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2009-04

A new and self-contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

by Gorenflo, N., Kunik, M..


Series: 2009-04, Preprints

MSC:
78A45 Diffraction, scattering
42A50 Conjugate functions, conjugate series, singular integrals
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and ~Wiener-Hopf type)
45H05 Miscellaneous special kernels

Abstract:
We present a new and self-contained theory for mapping properties of
the boundary operators for slit diffraction occurring in Sommerfeld’s
diffraction theory, covering two different cases of the polarisation of the
light. This theory is entirely developed in the context of the boundary
operators with a Hankel kernel and not based on the corresponding
mixed boundary value problem for the Helmholtz equation. For a
logarithmic approximation of the Hankel kernel we also study the corresponding
mapping properties and derive explicit solutions together
with certain regularity results.

Keywords:
Sommerfeld diffraction theory, Fourier analysis, Sobolev spaces, mapping properties, boundary integral equations, Hankel functions