Borel's Methods Of Summability

Summability methods are transformations that map sequences (or functions) to sequences (or functions). A prime requirement for a good summability method is that it preserves convergence. Unless it is the identity transformation, it will do more: it will transform some divergent sequences to convergent sequences.

An important type of theorem is called a Tauberian theorem. Here, we know that a sequence is summable. The sequence satisfies a further property that implies convergence. Borel's methods are fundamental to a whole class of sequences to function methods. The transformation gives a function that is usually analytic in a large part of the complex plane, leading to a method for analytic continuation.

These methods, dated from the beginning of the 20th century, have recently found applications in some problems in theoretical physics. Review: The treatment is careful and clear throughout. The Book will be a valuable work of reference in its field for many years to come. Mathematical Reviews The book is written in a very informative style providing proofs where they support the understanding and referring to the literature for technical details and further study; the reader will very soon notice and appreciate the authors' thorough way of referencing.

W Beekmann, Zentrallblatt for Mathematik, Band 840/96.