PREFACE(I)

The aim of this booklet is at three distinct targets:

 

(1)   To present to prospective mathematicians, an account of some elegant

but usually overlooked ideas from classical analysis,which are accessible at 

an early stage.We hope that this will help to provide them with a balanced

view of mathematics, and in particular, show that, at this level there are

attractive new ideas in analysis.

 

(2)   To provide a carrier for some mild propagenda for numerical analysis.

It must be made clear that this is not a main course in numerical analysis,

but rather an appetizer or relish. We have included occasional numerical

examples, for few scientists are not attracted by stricking numerical approx-

imations and computational tours de force. We point out, however, that the

clear-cut, best-possible results often available here are in sharp contrast to

the necessary vagueness in practical numerical analysis at levels near those

of current operations, althrough these have their own excitements.

 

(3)    To put on record, in a palatable context, some of the basic formulas 

and properties of the classical orthogonal polynomals, of which most

scentists have need from time to time.

 

      We consider that the core of the course is the Chebyshev theory sketched

in Chapter 3. A useful preliminary  to the some of the ideas used in this chapter

and Chapter 5 on Orthogonal Polynomial is a familiarity  with  the theory

of Sturm sequences. Althrough  no direct use is made of this theory, a good case

could be made for adding some account of this theory and its developments.

 

     We have developed in the text, or in the Problems, the basic formulae

concerning the Chebyshev and Legendre Polynomials--the corroesponding

results for the Laguerre and Hermite cases are recorded without proof.

 

     We have stressed the Aitken Interpolation Algorithm in the problems in 

Chapter 6-- this is a most valuable aid. In Chapter 7, we consider the account

of the Euler-Maclaurin Sum Formula and its application to the derivation of

the Stirling formula should not be omitted. Chapter 8 gives a brief intro-

duction to Funtional Analysis, so that we can present a reasonable proof of 

the existence of a best Chebyshev approximation. In Chapter 9 the idea of 

Gaussian quadratures is central. We have put considerable effort into the

collection of about 100 examples: the readers is recommended to do the 

same. Solutions to many are given at the end.

 

      In  preparing this printed version we have benefitted from comments by

T. K. Boehme, G. Baley Price, M. Lees and A. Sharples, who used the 

original mimeographed version in courses, and by Dieter Gaier, Walter 

Gustchi and J. Steinberg.