Properties of Orthogonal Complement Functions in Body Shape Reconstruction Problems

Computational aspects of constructing the approximating Herglotz functions in inverse scattering problems

“…Two ways of preparing the input data (scattering patterns) were used in independent simulations. The first way was choosing the currents mM j as segments of the Fourier series with random coefficients (diminishing as 1/ n as number n increases) and then calculating the scattering patterns by (3). The second way (similarly as in usual methods of inverse scattering theory) was solving the direct problem with incident plane waves coming from uniformly distributed directions.…”

“…Numerical experiments described in [2][3][4] show a good efficiency of the method not only for the bodies with boundaries for which the above Herglotz functions exist. It worked even for the case when interior eigenoscillation field of the body is continued with singularity outside its boundary.…”

“…A method of body shape reconstruction at its resonant frequencies was proposed in [1] and developed in [2][3][4]. The method uses the property that any set of scattering patterns can be noncomplete at these frequencies.…”

Dedication to the Memory of Prof. Nikolai Voitovich

“…Otherwise, the field of eigenoscillation within the region can be approximated by the sequence of Herglotz functions in the metric of the corresponding Sobolev space. The recent results related to this topic were published in [40] - [44].…”