Approximation problems in the variable exponent Lebesgue spaces Daniyal Israfilov & Ahmet Testici Balikesir University 25 August 2017 Fourier 2017 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
In this talk we discuss the approximation problems in the variable exponent Lebesgue spaces. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
1 CONSIDERED PROBLEMS Direct problems of approxmation theory in L p ( · ) ([ 0 , 2 π ]) 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
1 CONSIDERED PROBLEMS Direct problems of approxmation theory in L p ( · ) ([ 0 , 2 π ]) Inverse problems of approximation theory in L p ( · ) ([ 0 , 2 π ]) 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
1 CONSIDERED PROBLEMS Direct problems of approxmation theory in L p ( · ) ([ 0 , 2 π ]) Inverse problems of approximation theory in L p ( · ) ([ 0 , 2 π ]) Direct problems in variable exponent Smirnov classes 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
1 CONSIDERED PROBLEMS Direct problems of approxmation theory in L p ( · ) ([ 0 , 2 π ]) Inverse problems of approximation theory in L p ( · ) ([ 0 , 2 π ]) Direct problems in variable exponent Smirnov classes Inverse problems in variable exponent Smirnov classes 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
1 CONSIDERED PROBLEMS Direct problems of approxmation theory in L p ( · ) ([ 0 , 2 π ]) Inverse problems of approximation theory in L p ( · ) ([ 0 , 2 π ]) Direct problems in variable exponent Smirnov classes Inverse problems in variable exponent Smirnov classes Constructive characterization problems 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
2 INTRODUCTION The variable exponent Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p ( · ) . 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
2 INTRODUCTION The variable exponent Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p ( · ) . This space originates to: 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
2 INTRODUCTION The variable exponent Lebesgue spaces are a generalization of the classical Lebesgue spaces, replacing the constant exponent p with a variable exponent function p ( · ) . This space originates to: Orlicz W. : Über konjugierte Exponentenfolgen , Studia Math. 3, (1931), pp. 200-212. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Interest in the variable exponent Lebesgue spaces has increased since 1990s, because of their use in the different applications problems in mechanic, especially in fluid dynamic for the modelling of electrorheological fluids. These are fluids whose viscosity chances (often dramatically) when exposed to an electric field. The variable exponent Lebesgue spaces are also used in the study of image processing and some physical problems. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
See, for example the monographs: 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
See, for example the monographs: R˚ uˇ ziˇ cka M. : Elektrorheological Fluids: Modeling and Mathematical Theory , Springer, (2000). 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
See, for example the monographs: R˚ uˇ ziˇ cka M. : Elektrorheological Fluids: Modeling and Mathematical Theory , Springer, (2000). Cruz-Uribe D. V. and Fiorenza A. : Variable Lebesgue Spaces Foundation and Harmonic Analysis. Birkhäsuser, (2013), 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
See, for example the monographs: R˚ uˇ ziˇ cka M. : Elektrorheological Fluids: Modeling and Mathematical Theory , Springer, (2000). Cruz-Uribe D. V. and Fiorenza A. : Variable Lebesgue Spaces Foundation and Harmonic Analysis. Birkhäsuser, (2013), Diening L., Harjulehto P., Hästö P., Michael Ruzicka M.: Lebesgue and Sobolev Spaces with Variable Exponents , Springer, Heidelberg Dordrecht London New York(2011). 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Nowadays there are sufficiently wide investigations relating to the fundamental problems of these spaces, in view of potential theory, maximal and singular integral operator theory and others. The detailed presentation of the corresponding results can be found in the monographs mentioned above. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Nowadays there are sufficiently wide investigations relating to the fundamental problems of these spaces, in view of potential theory, maximal and singular integral operator theory and others. The detailed presentation of the corresponding results can be found in the monographs mentioned above. Some of the fundamental problems of approximation theory in the variable exponent Lebesgue spaces of periodic and non periodic functions defined on the intervals of real line were studied and solved by Sharapudinov. The detailed information can be found in the monograph: 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Nowadays there are sufficiently wide investigations relating to the fundamental problems of these spaces, in view of potential theory, maximal and singular integral operator theory and others. The detailed presentation of the corresponding results can be found in the monographs mentioned above. Some of the fundamental problems of approximation theory in the variable exponent Lebesgue spaces of periodic and non periodic functions defined on the intervals of real line were studied and solved by Sharapudinov. The detailed information can be found in the monograph: Sharapudinov I. I. : Some questions of approximation theory in the Lebesgue spaces with variable exponent: Vladikavkaz, 2012. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Nowadays there are sufficiently wide investigations relating to the fundamental problems of these spaces, in view of potential theory, maximal and singular integral operator theory and others. The detailed presentation of the corresponding results can be found in the monographs mentioned above. Some of the fundamental problems of approximation theory in the variable exponent Lebesgue spaces of periodic and non periodic functions defined on the intervals of real line were studied and solved by Sharapudinov. The detailed information can be found in the monograph: Sharapudinov I. I. : Some questions of approximation theory in the Lebesgue spaces with variable exponent: Vladikavkaz, 2012. Meanwhile, the approximation problems in these spaces, especially in the complex plane were not investigated sufficiently wide. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Let T : = [ 0 , 2 π ] and let p ( · ) : T → [ 0 , ∞ ) be a Lebesgue measurable 2 π periodic function such that p ( x ) : = p + < ∞ . 1 ≤ p − : = ess inf x ∈ T p ( x ) ≤ ess sup x ∈ T 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
Let T : = [ 0 , 2 π ] and let p ( · ) : T → [ 0 , ∞ ) be a Lebesgue measurable 2 π periodic function such that p ( x ) : = p + < ∞ . 1 ≤ p − : = ess inf x ∈ T p ( x ) ≤ ess sup x ∈ T In addition to this requirement if 2 π | p ( x ) − p ( y ) | ln | x − y | ≤ d , ∀ x , y ∈ [ 0 , 2 π ] with a positive constant d , then we say that p ( · ) ∈ P ( T ) . We also define P 0 ( T ) : = { p ( · ) ∈ P ( T ) : p − > 1 } . 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
The variable exponent Lebesgue space L p ( · ) ( T ) is defined as the set of all Lebesgue measurable 2 π periodic functions f such that 2 π � | f ( x ) | p ( x ) dx < ∞ . ρ p ( · ) ( f ) : = 0 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
The variable exponent Lebesgue space L p ( · ) ( T ) is defined as the set of all Lebesgue measurable 2 π periodic functions f such that 2 π � | f ( x ) | p ( x ) dx < ∞ . ρ p ( · ) ( f ) : = 0 Equipped with the norm � � � f � p ( · ) = inf λ > 0 : ρ p ( · ) ( f / λ ) ≤ 1 it becomes a Banach space. 25 August 2017 Fourier 2017 Daniyal Israfilov & Ahmet Testici Balikesir University () Approximation / 45
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