Constrained approximation and interpolation in Hilbert function spaces
Optimization problems in Hilbert function spaces play a significant role in many applications for instance in control theory and in signal processing. The paper considers a family of bounded extremal problems which emerge from two distinct orthogonal decompositions of a Hilbert space. Specifically, suppose that the Hilbert space is decomposed into \(H=H_+\oplus H_-=K_+\oplus K_-\), and we are given an element of \(K_+\), we seek an element of a bounded subset of \(H_+\), the projection of which onto \(K_+\) best approximates that element. Another question dealt with in the paper is that of extending the element of \(K_+\) to the whole of \(H\) (by elements of \(K_-\)) so that the resulting function is the closest to \(H_+\). Among the pairs of decompositions of \(H\) one should have in mind are \[ L^2(T)=H^2\oplus\overline H_0^2=L^2(K)\oplus L^2(T\setminus K), \] where \(K\) is a compact subset of the torus \(T\) such that both \(K\) and \(T\setminus K\) have positive measures, and \(L^2(K)\) is the set of functions in \(L^2(T)\) which vanish a.e. outside \(K\). The space \(H^2\) is the usual Hardy space of functions whose negative Fourier coefficients vanish and \(\overline H_0^2\), its complement. Another example is \[ L^2(R)=PW(\Omega)\oplus PW(R\setminus\Omega)=L^2(J)\oplus L^2(R\setminus J), \] where \(J\) and \(\Omega\) are compact subsets of the real line, of positive measures and \(PW(\Omega)\) is the Paley-Wiener space of functions in \(L^2(R)\) whose Fourier transform is supported on \(\Omega\).
optimization problems in Hilbert function spaces
scientific article; zbMATH DE number 1334722
scientific article