Problems of Convergence of Multiple Trigonometric Series and Spectral Decompositions. Ii

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

doi:10.1070/rm1977v032n01abeh001600

Cited by 23 publications

(20 citation statements)

References 1 publication

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“…The following result, also contained in [1], easily follows from the previous arguments. The example of characteristic functions of balls shows that for spherical partial sums of piecewise smooth functions, the localization may fail at least in dimension N > 3.…”

Section: Then For the Spherical Partial Sums Of F (X ) Localization Hmentioning

confidence: 60%

“…It was shown by Tonelli [23] for multiple Fourier series and by Bochner [4] for multiple Fourier integrals that localization may not hold when N > 1. See also [1]. The idea is that the spherical partial sum operators can be expressed as convolutions against kernels which are unbounded even outside the origin.…”

Section: Localization For Fourier Integrals In R Nmentioning

confidence: 99%

“…We want to mention that expansions in eigenfunctions of elliptic operators have been extensively studied (see, e.g., the survey by Alimov-II'in-Nikishin [1] and the references there). In particular, these authors proved definitive results for localization and pointwise convergence of expansions of functions in Sobolev spaces.…”

Section: < C If(t)ldtmentioning

confidence: 99%

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Localization and convergence of eigenfunction expansions

Brandolini

Colzani

1999

The Journal of Fourier Analysis and Applications

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ABSTRACT. We state a localization principle for expansions in eigenfunctions of a self-adjoint second order elliptic operator and we prove an equiconvergence result between eigenfunction expansions and trigonometric expansions. We then study the Gibbs phenomenon for eigenfunction expansions of piecewise smooth functions on two-dimensional manifolds.Two of the most elementary but basic results in harmonic analysis are the theorem of Dirichlet on the convergence of Fourier series and the localization principle of Riemann. The Riemann localization principle states that the behavior of the partial sums of a Fourier series at a given point depends only on the behavior of the function in an arbitrary small neighborhood of this point. The Dirichlet theorem, originally stated for functions with a finite number of maxima and minima, when applied to piecewise smooth functions guarantees the convergence at every point of the partial sums of Fourier series to the function expanded. At a discontinuity the convergence is to the midpoint of the jump but, as it was observed by Wilbraham and later by Michelson and Gibbs, in a neighborhood of the discontinuities, the partial sums have wild oscillations and overshoot the target by about 9% of the value of the jump.The classical trigonometric series is a quite faithful model for more general one-dimensional expansions. After suitable changes of variables, a regular Sturm-Liouville problem can be put in a canonical form:Moreover, if oty ~ 0 and if n ~ +oo, the sequences of eigenvalues {A n In=0 and normalized

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Section: Then For the Spherical Partial Sums Of F (X ) Localization Hmentioning

confidence: 60%

Section: Localization For Fourier Integrals In R Nmentioning

confidence: 99%

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Localization and convergence of eigenfunction expansions

Brandolini

Colzani

1999

The Journal of Fourier Analysis and Applications

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“…The first results on the different behaviour of the Riesz means of critical index 1 2 N s − = of the Fourier integral and the Fourier series in 1 L found by S. Bochner [15], where it is proved that the localization of the means (3) holds and for the means (2) the localization fails. In the same paper [15] it is proved validity of localization principle in 2 L for both expansions in the critical index …”

Section: Preliminariesmentioning

confidence: 99%

“…In N-dimensional case, > 1 N , localization principles, as well as equiconvergence, for the Fourier series and integral is not valid by the Pringsheim convergence [1]. In [2] it is given a review of recent results on equiconvergence of expansions in multiple trigonometric Fourier series and integral in the case of summation over rectangles.…”

Section: Introductionmentioning

confidence: 99%

On the Equiconvergence of the Fourier Series and Integral of Distributions

Rakhimov¹

2015

JAMP

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Reconstructing the wave speed and the source

Boumenir

Tuan

2021

Math Methods in App Sciences

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We are concerned with the inverse problem of recovering the unknown wave speed and also the source in a multidimensional wave equation. We show that the wave speed coefficient can be reconstructed from the observations of the solution taken at a single point. For the source, we may need a sequence of observation points due to the presence of multiple spectrum and nodal lines. This new method, based on spectral estimation techniques, leads to a simple procedure that delivers both uniqueness and reconstruction of the coefficients at the same time.

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Problems of Convergence of Multiple Trigonometric Series and Spectral Decompositions. Ii

Cited by 23 publications

References 1 publication

Localization and convergence of eigenfunction expansions

Localization and convergence of eigenfunction expansions

On the Equiconvergence of the Fourier Series and Integral of Distributions

Reconstructing the wave speed and the source

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