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Sequence transformations and their applications. (English) Zbl 0566.47018
Mathematics in Science and Engineering, Vol. 154. New York - London etc.: Academic Press. XIX, 257 p. (1981).
In this book a sequence transformation T is a mapping defined on sequences s in a Banach space B into sequences in B, with the following conditions:
i) \(T(\alpha s)=\alpha T(s)\), \(\alpha\in {\mathbb{C}}\); ii) \(T(s+c)=T(s)+T(c)\), c is a constant sequence; iii) T is regular; iv)T(s) converges more rapidly than s.
In the classical (Toeplitz) theory mostly linear T are considered but, as the author indicates, the classical methods restrict the field of applications considerably. Those who want to study the abstract theory of sequence transformations will not find what they look for because it is especially in the applications to numerical problems that the author is interested. (Therefore mostly \(B={\mathbb{C}})\). But in this direction the book is very complete. In fact it provides an abundance of information for those who are interested in the concrete evaluation of limits of any kind (sums of series, integrals,...). The author touches virtually every area of analysis, including interpolation and approximation, special functions, continued fractions and optimization methods. Also lots of examples are worked out. It is difficult to list here the algorithms described in the book. Not only because of the enormous variety of methods but also because of the difficulty in notation (this fact is the only - minor - disadvantage). The book is completed with an extended list of references.

MSC:
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A45 Sequence spaces (including Köthe sequence spaces)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
41A05 Interpolation in approximation theory
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)