### 2009-04

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