Chapter 1. Introduction ......................................... 1
1.1. A Brief Review .......................................... 1
1.2. Results and Organization of the Monograph ............... 5
1.3. Basic Notation and Some Preliminaries ................... 7
1.4. Classes of Weights and Basic Estimates .................. 8
1.5. Acknowledgements ....................................... 13
Chapter 2. Statement of Main Results ........................... 15
2.1. Limit Theorems of Polynomial Approximation with
Exponential Weights .................................... 15
2.2. Approximation of Entire Functions of Exponential
Type ................................................... 16
2.3. Polynomial Inequalities in the Complex Plane ........... 17
Chapter 3. Properties of Harmonic Functions .................... 19
3.1. The Poisson Integral Re H(ω) ........................... 19
3.2. The Function h(r) and the Constant bn .................. 24
3.3. The Functions ф(r) and ф1(r) ........................... 29
3.4. The Main Estimate for Re H(ω) .......................... 35
Chapter 4. Polynomial Inequalities with Exponential Weights .... 43
4.1. Nikolskii-type Inequalities ............................ 43
4.2. Extremal Polynomials ................................... 45
4.3. Polynomial Inequalities in the Complex Plane ........... 55
4.4. Proofs of Theorems 2.3.1 and 2.3.2 ..................... 57
Chapter 5. Entire Functions of Exponential Type and their
Approximation Properties ............................... 59
5.1. Entire Functions of Exponential Type ................... 59
5.2. Approximation Properties of Entire Functions of
Exponential Type ....................................... 62
Chapter 6. Polynomial Interpolation and Approximation of
Entire Functions of Exponential Type ................... 67
6.1. Interpolation on the Interval In ....................... 67
6.2. Interpolation on I\In .................................. 71
6.3. Proof of Theorem 2.2.1 ................................. 72
6.4. Proof of Theorem 2.2.2 ................................. 74
Chapter 7. Proofs of the Limit Theorems ........................ 77
7.1. Proof of Theorem 2.1.1 ................................. 77
7.2. Proof of Theorem 2.1.2 ................................. 80
7.3. Proofs of Theorems 2.1.3 and 2.1.4 ..................... 82
Chapter 8. Applications ........................................ 85
8.1. Approximation of Individual Functions and Proof of
Theorem 2.3.3 .......................................... 85
8.2. An Asymptotically Sharp Constant of Weighted
Approximation on the Class WrHλ[I] ..................... 96
8.3. Convergence of Polynomials and a Mehler-Heine
Formula for Orthonormal Polynomials ................... 100
Chapter 9. Multidimensional Limit Theorems of Polynomial
Approximation with Exponential Weights ................ 105
9.1. Multidimensional Limit Theorems with Exponential
Weights ............................................... 105
9.2. Proof of Theorem 9.1.3 ................................ 108
9.3. Proofs of Theorems 9.1.1 and 9.1.4 .................... 111
9.4. Approximation of A-Homogeneous Functions .............. 113
Chapter 10. Examples .......................................... 117
10.1. W(x) = exp(-|x|α), α > 1 .............................. 117
10.2. W(x) = exp(-|x|) ...................................... 121
10.3. W(x) = ехр(-|x|α), 0 < α <1 ........................... 127
10.4. W(x) = ехр(-|x|α), α → ∞ .............................. 132
10.5. Examples of Erdös Weights ............................. 134
Appendix A. Appendix. Negativity of a Kernel .................. 137
A.1. Statement of the Main Result .......................... 137
A.2. Some Technical Results ................................ 138
A.3. Proof of Proposition A. 1.1 ........................... 144
Bibliography .................................................. 155
Index ......................................................... 161
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