Convergence Problems of Multiple Trigonometric Series and Spectral Decompositions. I

Alimov, Sh. A.; Ильин, В. А.; Никишин, Е. М.

doi:10.1070/rm1976v031n06abeh001575

Cited by 52 publications

(11 citation statements)

References 34 publications

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“…By applying the Hörmander Tauberian theorem [2] to the function ϕ(λ) = E λ f (x), we obtain the following assertion. (Lemma 1 provides the necessary estimate for the jump of the function ϕ.…”

Section: Proof Of the Theoremmentioning

confidence: 93%

On Eigenfunction Expansions Associated with the Schrodinger Operator with a Singular Potential

Ashurov

Faiziev

2005

Diff Equat

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In the present paper, we prove the uniform convergence and convergence in mean of the Riesz means of eigenfunction expansions associated with the Schrödinger operator with a singular potential satisfying the Stummel condition. This is the widest class of potentials suitable for the analysis of uniform convergence of eigenfunction expansions, since if the potential does not satisfy the Stummel condition (i.e., has a stronger singularity), then all continuous eigenfunctions of the Schrödinger operator vanish at the point of singularity of the potential [1].Convergence and integrability issues for eigenfunction expansions associated with elliptic operators with smooth coefficients were studied by numerous authors (see the survey [2]). MAIN DEFINITIONS. STATEMENT OF RESULTSLet Ω be an arbitrary bounded domain in R N (N ≥ 3) with smooth boundary, and let q ∈ L 2 (Ω) be a nonnegative function. Consider the Schrödinger operator H = −∆ + q with domain C ∞ 0 (Ω) and denote an arbitrary nonnegative self-adjoint extension of H with discrete spectrum byĤ. Let 0 ≤ λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · be the eigenvalues ofĤ, and let {u n } ∞ n=1 be the corresponding complete orthonormal system of eigenfunctions.For each s ≥ 0, we define the sth-order Riesz mean of the eigenfunction expansion of a function f ∈ L 2 (Ω) by the formulaWe take a positive continuous function ω(t) defined for t > 0 such thatDefinition. We say that a positive function q ∈ L 2 (Ω) satisfies the generalized Stummel condition if there exists a constant C such that |x−y|≤1 |q(y)| 2 |x − y| N −4 ω(|x − y|) dy < C, x ∈ Ω.(2) (The function q is continued by zero outside Ω.)

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Section: Proof Of the Theoremmentioning

confidence: 93%

On Eigenfunction Expansions Associated with the Schrodinger Operator with a Singular Potential

Ashurov

Faiziev

2005

Diff Equat

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“…The following result gives a complete solution of the generalized localization problem for the spherical partial sums of the multiple Fourier series. It may be worth mentioning that this problem was first formulated in 1976 in a review paper [2]. Thus the problem of generalized localization for S λ is completely solved in classes Lp(T N ), p ≥ 1: if p ≥ 2 then we have the generalized localization and if p < 2, then the generalized localization fails.…”

Section: Generalized Localization Principle For the Multiple Fourier mentioning

confidence: 99%

“…In other words, does Carleson's theorem extend to spherical partial sums (1.1)? The answer to this question is unknown so far (see [1] and [2]). What is known is only that Hunt's theorem does not extend to partial sums (1.1) [3].…”

Section: Introductionmentioning

confidence: 99%

“…What is known is only that Hunt's theorem does not extend to partial sums (1.1) [3]. Namely, for each p ∈ [1,2) there exists a function f ∈ Lp(T N ) such that on a set of positive measure lim λ→∞ |S λ f (x)| = +∞.…”

Section: Introductionmentioning

confidence: 99%

“…In 1922 A.N.Kolmogorov constructed a remarkable example of a function f ∈ L1(T), whose Fourier series diverges a.e. and even at every point of T (see [1] and [2]). After this result some specialists have doubted the validity of the the Luzin's conjecture, since Kolmogorov's counterexample in L1 was unbounded in any interval, but it was thought to be only a matter of time before a continuous counterexample was found.…”

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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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2001

Encyclopaedia of Mathematics, Supplement III

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Convergence Problems of Multiple Trigonometric Series and Spectral Decompositions. I

Cited by 52 publications

References 34 publications

On Eigenfunction Expansions Associated with the Schrodinger Operator with a Singular Potential

On Eigenfunction Expansions Associated with the Schrodinger Operator with a Singular Potential

Generalized localization for spherical partial sums of multiple Fourier series

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