Książka - szczegóły

CONTENTS
Introduction..................................................................................................................................................................................... 3
CHAPTER I. Basic properties of the best polynomials
1. Existence, uniqueness and the characteristic properties of the best polynomials............................................................................... 6
2. The direct application of the theorem concerning the (n, F)-points to computing the best polynomials. The Chebyshev polynomials....... 15
3. The characteristic properties of pairs of successive best polynomials........................................................................................................................... 26
CHAPTER II. Estimation of error of the beet approximation
4. Basic theorems................................................................................................................................................................................................. 33
5. Vallée Poussin type estimations................................................................... 38
6. Estimation of the error of the best approximation for functions which are differentiable several times........................................ 53
7. Estimation of the (n + 1)-st error of the best approximation based upon the knowledge of the n-th best polynomial................... 57
8. The relation between the error of the best approximation and the interval of approximation.................................................. 65
CHAPTER III. The distribution of (n)-points in the interval of approximation
9. The estimates dependent upon the value of the ratio $ε_{n + 1}(ξ)ε_n(ξ)$....................................................................................... 70
10. Auxiliary theorems..................................................................... 84
11. The estimates dependent upon the properties of the derivatives of the function ξ. General theorems....................................... 92
12. The estimates dependent upon the properties of the derivatives of the function ξ. The choice of function γp............................. 111
CHAPTER IV. Methods of computing well-approximating and best polynomials
13. Well-approximating polynomials................................................................................................................................................................ 121
14. Approximation of families of functions. The Zolotarev polynomials................................................................................................. 140
15. The theory of the method of Remez................................................................................................................................................... 156
16. Other methods of computing the best polynomials................................................................................................................................ 169
References..................................................................................................................................................... 175