On totally monotonic families of sequences
The authors give some determinantal inequalities which are then used to derive convergence results for the \( E \)-algorithm. The determinants considered have, with \( i,j \geq 1 \), elements of the form [a] \( K(x(i), x(j)) \), where \( x(k)\) \((k \geq 1) \) is a strictly increasing real sequence and \( K \) is real valued, [b] \( g(i|n+j-1) \) where, for each \( i\geq 1\), \(g(i|n)\) \((n\geq 0) \) is a sequence whose members satisfy certain inequalities and, as a special case of this form, [c] \( g(i+n+j-2)\) \((n \geq 0) \) where \( g(n)\) \((n \geq 0) \) is a totally monotone sequence. The theory is illustrated by application to the transformation of the sum \( \{\sum (-1)^{k+1}/\ln(k+2) |k\geq 0\} \), it being necessary to obtain successive derivatives of the function \( f(t) = 1/\ln(1+t) \) which feature in the theory as \( g(1|n) = (-1)^{n}: f(n+1) \) and \( g(i|n) = (-1)^{n+1}: f^{(2i-3)}(n+1)\) \((i \geq 0) \). There is a not entirely successful attempt to contrast the method favorably with the epsilon algorithm. Firstly, application of the latter algorithm to the example considered is stable while the formation of cross products in use of the \( E \)-algorithm is not; secondly, the evaluation of derivatives is unnecessary when using the epsilon algorithm; thirdly, repeated application of the epsilon algorithm (which the authors do not consider) produces estimates which are in the present context considerably better than those displayed in the paper under review.
scientific article; zbMATH DE number 1398748