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**Chebyshev polynomials. From approximation theory to algebra and number theory.
2nd ed.**
*(English)*
Zbl 0734.41029

Pure and Applied Mathematics. New York: John Wiley & Sons, Inc. xvi, 249 p. (1990).

Summary: This is a much expected edition of Rivlin’s 1974 classic work (Zbl 0299.41015) on the Chebyshev polynomials \(\cos(n \arccos(x))\). About one-third of the material is new. The first chapter deals with elementary properties including orthogonality. The second chapter is exclusively concerned with extremal problems, while the third chapter discusses expansions in Chebyshev polynomials. The last two chapters cover a miscellany of topics, including iterative behaviour and reducibility. The unifying theme of this book is the Chebyshev polynomials, but by the end of the book the reader is introduced to virtually all of the central ideas of polynomial approximation theory and many of the techniques. This is not surprising given the central and unique nature of these ubiquitous polynomials. Written in an easy style and well punctuated with exercises, this book provides a wealth of material for the expert while remaining accessible and interesting to a wide audience. This is an attractive book with a long shelf life and is a very welcome addition to any library.

### MSC:

41A50 | Best approximation, Chebyshev systems |

41-02 | Research exposition (monographs, survey articles) pertaining to approximations and expansions |

41A05 | Interpolation in approximation theory |