Tangential Nevanlinna--Pick interpolation and its connection with Hamburger matrix moment problem (Q1882518)

The tangential Nevanlinna--Pick interpolation problem (TNPI) entails determining whether for given points \(z_1,\dots,z_{np}\) in the open upper-half complex plane and \(p\)-column vectors with complex entries \(x_1,\dots,x_{np}\) (non-null) and \(y_1,\dots,y_{np}\), there exists a Nevanlinna matrix function \(F(z)\) in the class \({\mathcal N}_p\) such that \(F(z_i)x_i=y_i,\;i=1,\dots,np\). The TNPI problem is solvable if and only if the information matrix \[ L:=\Bigl({x_j^*y_i-y_j^*x_i\over z_i-\bar{z}_j}\Bigr)_{i,j=1}^{np}\geq 0. \] If \((H_0,\dots,H_{2n-2})\) is the corresponding block Hankel vector, then the related truncated Hamburger matrix moment problem (THM) entails determining Hermitian measures \(\tau\) such that \(H_i=\int_{-\infty}^{\infty}u^id\tau(u),\;i=0,\dots,2n-3\) and \(H_{2n-2}\geq\int_{-\infty}^{\infty}u^{2n-2}d\tau(u)\). The THM problem is solvable if and only if the Hermitian block Hankel matrix \(H:=(H_{i+j})_{i,j=0}^{n-1}\) is Hermitian nonnegative. The authors provide an explicit description of a nonsingular \(np\times np\)-matrix \(W\) such that \(L=W^*HW\). A divisor-remainder connection between the solutions of the two related problems concludes the paper. A similar relationship between the TNPI problem with constraint \(\lim_{z\to\infty}{F(z)\over z}=0\) and the standard THM \[ H_i=\int_{-\infty}^{\infty}u^id\tau(u),\;i=0,\dots,2n-2, \] is also obtained.