Generalized inverses of Toeplitz operators and inverse approximation in \(H^ 2\)
Let \(H^ p\) \((p=2\) or \(\infty)\) be the Hardy space of the unit disk, and let \(\pi_ k\) be the linear space of all polynomials with degree at most k. For \(f\in H^{\infty}\), the least-squares inverse in \(\pi_ k\) of f was defined by \textit{C. K. Chui} [J. Math. Anal. Appl. 75, 149-163 (1980; Zbl 0438.41007)] as the unique \(g=g_ k\) such that the \(L^ 2\)-norm \(\| 1-fg\|_ 2\) on the unit circle is minimum when g runs over \(\pi_ k\). The double least-squares \(h_{n,k}\) in \(\pi_ n\) of f through \(\pi_ k\) was also defined as the least-squares inverse in \(\pi_ n\) of \(g_ k\). Using Moore-Penrose generalized inverses which are defined for range-closed Hilbert space operators, we can describe the least-squares inverse and the double least-squares inverse respectively as \[ g_ k=(T_ fE_ k)^{\dag}1\quad and\quad h_{n,k}=(T_{g_ k}E_ n)^{\dag}1. \] Here \(T_{\phi}\) is the Toeplitz operator with symbol \(\phi\) and \(E_ m\) is the orthogonal projection onto \(\pi_ m\). Hence the approximation problem of least-squares and double least-squares inverses is identical to the convergence problem of generalized inverses. Applying our results on convergence of generalized inverses [J. Approx. Theory 38, 269-278 (1983; Zbl 0519.41016)], we here study convergence of least-squares and double least-squares inverses. We extend some theorems due to Chui, and also settle a conjecture raised by him.
convergence problem of generalized inverses
scientific article; zbMATH DE number 3919126