Sadia Sadique, Emilia Blåsten and Lassi Päivärinta of the Institute of Cybernetics of the Tallinn University of Technology recently published a study in the scientific journal SIAM Journal on Mathematical Analysis. Their research area is inverse problems, a fascinating field within mathematics that involves the challenging task of deducing unknown parameters or structures from observed data.
This work is an innovative application of scattering of quantum or acoustic waves from a crack or screen in a two-dimensional domain. This study is the first step into understanding the singular behaviour of waves near the endpoint of cracks or screens in an acoustic medium. This work may also play an important role in the theory of antennas.
The travelling waves usually satisfy a hyperbolic equation called the Wave equation or in the Fourier image Helmholtz equation. The principal part of this equation being the Laplace operator they are lead to study of a logarithmic integral operator. Since the interest of the authors of the paper is in the singularities of the solution they take the derivative of this operator and end up with the Hilbert transform on a screen. They figured out that the poles of the Mellin symbol of this transform determine these singularities . They also established the connection of the Mellin transform to the Fourier transform, but not with the usual Fourier transform but with the one that is based on the locally compact multiplicative group on the half axis.