An acceleration property of the \(E\)-algorithm for alternate sequences
The double sequence \(g(i,n)\in \mathbb{C}\) \((i>0\), \(n\geq 0)\) being prescribed and \(S(n)\in \mathbb{C}\) \((n\geq 0)\) being a sequence to be transformed, \(L(S|r,m) \in\mathbb{C}\) is defined by eliminating the coefficients \(a(r,m|\nu)\) \((0< \nu\leq r)\) from the system of linear equations \(S(m+\tau) =L(S|r,m) +\{\sum a(r,m |\nu) g(\nu,m+ \tau) \mid [0<\nu \leq r]\}\) with \(0\leq\tau \leq r\). The various \(L(S|r,m)\) defined for \(r>0\), \(m\geq 0\) may be obtained by a process of recursive triangulation. Under certain conditions, \(L(S|r,m)\) is an estimate of \(\Lambda=\lim S(n)\) as \(n\) increases. As a representative of the sort of results obtained, it is shown that with \(g(i,n)= (-1)^n f(i,n)\), where \(f(i,n)>0\) \((i>0,\) \(n\geq 0)\) is bounded and further conditions hold, \[ \lim \biggl\{L \bigl(S|r+1, n\bigr) -\Lambda \biggr\} \Bigr/ \biggl\{L\bigl(S |r,n \bigr)- \Lambda\biggl\} =0 \] for \(r>0\). Determinantal conditions involving the \(f(i,n)\) and \(S(n)\) lead to the inequality \[ \biggl|L \bigl(S |r,m \bigr)- \Lambda \biggr|\leq\biggl|L\bigl(S |r+1,m \bigr)- L\bigl(S|r,m \bigr)\biggr| \] for \(r>0\) and \(m\geq 0\). The function \(h\) being prescribed, \(g\) being defined in terms of it by setting \(g(1,n)= (-1)^n h(n+1)\) and \[ g(i,n)= (-1)^{n+1} {\mathcal D}^{2i-3} h(n +1) \quad (i>1,n \geq 0) \] where \({\mathcal D}\) denotes differentiation with respect to \(n\), and \(S(n)\) \((n\geq 0)\) being the special sequence \(S(n)= \{\sum(-1)^\mu h(\mu+1) |[0\leq \mu\leq n]\}\), bounds and a sign rule are given for \(L(S|r,m)-\Lambda\). The inquiry is extended to the case in which \(S\) and \(g\) are in \(\mathbb{C}^p\).
scientific article; zbMATH DE number 947619