Hermite interpolation and Sobolev orthogonality (Q1591300)

The main aim of the authors is to reveal the interpretation of the real-valued polynomials, which are orthogonal with respect to a bilinear form given as \[ (f,g)_S= V(f) AV(g)^T+\langle u, f^{(N)} g^{(N)}\rangle,\tag{\(*\)} \] in connection with the theory of interpolation and approximation. In \((*)\) \(u\) stands for a given linear functional on the linear space of real polynomials while \(A\) is an \(N\times N\) real matrix with nonsingular principal submatrices. The vector-valued function \(V\) is defined by \[ V(f)= (f(c_0), f'(c_0),\dots, f^{(n_0-1)}(c_0),\dots, f(c_p), f'(c_p),\dots, f^{(n_p- 1)}(c_p)), \] where \(c_0,c_1,\dots, c_p\) refer to certain distinct real numbers while \(n_0,n_1,\dots, n_p\) are given positive integers such that \(n_0+ n_1+\cdots+ n_p= N\). They first give a description of the monic polynomials which are orthogonal with respect to \((*)\) and establish the kernel polynomials corresponding to them. Then they show that the polynomials in question are the error terms in some interpolation and approximation problems.