| Abstract: | In the paper N. Gorenflo, A new explicit solution method for the diffraction through a slit, ZAMP 53 (2002), 877–886] the problem of diffraction through a slit in a screen has been considered for arbitrary Dirichlet data,
prescribed in the slit, and under the assumption that the normal derivative of the diffracted wave vanishes on the screen
itself. For this problem certain functions with the following properties have been constructed: Each function f is defined on the whole of R and on the screen the values f(x), |x| ≥ 1, are the Dirichlet data of the diffracted wave which takes on the Dirichlet data f(x), |x| ≤ 1, in the slit. The problem of expanding arbitrary Dirichlet data, prescribed in the slit, into a series of functions
of the considered form has been addressed, but not solved in a satisfactory way (only the application of the Gram-Schmidt
orthogonalization process to such functions has been proposed). In this continuation of the aforementioned paper we choose
the remaining degrees of freedom in the earlier given representations of such functions in a certain way. The resulting concrete
functions can be expressed by Hankel functions and explicitly given coefficients. We suggest the expansion of arbitrary Dirichlet
data, prescribed in the slit, into a series of these functions, here the expansion coefficients can be expressed explicitly
by certain moments of the expanded data. Using this expansion, the diffracted wave can be expressed in an explicit form. In
the future it should be examined whether similar techniques as those which are presented in the present paper can be used
to solve other canonical diffraction problems, inclusively vectorial diffraction problems. |
