Conditional Optimization and One Inverse Boundary Value Problem

Abstract

Here we construct different approximate solutions of the plane inverse boundary value problem of aerohydrodynamics. In order to do this we solve some conditional optimization problems in the norms ‖ ⋅ ‖ 2 , ‖ ⋅ ‖ ∞ , and ‖ ⋅ ‖ 1 and some of their generalizations. We present the example clarifying the mathematical constructions and show that the supremum norm generalization seems to be the optimal one of all the functionals considered in the paper.

Functions of Minimal Norm with the Given Set of Fourier Coefficients

“…In this paper, we generalize the results in [12,13], where the problems of the existence and uniqueness of the solution of the problem of the minimum norm function f ∞ = ess sup…”

“…In connection with the above results, one can note related problems of approximation by a polynomial bounded in the norm • ∞ of an analytic function in the unit circle of the complex plane [10,11]. When solving a number of problems, for example, the inverse problem of aerohydrodynamics, one has to use other norms [12][13][14]. Note that purely integral norms of the spaces L r (a, b) produce large pointwise values of the solution [13] and discontinuities result in singularities of the reconstructed contours [14].…”

“…When solving a number of problems, for example, the inverse problem of aerohydrodynamics, one has to use other norms [12][13][14]. Note that purely integral norms of the spaces L r (a, b) produce large pointwise values of the solution [13] and discontinuities result in singularities of the reconstructed contours [14].…”