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Asymptotic expansions for Wiener–Hopf equations
Preview AbstractWiener–Hopf Equations are of the form
u(t)=f(t)+∫+∞0k(t−τ)u(τ)dτ,t>0.These equations arise in many physical problems such as radiative transport theory, reflection of an electromagnetive plane wave, sound wave transmission from a tube, and in material science. They are also known as the renewal equations on the half-line in Probability Theory. In this paper, we present a method of deriving asymptotic expansions for the solutions to these equations. Our method makes use of the Wiener–Hopf technique as well as the asymptotic expansions of Stieltjes and Hilbert transforms.Gevrey Asymptotics and Stieltjes Transforms of Algebraically Decaying Functions
Preview AbstractThe development of asymptotic expansions of Stieltjes transforms of exponentially decaying functions has been well established. In this paper, we are concerned with the more difficult case in which the functions decay only algebraically at infinity. By using a Gevrey-type condition, we obtain an exponentially improved asymptotic expansion, and give three representation theorems to show that the Stieltjes transform of algebraically decaying functions can be written as the difference of two integral transforms with exponentially decaying kernels, thus making the asymptotic theory developed for integral transforms with exponentially decaying kernels relevant to Stieltjes transforms of algebraically decaying functions, including the smoothing of the Stokes phenomenon.