The epsilon algorithm in a noncommutative algebra
Let \({\mathcal A}\) be a noncommutative Banach algebra with unit element I over a field \({\mathfrak K}\) and f be the formal power series \(f(t)=\sum^{\infty}_{i=0}c_ it^ i\) where \(c_ i\in {\mathcal A}\) (i\(\geq 0)\) and \(t\in {\mathfrak K}\). The approximating fraction \(R_ k^{(n)}\) of order k, \(n+k-1\) derived from f is given by \(R_ k^{(n)}(t)=N_ k^{(n)}(t)\{D_ n^{(k)}(t)\}^{-1}\) where \(D_ k^{(n)}\), \(N_ k^{(n)}\) are polynomials in t of degrees at most k, \(n+k-1\) respectively, \(D_ k^{(n)}(0)=I\) and \(f(t)-R_ k^{(n)}(t)=O(t^{n+2k}).\) The existence of the fractions \(R_ k^{(n)}\) and relationships obtaining among them are treated with reference to Hankel matrices with elements \(c_{n+i+j}\) (0\(\leq i,j\leq k)\). The theory is also presented in terms of application of the \(\epsilon\)-algorithm to the sequence \(S_ m\in {\mathcal A}\) (m\(\geq 0)\).
scientific article; zbMATH DE number 4026689