A Characterization of the range of a finite convolution operator with a hankel kernel

Gorenflo, Norbert

doi:10.1080/10652460108819331

Cited by 5 publications

(2 citation statements)

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“…Both the cases are considered in Sections 2-5 and show a quite different mathematical structure: in Sommerfeld's theory the E-polarization leads to an integro-differential equation for the boundary field e 1,0 , while the B-polarization is described by a boundary integral equation for e 2,0 . The boundary integral equation describing the B-polarization was investigated by Gorenflo and Gorenflo/Werner in a series of careful mathematical studies, see [14,15,17,18]. However, from the representation theorems in Sections 3 and 4 we also derive solutions with smooth boundary fields and finite energy E(0, H ), which are not covered by Sommerfeld's theory.…”

Section: Introductionmentioning

confidence: 99%

Diffraction of light revisited

Kunik

Skrzypacz

2007

Math Methods in App Sciences

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SUMMARYThe diffraction of monochromatic light is considered for a plane screen with an open infinite slit by solving the vectorial Maxwell-Helmholtz system in the upper half-space with the Fourier method. With this approach we can represent each solution satisfying an appropriate energy condition by its boundary fields in the Sobolev spaces H ±1/2 . We show that Sommerfeld's theory using a boundary integral equation with Hankel kernels for the so-called B-polarization is covered by our approach, but in general it violates a necessary energy condition. Our representation includes also solutions which are not covered by Sommerfeld's theory.

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Section: Introductionmentioning

confidence: 99%

Diffraction of light revisited

Kunik

Skrzypacz

2007

Math Methods in App Sciences

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“…In the papers [5], [6] and [7] the problem of extending the right-hand side f of (1.1) onto the whole of R has been treated. In [5] it has been shown that the consistent extension of f onto R , for which (1.1) holds for all x ∈ R , can be expressed by the values f (x) for |x| ≤ 1 and the solutions of the singular first-order system of ordinary differential equations…”

Section: Introductionmentioning

confidence: 99%

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Gorenflo

2002

Z angew Math Phys

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The two-dimensional time harmonic problem of diffraction through a slit is considered for arbitrary Dirichlet data, prescribed in the slit; on the screen itself the wavefield satisfies a homogeneous Neumann boundary condition. First of all, a sequence of special Dirichlet data is constructed, for each of these data the resulting wavefield can be expressed in closed form. The sequence can be constructed in a way which yields an orthonormal basis of Dirichlet data. After this has been done, arbitrary Dirichlet data can be expanded into a series of the orthonormal basis functions; this results in a representation for the searched wavefield. The presented method gives a deep insight into the mathematical structure of the problem.

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

Gorenflo

Kunik

2011

Mathematische Nachrichten

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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.

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A Characterization of the range of a finite convolution operator with a hankel kernel

Cited by 5 publications

References 10 publications

Diffraction of light revisited

Diffraction of light revisited

Untitled

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

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