On Wimp's vector sequence transformations (Q1893600)

A process of recursive triangularisation for deriving a system of vectors \(E(k,n)\) \((0 \leq k \leq N\), \(0 \leq n \leq N-k\)) in \(\mathbb{C}^p\) from two vector sequences \(s(n)\), \(\phi(n)\) \((0 \leq n \leq N)\) and a system of sequences \(g(i |n)\) \((0 \leq n \leq N)\) defined for \(0 < i \leq N\), is described. The process involves derived vectors \(g(i |k, n)\) (\(0 \leq k < N\), \(0 \leq n \leq N - k\), \(k < i \leq N)\) and frames \({\mathcal F} (k,n)\) whose components are vectors: the first component of \({\mathcal F} (k,n)\) is \(E(k,n)\) and the remaining components are \(g(i |k,n)\) \((k < i \leq N)\). Initially \(E(0,n) = s(n)\) \((0 \leq n \leq N)\) and \(g(i |0,n) = g(i |n)\) \((0 \leq n \leq N\), \(0 < i \leq N)\). Using the notation \(\Delta f(n) = f(n + 1) - f(n)\), set \(r(k,n) = \Delta \langle \phi (n), {\mathcal F}(k - 1, n)\rangle / \Delta \langle \phi(n), g(k |k-1, n)\rangle\) (the angular brackets denote scalar products; the denominator is in \(\mathbb{C}\); \(r(k,n)\) is in \(\mathbb{C}^{N - k + 1}\)). The \({\mathcal F} (k,n)\) are computed by setting \({\mathcal F} (k,n) = {\mathcal F} (k - 1, n) - r(k,n) \times g(k|k - 1,n)\) \((0 < k \leq N\), \(0 \leq n \leq N-k)\). (The product is formed from the complex number components of \(r\) and the common vector \(g(k |k - 1, n)\); at each stage except the very last, the second component of \({\mathcal F}(k,n)\) is \(g(k + 1 |k,n)\) and is used to compute \({\mathcal F} (k+1, n)).\) Formulae involving determinants, one of whose rows consists of vectors, are given for the \(E(k,n)\). It is shown that if the \(s(n)\) and \(g(i |n)\) are connected by the relationship \[ s(n) - S = \{\Sigma a(i) g(i|n) [0 < i \leq N]\} \] \((S\) is in \(\mathbb{C}^p\), the \(a(i)\) are in \(\mathbb{C}\)) then, subject to certain conditions involving the \(\phi (n)\), \(E(k,n) - S = \{\Sigma a (i) g(i |k,n)\;[k < i \leq N]\}\) for \(0 \leq k \leq N\), \(0 \leq n \leq N - k\). If \(s(n)\) \((n \geq 0)\) converges to \(S\), the \(E(k,n)\) offer approximations to \(S\) and \(E(N,0) = S\) gives this limit exactly. The convergence of various sequences taken from the \(E(k,n)\) is considered. The effect upon stability of choices of the \(\phi(n)\) is treated.