Multiple fourier series and integrals

Голубов, Б. И.

doi:10.1007/bf01305756

Cited by 14 publications

(6 citation statements)

References 164 publications

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“…Indeed, if Γ = T n and A = ∆ Γ , then (7.13) becomes the expansion of f into the n-multiple trigonometric series (as usual, T := {e iτ : 0 ≤ τ ≤ 2π}). It is known [22,Section 6] that this series is unconditionally uniformly convergent (on Γ) on every Hölder class C s (Γ) of order s > n/2. The exponent n/2 is critical here; namely, there exists a function f ∈ C n/2 (Γ) whose trigonometric series diverges at some point of T n .…”

Section: And 31 B]mentioning

confidence: 99%

An extended Hilbert scale and its applications

Mikhailets,

Murach,

Zinchenko

2021

Preprint

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We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of OR-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.

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Section: And 31 B]mentioning

confidence: 99%

An extended Hilbert scale and its applications

Mikhailets,

Murach,

Zinchenko

2021

Preprint

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“…is Hölder continuous in R 4 (because Φ ∈ C α (D)) and therefore the following fourdimensional Fourier series of ω, where ϕ ijkl (x , y , p , q ) = ϕ i (x )ϕ j (y )ϕ k (p )ϕ l (q ), converges uniformly [13]: Similarly, let w i denote the representative of γw φi with zero mean. By Theorem 2.1 and Lemma 3.10, ∂D w p,q φ i ds = w i (p)−w i (q) and λ ij = λ ji = ∂D w i φ j ds for all i, j = 0, 1, 2, .…”

Section: Conformally Mapped Neumann Problemsmentioning

confidence: 99%

Section: Bisweep Datamentioning

confidence: 99%

Point Electrode Problems in Piecewise Smooth Plane Domains

Seiskari¹

2014

SIAM J. Math. Anal.

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A conductivity equation is studied in piecewise smooth plane domains and with measure-valued current patterns (Neumann boundary values). This allows one to extend the recently introduced concept of bisweep data to piecewise smooth domains, which yields a new partial data result for the Calderón inverse conductivity problem. It is also shown that bisweep data are (up to a constant scaling factor) the Schwartz kernel of the relative Neumann-to-Dirichlet map. A numerical method for reconstructing the supports of inclusions from discrete bisweep data is also presented.

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“…There are much fewer proved results about the probabilistic behavior of the multiple trigonometric series such as the spectral expansions (29) or (62) (see [54][55][56][57][58][59][60] and the references therein), specifically regarding the conditions required for convergence of their sums to the limiting distributions. Nevertheless, the proposed connection to the spectral theory clearly implies the existence of the limiting distributions for such expansions and points out the origins of the spectral universality [29,30] from the perspective of the periodic orbit theory.…”

Section: Trigonometric Sums and Spectral Universalitymentioning

confidence: 99%

“…(48), (57), (60)) at the jth level of the hierarchy depend on the fluctuations on all the previous levels as well as on the oscillations introduced by the harmonic terms at the level j,…”

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confidence: 99%