《Bernstein Polynomials》求取 ⇩

INTRODUCTION3

0．1 Introduction3

Ⅰ．BERNSTEIN POLYNOMIALS IN REAL DOMAIN5

1．1 The theorem of Weierstrass5

1．2 Other proofs of the theorem of Weierstrass7

1．3 Generalizations of the theorem of Weierstrass9

1．4 Derivatives of the Bernstein polynomials12

1．5 Estimations and lemmas13

1．6 The degree of approximation by Bernstein polynomials19

1．6．1 Asymptotic formulae22

1．7 Monotone functions,convex functions,and functions of bounded variation23

1．8 Further theorems on derivatives25

1．9 Discontinuous functions27

Ⅱ．GENERALIZATIONS OF BERNSTEIN POLYNOMIALS30

2．1 Approximation of integrable functions30

2．2 Approximation of measurable functions34

2．3 Bernstein polynomials on an unbounded interval36

2．4 General methods of summation37

2．5 Euler methods of summation41

2．6 Degenerate Bernstein polynomials43

2．7 Divided differences and generalized pnv(x)44

2．8 Approximation by linear aggregates of functions xa46

2．9 Some further generalizations48

Ⅲ．SPACES OF FUNCTIONS AND MOMENT PROBLEMS52

3．1 Banach spaces52

3．2 Functionals and moment problems53

3．3 Moment problems with Stieltjes integrals57

3．4 Rearrangements of functions59

3．5 Spaces A and M．Other spaces of integrable functions65

3．6 Inequalities for spaces A68

3．7 Continuous linear functionals in spaces A and A×72

3．7．1 Spaces M(φ,p) as conjugate spaces75

3．8 Moment problems for integrable functions77

3．9 Hausdorff methods of summation81

Ⅳ．BERNSTEIN POLYNOMIALS OF ANALYTIC FUNCTIONS87

4．1 Preliminary theorems87

4．2 Contour integrals and asymptotic formulae92

4．3 The loop L？ and its properties94

4．3．1 Further properties of the loop100

4．4 The fundamental convergence theorem104

4．5 Sets of convergence and autonomous sets107

4．6 Examples of autonomous sets and sets of convergence110

4．7 Functions analytic on a part of the interval [0,1]116

4．8 Summation of power series117

4．9 Degenerate Bernstein polynomials in the complex domain121

BIBLIOGRAPHY125

INDEX129