INTRODUCTION 3
0．1 Introduction 3
Ⅰ．BERNSTEIN POLYNOMIALS IN REAL DOMAIN 5
1．1 The theorem of Weierstrass 5
1．2 Other proofs of the theorem of Weierstrass 7
1．3 Generalizations of the theorem of Weierstrass 9
1．4 Derivatives of the Bernstein polynomials 12
1．5 Estimations and lemmas 13
1．6 The degree of approximation by Bernstein polynomials 19
1．6．1 Asymptotic formulae 22
1．7 Monotone functions,convex functions,and functions of bounded variation 23
1．8 Further theorems on derivatives 25
1．9 Discontinuous functions 27
Ⅱ．GENERALIZATIONS OF BERNSTEIN POLYNOMIALS 30
2．1 Approximation of integrable functions 30
2．2 Approximation of measurable functions 34
2．3 Bernstein polynomials on an unbounded interval 36
2．4 General methods of summation 37
2．5 Euler methods of summation 41
2．6 Degenerate Bernstein polynomials 43
2．7 Divided differences and generalized pnv(x) 44
2．8 Approximation by linear aggregates of functions xa 46
2．9 Some further generalizations 48
Ⅲ．SPACES OF FUNCTIONS AND MOMENT PROBLEMS 52
3．1 Banach spaces 52
3．2 Functionals and moment problems 53
3．3 Moment problems with Stieltjes integrals 57
3．4 Rearrangements of functions 59
3．5 Spaces A and M．Other spaces of integrable functions 65
3．6 Inequalities for spaces A 68
3．7 Continuous linear functionals in spaces A and A× 72
3．7．1 Spaces M(φ,p) as conjugate spaces 75
3．8 Moment problems for integrable functions 77
3．9 Hausdorff methods of summation 81
Ⅳ．BERNSTEIN POLYNOMIALS OF ANALYTIC FUNCTIONS 87
4．1 Preliminary theorems 87
4．2 Contour integrals and asymptotic formulae 92
4．3 The loop L？ and its properties 94
4．3．1 Further properties of the loop 100
4．4 The fundamental convergence theorem 104
4．5 Sets of convergence and autonomous sets 107
4．6 Examples of autonomous sets and sets of convergence 110
4．7 Functions analytic on a part of the interval [0,1] 116
4．8 Summation of power series 117
4．9 Degenerate Bernstein polynomials in the complex domain 121
BIBLIOGRAPHY 125
INDEX 129