scholarly journals The Behavior, onḂ0,11, of an Oscillatory Integral with Polynomial Phase Function

1998 ◽  
Vol 221 (2) ◽  
pp. 658-671
Author(s):  
Philip J Gloor
Keyword(s):  
Phase Function ◽  
Oscillatory Integral ◽  
Polynomial Phase

scholarly journals Formal oscillatory distributions

2021 ◽  
pp. 1-19
Author(s):  
Alexander Karabegov
Keyword(s):  
Asymptotic Expansion ◽  
Critical Point ◽  
Phase Function ◽  
Star Product ◽  
Oscillatory Integral ◽  
Poisson Manifold ◽  
Formal Asymptotic Expansion ◽  
Formal Distribution ◽  
Nondegenerate Critical Point

The formal asymptotic expansion of an oscillatory integral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ⋆ on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f ⊗ g ↦ ( f ⋆ g ) ( x ) is oscillatory for every x ∈ M.

A method for the separation of interfering surface waves and the computation of their dispersion

1977 ◽  
Vol 67 (2) ◽  
pp. 383-392
Author(s):  
Paul Michaels
Keyword(s):  
Surface Waves ◽  
Least Squares ◽  
Phase Function ◽  
Frequency Dispersion ◽  
Generalized Least Squares ◽  
Dispersion Curves ◽  
Modulation Equation ◽  
Polynomial Phase ◽  
Phase Velocity Dispersion ◽  
Wave Forms

Abstract A practical technique for separating two surface waves that interfere, producing beats, is accomplished by a generalized least-squares inversion. A sinusoid consisting of a polynomial phase function is fit via least squares to both the envelope and the carrier wave forms of a digitized signal. The phase functions of the two interfering waves are then determined via a modulation equation. The derivative with respect to time of each phase function yields the frequency dispersion curves. Determinations of both group- and phase-velocity dispersion curves are discussed and a numerical example of the technique is given.

scholarly journals Remarks on Fujiwara’s stationary phase method on a space of large dimension with a phase function involving electromagnetic fields

1994 ◽  
Vol 136 ◽  
pp. 157-189 ◽  
Author(s):  
Tetsuo Tsuchida
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
Stationary Phase ◽  
Phase Function ◽  
Remainder Term ◽  
Oscillatory Integral ◽  
Phase Method ◽  
Stationary Phase Method ◽  
Action Integral ◽  
Electric And Magnetic Fields

We consider an oscillatory integral of the formHere each xj, j = 0, 1,…, L, runs in Rd, ν > 1 is a constant and tj, j = 1,…,L, are positive constants. Fujiwara [5] discussed this integral for L large and developed the stationary phase method with an estimate of the remainder term for the phase function S(xL,…, x0) coming from the action integral for a particle in an electric field. But his results cannot be applied to the integral which naturally arises in the discussion of quantum mechanics of a charged particle moving in a magnetic field. In this paper we extend his results to the case for the phase function involving both electric and magnetic fields.

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