Korovkin-type theorems and their statistical versions in grand Lebesgue spaces

Zeren, Yusuf; Исмайлов, М. И.; Karaçam, Cemil

doi:10.3906/mat-2003-21

Cited by 9 publications

(6 citation statements)

References 32 publications

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“…Let us extend the function f by zero to the entire axis R , i.e. [34,35]). Next, we need the fact that the singular integral is bounded in grand Lebesgue spaces.…”

Section: Some Concepts and Auxiliary Resultsmentioning

confidence: 99%

“…Proof. Let's take an arbitrary η > 0 and an arbitrary function [34], Lemma 3.1) there exists a function…”

Section: Theorem 32 the Exponential System {Ementioning

confidence: 99%

“…Questions of solvability of Dirichlet problems for elliptic equations were considered in [7,8]. Korovkin-type theorems, as well as spectral problems with a spectral parameter in boundary conditions in grand Lebesgue spaces, were studied in [34] and [35], respectively. Note that the problem considered in [35] in Morrey-type spaces was studied in [3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

Исмайлов

ACAR

Zeren

et al. 2022

Hacettepe Journal of Mathematics and Statistics

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Basis properties of exponential and trigonometric systems in grand Lebesgue spaces $ L_{p)} (-\pi,\pi) $ are studied. Based on a shift operator, we consider the subspace $G_{p)} (-\pi,\pi)$ of the space $ L_{p)} (-\pi,\pi) $, where continuous functions are dense, and the boundedness of the singular operator in this subspace is proved. We establish the basicity of exponential system $ \{{e^{int}}\}_{n\in Z}$ for $G_{p)} (-\pi,\pi)$ and the basicity of trigonometric systems $ \{{\sin{nt}}\}_{n\in N }$ and $ \{{\cos{nt}}\}_{n\in N_0}$ for $G_{p)} (0,\pi)$.

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“…Let us extend the function f by zero to the entire axis R , i.e. [34,35]). Next, we need the fact that the singular integral is bounded in grand Lebesgue spaces.…”

Section: Some Concepts and Auxiliary Resultsmentioning

confidence: 99%

“…Proof. Let's take an arbitrary η > 0 and an arbitrary function [34], Lemma 3.1) there exists a function…”

Section: Theorem 32 the Exponential System {Ementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

Исмайлов

ACAR

Zeren

et al. 2022

Hacettepe Journal of Mathematics and Statistics

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show abstract

“…The problems of analysis and approximation theory have been relatively well studied in Lebesgue spaces with variable summability index and Morrey spaces (see [7][8][9][10][11][12][13][14]). The above mentioned problems have begun to be studied in Grand Lebesgue spaces, and valuable results have been obtained in this direction (see [15,16]). The solvability problems of partial differential equations have also begun to be studied in the Sobolev spaces generated by these spaces (see [17][18][19][20][21][22][23][24][25][26][27]).…”

Section: Introductionmentioning

confidence: 99%

Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces

Bilalov¹,

Sadigova²,

Kasumov³

2023

Nonlinear Systems - Recent Developments and Advances

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In this chapter an m-th order elliptic equation is considered in Sobolev spaces generated by the norm of a grand Lebesgue space. Subspaces are determined in which the shift operator is continuous, and local solvability (in the strong sense) is established in these subspaces. It is established an interior and up-to boundary Schauder-type estimates with respect to these Sobolev spaces for m-th order elliptic operators, the trace of functions and trace operator are determined, the boundedness of trace operator and the extension theorem are proved, the properties of the Riesz potential are studied regarding these Sobolev spaces, etc. It is considered a second-order elliptic equation, and we study the Fredholmness of the Dirichlet problem in the Sobolev space generated by a separable subspace of the grand Lebesgue space. It is also considered one spectral problem for a discontinuous second-order differential operator and proved the theorem on the basicity of eigenfunctions of this operator in subspace of Morrey space, in which the infinitely differentiable functions with compact support are dense.

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“…The situation is different with the case of Morrey-type and grand Lebesgue spaces, and only recently the approximation matters began to be studied in these spaces. In this direction, various issues were studied in [2], [4]- [8], [11], [15]- [17], [26], [27].…”

Section: Introductionmentioning

confidence: 99%

On some properties of functions of the grand Hardy classes

ALIYAROVA¹,

KARACAM²,

ISMAILOV³

2023

BMJ

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Grand Hardy class H+p), p > 1, is defined and some properties of functions belonging to this class are studied in this work. Namely, the analogs of the Riesz and Smirnov theorems as well as the Cauchy’s formula for representation of function are proved. Necessary and sufficient condition for the validity of Riesz theorem in grand Hardy spaces H+p), p > 1, is found. Subspace GH+p)of the grand Hardy space H+p)generated by this condition is defined.

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Korovkin-type theorems and their statistical versions in grand Lebesgue spaces

Cited by 9 publications

References 32 publications

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

Some Solvability Problems of Differential Equations in Non-standard Sobolev Spaces

On some properties of functions of the grand Hardy classes

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