ 0 µ    2 ≤ 2M 2µ₋₁ X s=2µ−1 |∆µ,s|2|λs− λs+1| + |∆µ|2|λ2µ| ! .

Since p (x) ≤ p+ by Lemma 3 and (2.2)

ρp(·)      ∞ X µ=1   ∆ 0 µ    2   1/2  = Z 2π 0   ∞ X µ=1   ∆ 0 µ    2   p(x)/2 dx ≤ Z 2π 0   ∞ X µ=1 2M 2µ−1 X s=2µ−1 |∆_µ,s|2|λ_s− λ_s+1| + |∆_µ|2|λ₂µ| !  p(x)/2 dx = Z 2π 0 (2M )p(x)/2   ∞ X µ=1 2µ₋₁ X s=2µ−1 |∆_µ,s|2|λ_s− λ_s+1| + |∆_µ|2|λ₂µ| !  p(x)/2 dx ≤ c Z 2π 0 (2M )p(x)/2   ∞ X µ=1 |∆_µ|2 2µ₋₁ X s=2µ−1 |λ_s− λ_s+1| + |λ₂µ| !  p(x)/2 dx ≤ c Z 2π 0 (2M )p+   ∞ X µ=1 |∆_µ|2   p(x)/2 dx ≤ c3 Z 2π 0   ∞ X µ=1 |∆_µ|2   p(x)/2 dx = c3ρp(·)      ∞ X µ=1 |∆_µ|2   1/2  ,

which by Lemma 2 implies that kF k_p(·) ≤   ∞ X µ=1   ∆ 0 µ    2   1/2 p(·) ≤ c3   ∞ X µ=1 |∆µ|2   1/2 p(·) ≤ c4kf kp(·). 

3. PROOFS OF MAIN RESULTS

Proof of Theorem 1. Let f ∈ Lp(·)_2π , m ∈ N and let S2m+1 be the 2m+1th

partial sum of its Fourier series. Let also h ≤ 2−m−1. Then D (f, σ, h, p(·)) ≤ Z ∞ −∞ [(σhf ) (x, u) − (σhS2m+1f ) (x, u)]dσ (u) p(·) + Z ∞ −∞ (σhS2m+1f ) (x, u)]dσ (u) p(·) . (3.1)

By applying Theorems B and C and Lemma 1 in the first term, we have Z ∞ −∞ [(σhf ) (x, u) − (σhS2m+1f ) (x, u)]dσ (u) p(·) ≤ K_p(·) Z ∞ −∞ k(σhf ) (·, u) − (σhS2m+1f ) (·, u)k_p(·)dσ (u) = K_p(·) Z ∞ −∞ k(σ_h(f − S₂m+1f )) (·, u) k_p(·)dσ (u) ≤ c5Kp(·) Z ∞ −∞ kf − S₂m+1f k_p(·)dσ (u) ≤ c5Kp(·)E2m+1(f )_p(·) Z ∞ −∞ dσ (u) ≤ c6E2m+1(f )_p(·).

Without loss of generality, we suppose that the Fourier series of f is

∞ X k=1 ˆ Ak(x) := ∞ X k=1 ˆ fkeikx. Then Z ∞ −∞ (σhS2m+1f ) (x, u) dσ (u)

= Z ∞ −∞ [ 1 2h Z h −h S₂m+1f (x + tu) dt]dσ (u) = Z ∞ −∞ [ 1 2h Z h −h 2m+1₋₁ X k=1 ˆ fkeik(x+tu)dt]dσ (u) = Z ∞ −∞ [ 1 2h 2m+1₋₁ X k=1 ˆ fkeikx Z h −h eiktudt]dσ (u) = Z ∞ −∞ [ 1 2h 2m+1₋₁ X k=1 ˆ Ak(x) Z h −h eiktudt]dσ (u) = 2m+1₋₁ X k=1 ˆ Ak(x) Z ∞ −∞ eikhu− e−ikhu 2ikhu dσ (u) = 2m+1₋₁ X k=1 ˆ Ak(x) ˆσ (kh) , (3.2)

and hence by (3.2) and (3.1)

(3.3) D (f, σ, h, p(·)) ≤ 2m+1₋₁ X k=1 ˆ Ak(x) ˆσ (kh) p(·) + c6E2m+1(f )_p(·).

Now, by inequality (a + b)p < ap+ bp, which holds for every positive numbers a and b in the case of 0 < p < 1 and by Lemma 2

2m+1₋₁ X k=1 ˆ Ak(x) ˆσ (kh) p(·) ≤ c   m X k=0       2k+1−1 X l=2k ˆ Al(x) ˆσ (lh)       2  1/2 p(·) : = c m X k=0 ∆2_k,σ !1/2 p(·) < c m X k=0 ∆k,σ p(·) ≤ c m X k=0 k∆_k,σk_p(·).

On the other hand, applying the Abel transformation to the sum ∆k,σ= 2k+1₋₁ X l=2k ˆ Al(x) ˆσ (lh) we have ∆k,σ = 2k+1₋₁ X l=2k S_l(f, x) − S₂k+1₋₁(f, x) [ˆσ (lh) − ˆσ ((l + 1) h)] +S₂k+1₋₁(f, x) − S₂k₋₁(f, x) ˆσ  2kh  and then k∆_k,σk_p(·)≤ 2k+1−1 X l=2k kS_l(f ) − S₂k+1(f )k_p(·)|ˆσ (lh) − ˆσ ((l + 1) h)| + S₂k+1₋₁(f ) − S₂k₋₁(f ) p(·)   σˆ  2kh    = kS₂k(f ) − S₂k+1(f )k_p(·)   σˆ  2kh− ˆσ2k+ 1h  +... + S₂k₊₁(f ) − S₂k+1(f ) p(·)   σˆ  2k+ 1  h  − ˆσ  2k+ 2  h    + S₂k+1₋₁(f ) − S₂k₋₁(f ) p(·)   σˆ  2kh  ≤hkS₂k(f ) − f k_p(·)+ kS₂k+1(f ) − f k_p(·) i   σˆ  2kh  − ˆσ  2k+ 1  h   + ... + h S₂k₊₁(f ) − f p(·)+ kS2k+1(f ) − f k_p(·) i   σˆ  2k+1− 1h  − ˆσ  2kh    +h S₂k+1₋₁(f ) − f p(·)+ S₂k₋₁(f ) − f p(·) i   ˆσ  2kh  ≤ cE₂k₋₁(f )_p(·)δ₂k_,h. Hence 2m+1₋₁ X k=1 ˆ Ak(x) ˆσ (kh) p(·) ≤ c m X r=0 E₂k₋₁(f )_p(·)δ₂k_,h

and by (3.3) we obtain the required inequality. 

Proof of Theorem 2. Let f ∈ Lp(·)_2π . Repeating the techniques used for the estimation of the quantity D (f, σ, h, p(·)) from Theorem 1, we have

(3.4) D (f, σ1, h, p(·)) ≤ Z ∞ −∞ (σhS2m+1f ) (·, u) dσ₁(u) p(·) + cp(·)E2m+1(f )_p(·).

On the other by (3.2) and Lemma 1 Z ∞ −∞ (σhS2m+1f ) (x, u) dσ₁(u) p(·) = 2m+1−1 X k=1 ˆ Ak(x) ˆσ1(kh) p(·) = 2m+1₋₁ X k=1 ˆ Ak(x) ˆσ2(kh) F (kh) p(·) = 2m+1₋₁ X k=1 ˆ fkeikxσˆ2(kh) F (kh) p(·) ≤ c 2m+1₋₁ X k=1 ˆ fkeikxˆσ2(kh) p(·) = c Z ∞ −∞ (σhS2m+1f ) (x, u) dσ₂(u) p(·) = c Z ∞ −∞ S₂m+1(σ_hf ) (x, u) dσ₂(u) p(·) = c S2m+1 Z ∞ −∞ (σhf ) (x, u) dσ2(u) p(·) ≤ Z ∞ −∞ σhf (x, u) dσ2(u) p(·) . The last inequality together with (3.4) implies the required relation.  Acknowledgments. The authors sincerely thank the anonymous reviewers for their careful reading and constructive comments. This work was supported by TUBITAK grant 114F422: Approximation Problems in the Variable Exponent Lebesgue Spaces.

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Received 6 March 2015 Balikesir University,

Faculty of Art and Science, Department of Mathematics,

10145, Balikesir, Turkey ANAS Institute of Mathematics

and Mechanics, Baku, Azerbaijan mdaniyal@balikesir.edu.tr

Balikesir University, Faculty of Art and Science, Department of Mathematics,

10145, Balikesir, Turkey elife.yirtici@gmail.com