Moving approximation and the absorption principle
In this technical note, the problem is to estimate the value of a \(C^{k+1}\) signal \(x(t)\) at \(t= nT\), \(x_n\) when one knows \(x_{n-1},\dots, x_{n-k-1}\) assuming that the \((k+1)\)th derivative of \(x(t)\) is uniformly bounded by \(M\). The answer is \[ \widehat x_n= \sum^{s=k+1}_{s=1} (-1)^{s-1} C^s_{k+1} x_{n-s}. \] It is obtained by using the binomial expansion of the \((k+1)\)th difference \(\Delta^{(k+1)}x_n\) and an expression of this quantity via a mean-value theorem. The error \(x_n- \widehat x_n\) has a norm bounded by \(T^{(k+1)}M\) and can be made arbitrarily small by choosing the sampling interval \(T\) small enough. The term absorption from the title occurs in this sense. The author states that the result is original in reference to previous work of his and others because the above formula is universal in the sense that the coefficients are fixed and need not be chosen adaptively.
scientific article; zbMATH DE number 1568575