The generalized localization for multiple Fourier integrals

Ashurov, Ravshan; Ahmedov, Anvarjon; Mahmud, Ahmad Rodzi b.

doi:10.1016/j.jmaa.2010.06.014

Cited by 14 publications

(5 citation statements)

References 11 publications

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“…This localization principle generalizes the classical Riemann localization principle and for L p functions was intensively investigated by Sjölin 2 , Carbery and Soria 3, 4 , Bastis 5-7 , and Ashurov et al8 . It was established that R n localization holds true in L p , where p ∈ 2, 2n/ n − 1 and fails otherwise.…”

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confidence: 74%

On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

Ashurov¹,

Butaev²,

Pradhan

2012

Abstract and Applied Analysis

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We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classicalLpspaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions.

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confidence: 74%

On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

Ashurov¹,

Butaev²,

Pradhan

2012

Abstract and Applied Analysis

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“…Note, if A(ξ) = |ξ| 2 , then the partial sums E(A, λ, f )(x) coincides with the spherical partial sums E √ λ f (x). In the paper [22], using the method of Carbery-Soria [13], the following theorem is proved. So in case of an arbitrary elliptic polynomial A(ξ) we have the same result as for the spherical partial sums E λ f (x).…”

Section: Generalized Localization Principle For the Multiple Fourier ...mentioning

confidence: 99%

“…The formulated theorems are easily transferred to the case of non-spherical partial sums of multiple Fourier series (see [22], [25]).…”

Section: Generalized Localization Principle For the Multiple Fourier mentioning

confidence: 99%

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Generalized localization for spherical partial sums of multiple Fourier series

Ashurov

2019

Доклады Академии наук

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It is well known, that Luzin's conjecture has a positive solution for one dimensional trigonometric Fourier series and it is still open for the spherical partial sums S λ f (x), f ∈ L 2 (T N ), of multiple Fourier series, while it has the solution for square and rectangular partial sums. Historically progress with solving Luzin's conjecture has been made by considering easier problems. One of such easier problems for S λ f (x) was suggested by V. A. Il'in in 1968 and this problem is called the generalized localization principle. In this paper we first give a short survey on convergence almost-everywhere of Fourier series and on generalized localization of Fourier integrals, then present a positive solution for the generalized localization problem for S λ f (x).

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“…Note that the localization problems of the spectral expansions of the distribution are investigated in [10]. For more references we refer the reader to [11]- [17].…”

Section: Introductionmentioning

confidence: 99%

The Uniform Convergence of Eigenfunction Expansions of Schrödinger Operator in the Nikolskii Classes ${H}_{p}^{\alpha }(\bar{\Omega })$

Jamaludin

Ahmedov

2017

J. Phys.: Conf. Ser.

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Abstract. Many boundary value problems in the theory of partial differential equations can be solved by separation methods of partial differential equations. When Schrödinger operator is considered then the influence of the singularity of potential on the solution of the partial differential equation is interest of researchers. In this paper the problems of the uniform convergence of the eigenfunction expansions of the functions from corresponding to the Schrödinger operator with the potential from classes of Sobolev are investigated. The spectral function corresponding to the Schrödinger operator is estimated in closed domain. The isomorphism of the Nikolskii classes is applied to prove uniform convergence of eigenfunction expansions of Schrödinger operator in closed domain.

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The generalized localization for multiple Fourier integrals

Cited by 14 publications

References 11 publications

On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

On Generalized Localization of Fourier Inversion Associated with an Elliptic Operator for Distributions

Generalized localization for spherical partial sums of multiple Fourier series

The Uniform Convergence of Eigenfunction Expansions of Schrödinger Operator in the Nikolskii Classes ${H}_{p}^{\alpha }(\bar{\Omega })$

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