A weak approximation for the Wiener–Hopf factorization

Amir T Payandeh Najafabadi,Dan Z Kucerovsky

doi:10.1080/23311835.2015.1074773

Amir T Payandeh Najafabadi, Dan Z Kucerovsky

Open Access

https://doi.org/10.1080/23311835.2015.1074773

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Abstract

The Wiener–Hopf factorization plays a crucial role in studying various mathematical problems. Unfortunately, in many situations, the Wiener–Hopf factorization cannot provide closed form solutions and one has to employ some approximation techniques to find its solutions. This article provides several weak, approximation for a given Wiener–Hopf factorization problem. Application of our finding in spectral factorization and Levy processes have been given.

Highlights

IntroductionThe key steps to solve a Wiener–Hopf factorization is decomposing of the kernel g into a product of two terms, g+ and g−, where g+ and g− are analytic and bounded in the upper and the lower complex half planes, respectively
Speaking, the Wiener–Hopf factorization problem is a technique to find a single complexvalued function Φ in which its radial limits, say Φ±, are respectively analytic and bounded separately in the upper and lower complex half planes (i.e. C+: = { ∈ C:I( ) ≥ 0}and C−: = { ∈ C:I( ) ≤ 0}) and satisfy Φ+( )Φ−( ) = g( ), where ∈ R and g is a zero index function which satisfies the Hölder condition.The Wiener–Hopf factorization has proved remarkably useful in solving an enormous variety of model problems in a wide range of branches of physics, mathematics, and engineering
The key steps to solve a Wiener–Hopf factorization is decomposing of the kernel g into a product of two terms, g+ and g−, where g+ and g− are analytic and bounded in the upper and the lower complex half planes, respectively

Summary

Introduction

The key steps to solve a Wiener–Hopf factorization is decomposing of the kernel g into a product of two terms, g+ and g−, where g+ and g− are analytic and bounded in the upper and the lower complex half planes, respectively Such decomposition can be expressed in terms of a Sokhotski–Plemelj integral (see Equation, 1), but this form presents some difficulties in numerical work due to slow evaluation and numerical problems caused by singularities near the integral contour (see Kucerovsky & Payandeh Najafabadi, 2009, for more details). Using the Hausdorff–Young theorem, Payandeh Najafabadi and Kucerovsky (2014a) established that Hilbert transform of an Lp(R), 1 < p ≤ 2 function s, say Hs, satisfies Form this observation, one may conclude that “if {fn}, n ≥ 1, is a sequence of functions which converge in Lp(R), 1 < p ≤ 2, to f.

Main results

Application to spectral factorization

Application to Lévy processes

Conclusion and suggestion