Algorithm for the \(E\)-prediction
Let \(k>0\) be a fixed integer, and the complex number sequence \({\mathcal S}:= s(0)\), \(s(1), \dots\) and suitable sequences \({\mathcal G} (i):= g(i\mid 0)\), \(g(i\mid 1), \dots\) with \(0 <i \leq k\) being provided, it is possible to derive numbers \(e(k,m)\) \((m\geq 0)\) from the systems of linear equations \[ e(k,m)+ \bigl\{\sum \gamma (k,m|i) g(i|n) \mid [0<i \leq k]\bigr\} = s(n) \tag{*} \] with \(m\leq n\leq m+k\). It is also possible to obtain from the first \(k+1\) members of \({\mathcal S}\) a \(g\)-basal sequence \(s(g,k|0)\), \(s(g,k|1), \dots\) whose first \(k+1\) members are those of \({\mathcal S}\) and for which a system of linear equations of the form (*) hold where (a) \(e(k,m)\) is replaced by \(e(k,0)\) consistently, (b) the \(\gamma (k,m|i)\) are possibly replaced by other coefficients \(\gamma' (k,m |i)\) and (c) the \(s(n)\) are replaced by \(s(g,k |n)\). A simple recursive process for constructing the \(g\)-basal sequence is given; it is analogous to the construction of successive values of a polynomial by use of a forward difference table containing a column of equal entries. The question of the convergence of \({\mathcal S}\) and a \(g\)-basal sequence to the same limit is touched upon.
scientific article; zbMATH DE number 1100608