On the best approximation matrix problem for integrable matrix functions (Q2721745)

Let us denote by \(\mathcal{P}_n[x]\) the set of matrix polynomials \(A_0+A_1x+\dots+a_nx^n\) (\(A_i\in \mathbf{M}_n(\mathbb{C})\), \(i=0,1,\dots,n\)) of degree less or equal to \(n\). Given \(W\) an Hermitian positive definite matrix function, we denote NEWLINE\[NEWLINE L^2_W([a,b],\mathbf{M}_n(\mathbb{C})):=\{f:\int_a^bf(x)W(x)f^H(x)dx<\infty\},NEWLINE\]NEWLINE where the \(f\) denote matrix functions and let \(\mathcal{L}: L^2_W([a,b],\mathbf{M}_n(\mathbb{C}))\times L^2_W([a,b],\mathbf{M}_n(\mathbb{C})) \rightarrow \mathbf{M}_n(\mathbb{C})\) be a positive definite Herminian bilinear matrix functional. The main goal of this paper is to study the following approximation matrix problem: Given \(f\in L^2_W([a,b],\mathbf{M}_n(\mathbb{C}))\), find \(Q\in\mathcal{P}_n[x]\) such that NEWLINE\[NEWLINE \mathcal{L}(f-Q,f-Q)=min_{q\in\mathcal{P}_n[x]}\mathcal{L}(f-q,f-q). NEWLINE\]NEWLINE Associated to the functional \(\mathcal{L}\) there exists a concept of orthogonality. The authors solve the problem by considering the corresponding theory of orthogonal expansions (existence of sequence of orthogonal polynomials, the Fourier coefficients, the completeness of the system, the Riemann-Lebesgue theorem, etc). The results are paralell to the classical theory of orthogonal expansions of functions. The proofs are more or less straightfoward. The paper is quite readable and (to the reviewer) of certain interest.