Basicity of a system of exponents with a piecewise linear phase in Morrey-typespaces

Bilalov, B. T.; Seyidova, Fidan

doi:10.3906/mat-1901-113

Cited by 18 publications

(6 citation statements)

References 22 publications

(51 reference statements)

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“…This directly implies the nonseparability of the space L p) (0; 1) . Similar to Morrey space (see [9][10][11][12]49]) we construct the needed separable subcpace of L p) (−π, π) as follows. For ∀f ∈ L p) (−π; π) and ∀δ > 0 we set…”

Section: Preliminaries and Some Related Resultsmentioning

confidence: 99%

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

Zeren¹,

Исмайлов²,

Karaçam³

2020

Turk J Math

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The analogs of Korovkin theorems in grand-Lebesgue spaces are proved. The subspace G p) (−π; π) of grand Lebesgue space is defined using shift operator. It is shown that the space of infinitely differentiable finite functions is dense in G p) (−π; π). The analogs of Korovkin theorems are proved in G p) (−π; π). These results are established in G p) (−π; π) in the sense of statistical convergence. The obtained results are applied to a sequence of operators generated by the Kantorovich polynomials, to Fejer and Abel-Poisson convolution operators.

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Section: Preliminaries and Some Related Resultsmentioning

confidence: 99%

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

Zeren¹,

Исмайлов²,

Karaçam³

2020

Turk J Math

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“…Compared with other areas of mathematics, the apparatus of harmonic analysis has been fairly well studied in relation to these spaces. 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]).…”

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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“…Approximation properties are also of interest in suchlike spaces. These properties have been relatively well studied in generalized Lebesgue spaces in the works [2,4,9,13,15,17,29,[31][32][33]. These properties are different with the cases of Morrey-type and grand Lebesgue spaces: only recently the approximation matters began to be studied in these spaces, and many problems are still open.…”

Section: Introductionmentioning

confidence: 99%

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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Basicity of a system of exponents with a piecewise linear phase in Morrey-typespaces

Cited by 18 publications

References 22 publications

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

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

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

On basicity of exponential and trigonometric systems in grand Lebesgue spaces

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