# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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) $T\left(\alpha s\right)=\alpha T\left(s\right)$, $\alpha \in ℂ$; ii) $T\left(s+c\right)=T\left(s\right)+T\left(c\right)$, 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 $B=ℂ\right)$. 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 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46A45 Sequence spaces 47-02 Research monographs (operator theory) 41A05 Interpolation (approximations and expansions) 41A65 Abstract approximation theory
##### Keywords:
sequence transformation