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.
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.
1: Historical Overview
2: Summability Methods in General
3: Borel's Methods of Summability
4: Relations with the family of circle methods
6: Albelian Theorems
7: Tauberian Theorems - I
8: Tauberian Theorems - II
9: Relationships with other methods
10: Applications of Borel's Methods