Wave fronts via Fourier series coefficients

Maksimović, Snježana; Pilipović, Stevan; Sokoloski, Petar; Vindas, Jasson

doi:10.2298/pim150107001m

Cited by 2 publications

(4 citation statements)

References 12 publications

(14 reference statements)

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“…To this end, we also study Fourier series expansions of ultradistributions. The paper refines and extends earlier results on toroidal wave front sets from [19,30].…”

Section: Introductionsupporting

confidence: 86%

“…where W F (f ) stands for the classical Hörmander wave front set when regarding f as a distribution on R d . The latter equality further extends to Sobolev-type and Gevrey wave front sets, as recently shown in [8,19]. It should also be mentioned that Rodino and Wahlberg [28] and Johansson et al [15] have investigated discrete definitions of microregularity properties of distributions via Gabor frames.…”

Section: Introductionmentioning

confidence: 57%

“…Furthermore, Corollary 4.6 also includes results by Ruzhansky and Turunen on the toroidal C ∞ -wave front set [30,Thm. 7.4], while Theorem 4.5 covers all results from [8,19].…”

Section: Choose An Open Conic Neighborhood γmentioning

confidence: 84%

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Discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type

Debrouwere

Vindas

2016

Journal of Mathematical Analysis and Applications

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We obtain discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in R d . In particular, we prove the following discrete characterization of the analytic wave front set of a distribution f ∈ D ′ (Ω). Let Λ be a lattice in R d and let U be an open convex neighborhood of the origin such that U ∩ Λ * = {0}. The analytic wave front set W F A (f ) coincides with the complement in Ω × (R d \ {0}) of the set of points (x 0 , ξ 0 ) for which there are an open neighborhood V ⊂ Ω ∩ (x 0 + U ) of x 0 , an open conic neighborhood Γ of ξ 0 , and a bounded sequence (2010 Mathematics Subject Classification. Primary 35A18, 42B05. Secondary 46F05.

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“…To this end, we also study Fourier series expansions of ultradistributions. The paper refines and extends earlier results on toroidal wave front sets from [19,30].…”

Section: Introductionsupporting

confidence: 86%

Section: Introductionmentioning

confidence: 57%

See 1 more Smart Citation

Discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type

Debrouwere

Vindas

2016

Journal of Mathematical Analysis and Applications

Self Cite

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“…Many variations have been devised: analytic wavefront set (see [77] and [78] for a recent comparison of various definitions), homogeneous wavefront set [79], Gabor wavefront set [80], global wavefront set [81][82][83], discrete wavefont set [84], etc. Specific techniques can be applied to the (Sobolev) wavefront set of periodic distributions [85].…”

Section: Properties Of the Wavefront Set 9 Conclusionmentioning

confidence: 99%

A smooth introduction to the wavefront set

Brouder¹,

Dang²,

Hélein³

2014

J. Phys. A: Math. Theor.

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Abstract. The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.

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Wave fronts via Fourier series coefficients

Abstract: Abstract. Motivated by the product of periodic distributions, we give a new description of the wave front and the Sobolev-type wave front of a distribution f ∈ D ′ (R d ) in terms of Fourier series coefficients.

Cited by 2 publications

References 12 publications

Discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type

Discrete characterizations of wave front sets of Fourier–Lebesgue and quasianalytic type

A smooth introduction to the wavefront set

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