scholarly journals Global wave-front sets of Banach, Fréchet and modulation space types, and pseudo-differential operators

2013 ◽  
Vol 254 (8) ◽  
pp. 3228-3258 ◽  
Author(s):  
Sandro Coriasco ◽  
Karoline Johansson ◽  
Joachim Toft
Keyword(s):  
Wave Front ◽  
Differential Operators ◽  
Modulation Space ◽  
Pseudo Differential Operators ◽  
Wave Front Sets

Local wave-front sets of Banach and Fréchet types, and pseudo-differential operators

2012 ◽  
Vol 169 (3-4) ◽  
pp. 285-316 ◽  
Author(s):  
Sandro Coriasco ◽  
Karoline Johansson ◽  
Joachim Toft
Keyword(s):  
Wave Front ◽  
Differential Operators ◽  
Pseudo Differential Operators ◽  
Local Wave ◽  
Wave Front Sets

scholarly journals Beyond gevrey regularity: Superposition and propagation of singularities

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2763-2782 ◽  
Author(s):  
Stevan Pilipovic ◽  
Nenad Teofanov ◽  
Filip Tomic
Keyword(s):  
Wave Front ◽  
Regularity Condition ◽  
Differential Operators ◽  
Ultradifferentiable Functions ◽  
Gevrey Regularity ◽  
Gevrey Classes ◽  
Partial Differential Operators ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Two Parameters

We propose the relaxation of Gevrey regularity condition by using sequences which depend on two parameters, and define spaces of ultradifferentiable functions which contain Gevrey classes. It is shown that such a space is closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients are less regular then Gevrey-type ultradifferentiable functions. To that aim we introduce appropriate wave front sets and prove a theorem on propagation of singularities. This extends related known results in the sense that assumptions on the regularity of the coefficients are weakened.

scholarly journals Micro-local analysis in some spaces of ultradistributions

2012 ◽  
Vol 92 (106) ◽  
pp. 1-24 ◽  
Author(s):  
Karoline Johansson ◽  
Stevan Pilipovic ◽  
Nenad Teofanov ◽  
Joachim Toft
Keyword(s):  
Wave Front ◽  
Modulation Space ◽  
Gabor Frames ◽  
Local Analysis ◽  
Wave Front Sets

We extend some results from [14] and [19], concerning wave-front sets of Fourier-Lebesgue and modulation space types, to a broader class of spaces of ultradistributions. We relate these wave-front sets one to another and to the usual wave-front sets of ultradistributions. Furthermore, we give a description of discrete wave-front sets by introducing the notion of discretely regular points, and prove that these wave-front sets coincide with corresponding wave-front sets in [19]. Some of these investigations are based on the properties of the Gabor frames.

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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