approximation problems in the variable exponent lebesgue...

Approximation problems in the variable exponentLebesgue spaces

Daniyal Israfilov & Ahmet TesticiBalikesir University

25 August 2017 Fourier 2017

Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation25 August 2017 Fourier 2017 1

In this talk we discuss the approximation problems in the variableexponent Lebesgue spaces.

1 CONSIDERED PROBLEMS

Direct problems of approxmation theory in Lp(·)([0, 2π])

Inverse problems of approximation theory in Lp(·)([0, 2π])

Direct problems in variable exponent Smirnov classes

Inverse problems in variable exponent Smirnov classes

Constructive characterization problems

2 INTRODUCTION

The variable exponent Lebesgue spaces are a generalization of theclassical Lebesgue spaces, replacing the constant exponent p with avariable exponent function p(·).

This space originates to:

Orlicz W. : Über konjugierte Exponentenfolgen, Studia Math. 3,(1931), pp. 200-212.

2 INTRODUCTION

The variable exponent Lebesgue spaces are a generalization of theclassical Lebesgue spaces, replacing the constant exponent p with avariable exponent function p(·).This space originates to:

2 INTRODUCTION

The variable exponent Lebesgue spaces are a generalization of theclassical Lebesgue spaces, replacing the constant exponent p with avariable exponent function p(·).This space originates to:

Interest in the variable exponent Lebesgue spaces has increased since1990s, because of their use in the different applications problems inmechanic, especially in fluid dynamic for the modelling ofelectrorheological fluids. These are fluids whose viscosity chances(often dramatically) when exposed to an electric field. The variableexponent Lebesgue spaces are also used in the study of imageprocessing and some physical problems.

See, for example the monographs:

Ruzicka M. : Elektrorheological Fluids: Modeling and MathematicalTheory, Springer, (2000).

Cruz-Uribe D. V. and Fiorenza A. : Variable Lebesgue SpacesFoundation and Harmonic Analysis. Birkhäsuser, (2013),

Diening L., Harjulehto P., Hästö P., Michael Ruzicka M.: Lebesgueand Sobolev Spaces with Variable Exponents, Springer, HeidelbergDordrecht London New York(2011).

Nowadays there are suffi ciently wide investigations relating to thefundamental problems of these spaces, in view of potential theory,maximal and singular integral operator theory and others. Thedetailed presentation of the corresponding results can be found in themonographs mentioned above.

Some of the fundamental problems of approximation theory in thevariable exponent Lebesgue spaces of periodic and non periodicfunctions defined on the intervals of real line were studied and solvedby Sharapudinov. The detailed information can be found in themonograph:

Sharapudinov I. I. : Some questions of approximation theory in theLebesgue spaces with variable exponent:Vladikavkaz, 2012.

Meanwhile, the approximation problems in these spaces, especially inthe complex plane were not investigated suffi ciently wide.

Let T := [0, 2π] and let p (·) : T→ [0,∞) be a Lebesguemeasurable 2π periodic function such that

1 ≤ p− := ess infx∈T

p (x) ≤ ess supx∈T

p (x) := p+ < ∞.

In addition to this requirement if

|p (x)− p (y)| ln 2π

|x − y | ≤ d , ∀x , y ∈ [0, 2π]

with a positive constant d , then we say that p (·) ∈ P (T). We alsodefine P0 (T) := p (·) ∈ P (T) : p− > 1.

Let T := [0, 2π] and let p (·) : T→ [0,∞) be a Lebesguemeasurable 2π periodic function such that

1 ≤ p− := ess infx∈T

p (x) ≤ ess supx∈T

p (x) := p+ < ∞.

In addition to this requirement if

|p (x)− p (y)| ln 2π

|x − y | ≤ d , ∀x , y ∈ [0, 2π]

with a positive constant d , then we say that p (·) ∈ P (T). We alsodefine P0 (T) := p (·) ∈ P (T) : p− > 1.

The variable exponent Lebesgue space Lp(·) (T) is defined as the setof all Lebesgue measurable 2π periodic functions f such that

ρp(·) (f ) :=2π∫0

|f (x)|p(x ) dx < ∞.

Equipped with the norm

‖f ‖p(·) = inf

λ > 0 : ρp(·) (f /λ) ≤ 1

it becomes a Banach space.

The variable exponent Lebesgue space Lp(·) (T) is defined as the setof all Lebesgue measurable 2π periodic functions f such that

ρp(·) (f ) :=2π∫0

|f (x)|p(x ) dx < ∞.

‖f ‖p(·) = inf

λ > 0 : ρp(·) (f /λ) ≤ 1

it becomes a Banach space.

One of the main problem observed in the investigations on theapproximation theory is the correct definition of themodulus of smoothness. It is a fact that Lp(·) (T) is noninvariant withrespect to the usual shift operator f (·+ h), in general.

Nevertheless, the Steklov mean value operator

σh (f ) :=1h

0f (x + t) dt, h > 0

is bounded in Lp(·) (T). See,

Diening L., Ruzicka M. : Calderon-Zigmund operators on generalizedLebesgue spaces Lp(x ) and problems related to fluid dynamic, J. ReineAngew. Math., Vol. 563, (2003), pp. 197-220).

By using this boundedness was constructed by us the first ordermodulus of smoothness

Ωp(·) (f , δ) := sup0<h≤δ

∥∥∥∥1h∫ h

0|f (·)− f (·+ t)| dt

∥∥∥∥p(·)

and was obtained the direct theorem of approximation theory in Lp(·) (T),p (·) ∈ P0 (T), and also some results on the approximation by theNörlund means of Fourier series in Lp(·) (T). See:

Guven A. and Israfilov D. M. : Trigonometric Approximation inGeneralized Lebesgue Spaces Lp(x ), Journal of Math. Inequalities,Vol. 4, No: 2, (2010), pp. 285-299.

By using this boundedness was constructed by us the first ordermodulus of smoothness

Ωp(·) (f , δ) := sup0<h≤δ

∥∥∥∥1h∫ h

0|f (·)− f (·+ t)| dt

∥∥∥∥p(·)

and was obtained the direct theorem of approximation theory in Lp(·) (T),p (·) ∈ P0 (T), and also some results on the approximation by theNörlund means of Fourier series in Lp(·) (T). See:

Guven A. and Israfilov D. M. : Trigonometric Approximation inGeneralized Lebesgue Spaces Lp(x ), Journal of Math. Inequalities,Vol. 4, No: 2, (2010), pp. 285-299.

Similar results under the condition of p (·) ∈ P0 (T) using some othermodulus of smoothness were stated or proved in the papers:

Israfilov D., Kokilashvili V., Samko S. : Approximation In WeightedLebesgue and Smirnov Spaces With Variable Exponents, Proceed. ofA. Razmadze Math. Institute, Vol 143, (2007), pp 25-35.

Akgun R. : Trigonometric Approximation of Functions in GeneralizedLebesgue Spaces With Variable Exponent, Ukranian Math. Journal,Vol. 63, No:1, (2011), pp. 3-23.

Akgun R. : Polynomial approximation of functions in weightedLebesgue and Smirnov spaces with nonstandard growth, GeorgianMath. Journal, 18, (2011), pp. 203-235.

Akgun R. and Kokilashvili V. M. : The refined direct and converseinequalities of trigonometric approximation in weighted variableexponent Lebesgue spaces, Georgian Math. Journal, 18, (2011), pp.399-423.

In the more general case, i.e. in the case of p (·) ∈ P (T) ⊃ P0 (T)using the modulus

Ω (f , δ)p(·) := sup0<h≤δ

∥∥∥∥1h∫ h

0[f (·)− f (·+ t)] dt

∥∥∥∥p(·)

which is more sensitive than Ωp(·) (f , δ) , the direct and inversetheorems were proved by Sharapudinov in the above cited hismonograph.

In term of Ω (f , δ)p(·) with p (·) ∈ P (T), one general inversetheorem which generalizes the inverse theorem obtained bySharapudinov was proved in the work:

Israfilov D. M. and Testici A. : Approximation in Smirnov Classeswith Variable Exponent, Complex Variables and Elliptic Equations,Vol. 60, No: 9, (2015), pp.1243-1253.

In term of Ω (f , δ)p(·) with p (·) ∈ P (T), one general inversetheorem which generalizes the inverse theorem obtained bySharapudinov was proved in the work:

Israfilov D. M. and Testici A. : Approximation in Smirnov Classeswith Variable Exponent, Complex Variables and Elliptic Equations,Vol. 60, No: 9, (2015), pp.1243-1253.

3 NEW RESULTS

We discuss some results obtained by us on the approximationproblems in Lp(·) (T), p (·) ∈ P (T), in the term of the r th( r = 1, 2, ...) modulus of smoothness Ωr (f , δ)p(·).

Let f ∈ Lp(·) (T) with p (·) ∈ P (T) and let

∆rt f (x) :=r

∑s=0(−1)r+s

)f (x + st) , r = 1, 2, ... .

Definition (1)We define the r -th modulus of smoothness as

Ωr (f , δ)p(·) := sup0<h≤δ

∥∥∥∥∥∥1hh∫0

∆rt fdt

∥∥∥∥∥∥p(·)

, δ > 0.

3 NEW RESULTS

We discuss some results obtained by us on the approximationproblems in Lp(·) (T), p (·) ∈ P (T), in the term of the r th( r = 1, 2, ...) modulus of smoothness Ωr (f , δ)p(·).

Let f ∈ Lp(·) (T) with p (·) ∈ P (T) and let

∆rt f (x) :=r

∑s=0(−1)r+s

)f (x + st) , r = 1, 2, ... .

Definition (1)We define the r -th modulus of smoothness as

Ωr (f , δ)p(·) := sup0<h≤δ

∥∥∥∥∥∥1hh∫0

∆rt fdt

∥∥∥∥∥∥p(·)

, δ > 0.

For f ∈ Lp(·) (T) we define the best approximation number

En (f )p(·) := inf‖f − Tn‖p(·) : Tn ∈ Πn

in the class Πn of the trigonometric polynomials of degree notexceeding n.

Throughout this talk by c(·), c(·, ·), c1(·, ·), c2(·, ·),... we denote theconstants (which can be different in different relations) dependingonly on the parameters given in the corresponding brackets.

The main direct and inverse results obtained in the spacesLp(·)([0, 2π]) are following.

Theorem (1)

Let p (·) ∈ P (T), r ∈N. Then there exists a positive constant c (p, r)such that for every f ∈ Lp(·) (T) and n ∈N the inequality

En (f )p(·) ≤ c(p, r)Ωr (f , 1/n)p(·)

holds.

Theorem (1)

En (f )p(·) ≤ c(p, r)Ωr (f , 1/n)p(·)

holds.

Theorem (1)

En (f )p(·) ≤ c(p, r)Ωr (f , 1/n)p(·)

holds.

Theorem (2)

Ωr (f , 1/n)p(·) ≤c(p, r)nr

∑k=0

(k + 1)r−1 Ek (f )p(·)

holds.

Denoting by

W p(·)k (T):=

f : f (k−1) is absolutely continuous and f (k ) ∈ Lp(·) (T)

k = 1, 2, ..., the variable exponent Sobolev space and combiningTheorem 1 with the estimation

En (f )p(·) ≤c(p)nk

En(f (k )

)p(·),

which can be deduced from Sharapudinov’s work : On Direct andInverse Theorems of Approximation Theory In Variable LebesgueSpace And Sobolev Spaces, Azerbaijan Journal of Math., Vol. 4, No1, (2014), pp. 55-72., we have

Corollary (1)

Let p (·) ∈ P (T), k ∈N. Then there exists a positive constant c (p, r)

such that for every f ∈ W p(·)k (T) and, n ∈N the following inequality

En (f )p(·) ≤c(p, r)nk

(f (k ), 1/n

)p(·).

On the other hand, Theorem 2 implies

Corollary (2)

If En (f )p(·) = O (n−α), α > 0, then under the conditions of Theorem 2

Ωr (f , δ)p(·) =

O (δα) , r > α

O (δα log (1/δ)) , r = αO (δr ) , r < α.

Hence, if we define a generalized Lipschitz class Lipp(·)α (T) for α > 0and r := [α] + 1 ([α] is the integer part of α) as

Lipp(·)α (T) :=f ∈ Lp(·) (T) : Ωr (f , δ)p(·) = O (δ

α) , δ > 0,

then we have

Corollary (3)

If En (f )p(·) = O (n−α), α > 0, then under the conditions of Theorem 2,

f ∈ Lipp(·)α (T).

On the other hand, from Theorem 1 we also get

Corollary (4)

If f ∈ Lipp(·)α (T) with p (·) ∈ P (T) and for some α > 0, thenEn (f )p(·) = O (n−α).

Now Corollaries 3 and 4 imply

Theorem (3)

Let f ∈ Lp(·) (T), p (·) ∈ P (T), and let α > 0. The following statementsare equivalent:

i)f ∈ Lipp(·)α (T) ,

ii)En (f )p(·) = O(n−α

), n ∈N.

Note that when p (·) =constant these results coincide with theclassical results, proved by different authors.

Now Corollaries 3 and 4 imply

Theorem (3)

Let f ∈ Lp(·) (T), p (·) ∈ P (T), and let α > 0. The following statementsare equivalent:

i)f ∈ Lipp(·)α (T) ,

ii)En (f )p(·) = O(n−α

), n ∈N.

Note that when p (·) =constant these results coincide with theclassical results, proved by different authors.

4 NEW RESULT IN THE COMPLEX DOMAINS

Let G ⊂ C be a finite domain in the complex plane, bounded by arectifiable Jordan curve Γ and let G−:= Ext Γ. Let alsoT:= w ∈ C : |w | = 1, D := Int T and D−:= Ext T.

Definition (2)

The variable exponent Lebesgue spaces Lp(·)(Γ) for a given nonnegativeLebesgue measurable variable exponent p(z) ≥ 1 on Γ we define as the setof Lebesgue measurable functions f , such that∫

|f (z)|p(z ) |dz |< ∞.

4 NEW RESULT IN THE COMPLEX DOMAINS

Let G ⊂ C be a finite domain in the complex plane, bounded by arectifiable Jordan curve Γ and let G−:= Ext Γ. Let alsoT:= w ∈ C : |w | = 1, D := Int T and D−:= Ext T.

Definition (2)

The variable exponent Lebesgue spaces Lp(·)(Γ) for a given nonnegativeLebesgue measurable variable exponent p(z) ≥ 1 on Γ we define as the setof Lebesgue measurable functions f , such that∫

|f (z)|p(z ) |dz |< ∞.

‖f ‖Lp(·)(Γ) := inf

λ ≥ 0 :∫Γ

∣∣∣∣ f (z)λ

∣∣∣∣p(z ) |dz | ≤ 1< ∞

Lp(·)(Γ) becomes a Banach spaces.

In the case of Γ := T we obtain the variable exponent Lebesgue spaceLp(·)(T) with the norm

‖f ‖Lp(·)(T) := inf

λ ≥ 0 :2π∫0

∣∣∣∣ f (e it )λ

∣∣∣∣p(eit )

|dt| ≤ 1

=: ‖f ‖Lp(·)([0,2π]) .

Let E 1(G ) be the classical Smirnov class of analytic functions in G .The Smirnov classes in detail were investigated in the monograph:

Goluzin G. M. : Geometric Theory of Functions of a ComplexVariable. Translation of Mathematical Monographs, Vol. 26, AMS1969.

Definition (3)

Let p (·) : Γ→ [1,∞) be a Lebesgue measurable function. The set

E p(·)(G ):=f ∈ E 1(G ) : f ∈ Lp(·)(Γ)

is called the variable exponent Smirnov class of analytic functions in G .

Let E 1(G ) be the classical Smirnov class of analytic functions in G .The Smirnov classes in detail were investigated in the monograph:

Goluzin G. M. : Geometric Theory of Functions of a ComplexVariable. Translation of Mathematical Monographs, Vol. 26, AMS1969.

Definition (3)

Let p (·) : Γ→ [1,∞) be a Lebesgue measurable function. The set

E p(·)(G ):=f ∈ E 1(G ) : f ∈ Lp(·)(Γ)

is called the variable exponent Smirnov class of analytic functions in G .

In particular if G := D, then we have variable exponent Hardy spacesHp(·)(D).Let Γ be a Jordan rectifiable curve in the complex plane C and letp (·) : Γ→ R+ be a measurable function defined on Γ such that

1 ≤ p− := ess infz∈Γ

p(z) ≤ ess supz∈Γ

p(z) := p+ < ∞. (1)

Definition (4)

We say that p (·) ∈ P(Γ), if p (·) satisfies the conditions (1 ) and

|p(z1)− p(z2)| ln|Γ|

|z1 − z2|≤ c , ∀z1, z2∈ Γ

with a positive constant c , where |Γ| is the Lebesgue measure of Γ.

If p (·) ∈ P(Γ) with p− > 1, then we say that p (·) ∈ P0(Γ).

Let Γ be a smooth Jordan curve and let θ (s) be the angle betweenthe tangent and the positive real axis expressed asa function of arclength s. If Γ has a modulus of continuity ω (θ, s),satisfying the Dini-smooth condition

δ∫0

ω (θ, s) /s ds < ∞, δ > 0,

then we say that Γ is a Dini smooth curve and the set of Dini-smoothcurves we denote by D.

By ϕ we denote the conformal mapping of G− onto D−, normalizedby the conditions

ϕ (∞) = ∞ and limz→∞

ϕ (z) /z > 0.

Let ψ be the inverse mapping of ϕ.

The mappings ϕ and ψ have continuous extensions to Γ and T,respectively. Their derivatives ϕ′ and ψ′ have definite nontangentiallimit values a.e. on Γ and T, and the limit functions are integrablewith respect to Lebesgue measure on Γ and T, respectively.

For a given function f ∈ Lp(·)(Γ) with p ∈ P(Γ) we set

f0 (w) := f [ψ (w)]

p0(w) := p(ψ (w)).

ϕ (z) /z > 0.

f0 (w) := f [ψ (w)]

p0(w) := p(ψ (w)).

ϕ (z) /z > 0.

f0 (w) := f [ψ (w)]

p0(w) := p(ψ (w)).

If Γ ∈ D, then as follows from Warschawski’s works, there are thepositive constants ci > 0, i = 1, 2, 3, 4 such that

0 < c1 ≤∣∣∣ψ′ (w)∣∣∣ ≤ c2 < ∞,

0 < c3 ≤∣∣∣ϕ′ (z)∣∣∣ ≤ c4 < ∞,

a.e. on T and on Γ, respectively.

Therefore if Γ ∈ D, then

f ∈ Lp(·)(Γ)⇔ f0 ∈ Lp0(·)(T).

If Γ ∈ D, then as follows from Warschawski’s works, there are thepositive constants ci > 0, i = 1, 2, 3, 4 such that

0 < c1 ≤∣∣∣ψ′ (w)∣∣∣ ≤ c2 < ∞,

0 < c3 ≤∣∣∣ϕ′ (z)∣∣∣ ≤ c4 < ∞,

a.e. on T and on Γ, respectively.Therefore if Γ ∈ D, then

f ∈ Lp(·)(Γ)⇔ f0 ∈ Lp0(·)(T).

Moreover,

‖f0‖Lp0 (T)≤ c9 ‖f ‖Lp(·)(Γ)≤ c10 ‖f0‖Lp0 (·)(T)

It is also clear that if Γ ∈ D, then

p0(·) ∈ P(T)⇔ p(·) ∈ P(Γ).

For a given function f ∈ Lp(·) (Γ) we define the Cauchy type integral

f +0 (w) :=12πi

f0 (τ)τ − w dτ

which are analytic in D.

Moreover,

‖f0‖Lp0 (T)≤ c9 ‖f ‖Lp(·)(Γ)≤ c10 ‖f0‖Lp0 (·)(T)

p0(·) ∈ P(T)⇔ p(·) ∈ P(Γ).

f +0 (w) :=12πi

f0 (τ)τ − w dτ

Moreover,

‖f0‖Lp0 (T)≤ c9 ‖f ‖Lp(·)(Γ)≤ c10 ‖f0‖Lp0 (·)(T)

p0(·) ∈ P(T)⇔ p(·) ∈ P(Γ).

f +0 (w) :=12πi

f0 (τ)τ − w dτ

For a given f ∈ Lp(·)(T), defining the mean value function on theunit circle T as

σhf (w) :=1h

f(we it

)dt, w ∈T

we obtain the following modification of the modulus of smoothness off on T:

Ω (f , δ)T,p(·) := sup0<h≤δ

‖f (w)− σhf (w)‖Lp(·)(T) .

If f ∈ E p(·)(G ), then we define the modulus of smoothness

Ω (f , δ)G ,p(·) := Ω(f +0 , δ

)T,p0(·) , δ > 0

for f .

For a given f ∈ Lp(·)(T), defining the mean value function on theunit circle T as

σhf (w) :=1h

f(we it

)dt, w ∈T

we obtain the following modification of the modulus of smoothness off on T:

Ω (f , δ)T,p(·) := sup0<h≤δ

‖f (w)− σhf (w)‖Lp(·)(T) .

If f ∈ E p(·)(G ), then we define the modulus of smoothness

Ω (f , δ)G ,p(·) := Ω(f +0 , δ

)T,p0(·) , δ > 0

for f .

The best approximation number of f ∈ E p(·)(G ) is defined by

En (f )G ,p(·) := inf‖f − Pn‖Lp(·)(Γ) : Pn ∈ Πn

where Πn is the class of algebraic polynomials of degree notexceeding n.

For simplicity we are formulate the results only for the first modulus.For the higher moduli the appropriate results also are true.

Then the direct and inverse results obtained in the classes E p(·)(G )can be formulated as following:

Theorem (4)

Let Γ ∈ D. If f ∈ E p(·)(G ) with p(·) ∈ P0(Γ), then

En (f )G ,p(·) ≤ c (p) Ω (f , 1/n)G ,p(·)

with a constant c > 0 independent of n.

Theorem (5)

Ω (f , 1/n)G ,p(·) ≤c (p)n

∑v=0

Ev (f )G ,p(·) n = 1, 2, ...,

with a constant c > 0 independent of n.Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation25 August 2017 Fourier 2017 36

For simplicity we are formulate the results only for the first modulus.For the higher moduli the appropriate results also are true.Then the direct and inverse results obtained in the classes E p(·)(G )can be formulated as following:

Theorem (4)

En (f )G ,p(·) ≤ c (p) Ω (f , 1/n)G ,p(·)

with a constant c > 0 independent of n.

Theorem (5)

Ω (f , 1/n)G ,p(·) ≤c (p)n

∑v=0

Ev (f )G ,p(·) n = 1, 2, ...,

with a constant c > 0 independent of n.Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation25 August 2017 Fourier 2017 36

Defining the generalized Lipschitz class Lipp(·) (G , α) with α ∈ (0, 1)by

Lipp(·) (G , α) :=f ∈ E p(·)(G ) : Ω (f , δ)G ,p(·) = O (δ

α) , δ > 0,

from Theorem (5) after simple computations we obtain:

Corollary (5)

Let Γ ∈ D and p(·) ∈ P0(Γ). If En (f )G ,p(·) = O (n−α) with α ∈ (0, 1),then f ∈ Lipp(·) (G , α).

At the same time Theorem (4) implies

Corollary (6)

If f ∈ Lipp(·) (G , α) with p(·) ∈ P0(Γ) and α ∈ (0, 1), then

En (f )G ,p(·)= O(n−α

Defining the generalized Lipschitz class Lipp(·) (G , α) with α ∈ (0, 1)by

Lipp(·) (G , α) :=f ∈ E p(·)(G ) : Ω (f , δ)G ,p(·) = O (δ

α) , δ > 0,

from Theorem (5) after simple computations we obtain:

Corollary (5)

Let Γ ∈ D and p(·) ∈ P0(Γ). If En (f )G ,p(·) = O (n−α) with α ∈ (0, 1),then f ∈ Lipp(·) (G , α).

At the same time Theorem (4) implies

Corollary (6)

If f ∈ Lipp(·) (G , α) with p(·) ∈ P0(Γ) and α ∈ (0, 1), then

En (f )G ,p(·)= O(n−α

The corollaries (5) and (6) imply the following constructivecharacterization of Lipp(·) (G , α):

Theorem (6)

Let Γ ∈ D and p(·) ∈ P0(Γ), and let α ∈ (0, 1). The following statementsare equivalent:

i f ∈ Lipp(·) (G , α) , ii) En (f )G ,p(·) = O(n−α

), n = 1, 2, 3, ..

Acknowledgement

This work was supported by TUBITAK grant 114F422: "ApproximationProblems in the Variable Exponent Lebesgue Spaces".

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