Finite Blaschke product interpolation on the closed unit disc
Let \(\{z_j\}_{j=1}^n\) be distinct points and \(\{w_j\}_{j=1}^n\) be complex numbers such that \(|z_j|= 1\), \(|w_j|=1\), \(j =1,\dots,n_1\), \((n_1\geqslant 1)\), and \(|z_j|< 1\), \(|w_j|< 1\), \(j = n_1+1,\dots,n_1 + n_2=n\). Suppose that the Pick matrix \(Q_{n_2}\) defined by \(Q_{ij}=(1-w_i\overline{w}_j)/(1-z_i\overline{z}_j)\), \(i,j = n_1+1, \dots,n\) is positive semi-definite and of full rank. Based on a result of W.~B.~Jones and S.~Ruscheweyh the authors show that there exists a finite Blaschke product \(B(z)\) of degree at most \(n-1\) that satisfies \[ B(z_j) =w_j,\quad 1\leqslant j\leqslant n.\tag{1} \] A Blaschke product interpolation problem of this kind occurs in the design of digital filters. All finite Blaschke products \(B_n(z)\) of degree \(n\) in the form \[ \frac{\alpha_0+\alpha_1z+\dots+\alpha_nz^n}{\overline{\alpha}_0+ \overline{\alpha}_1z+\dots+\overline{\alpha}_nz^n},\quad\alpha_j\in\mathbb C, \quad j=0,\dots,n, \tag{2} \] that satisfy (1) for given Nevanlinna-Pick data \(\{z_j\}_{j=1}^n\), \(\{w_j\}_{j=1}^n\) in the open unit disc \(D\) with the positive semi-definite Pick matrix \(Q_n\), and the scaled finite Blaschke product \(cB_m(z)\) with \(m\leqslant n-1\) of minimal norm on \(D\), that satisfies (1) for the data \(\{z_j\}_{j=1}^n\subset D\) and \(\{w_j\}_{j=1}^n\subset\mathbb C\), are constructed by solving eigenvalue problems of the interpolation data. Finally, a numerical method for determining an interpolating finite Blaschke product of degree at most \(n-1\) is discussed.
scientific article; zbMATH DE number 1866576