Limit Theorems of Polynomial Approximation with Exponential Weights.

Intro -- Contents -- Chapter 1. Introduction -- 1.1. A Brief Review -- 1.2. Results and Organization of the Monograph -- 1.3. Basic Notation and Some Preliminaries -- 1.4. Classes of Weights and Basic Estimates -- 1.5. Acknowledgements -- Chapter 2. Statement of Main Results -- 2.1. Limit Theorems of Polynomial Approximation with Exponential Weights -- 2.2. Approximation of Entire Functions of Exponential Type -- 2.3. Polynomial Inequalities in the Complex Plane -- Chapter 3. Properties of Harmonic Functions -- 3.1. The Poisson Integral Re H(w) -- 3.2. The Function h(r) and the Constant b[sub(n)] -- 3.3. The Functions φ(r) and φ[sub(1)](r) -- 3.4. The Main Estimate for Re H(w) -- Chapter 4. Polynomial Inequalities with Exponential Weights -- 4.1. Nikolskii-type Inequalities -- 4.2. Extremal Polynomials -- 4.3. Polynomial Inequalities in the Complex Plane -- 4.4. Proofs of Theorems 2.3.1 and 2.3.2 -- Chapter 5. Entire Functions of Exponential Type and their Approximation Properties -- 5.1. Entire Functions of Exponential Type -- 5.2. Approximation Properties of Entire Functions of Exponential Type -- Chapter 6. Polynomial Interpolation and Approximation of Entire Functions of Exponential Type -- 6.1. Interpolation on the Interval I[sub(n)] = […a[sub(n)](1+δ[sub(n)]), a[sub(n)](1+δ[sub(n)])] -- 6.2. Interpolation on I\I[sub(n)] -- 6.3. Proof of Theorem 2.2.1 -- 6.4. Proof of Theorem 2.2.2 -- Chapter 7. Proofs of the Limit Theorems -- 7.1. Proof of Theorem 2.1.1 -- 7.2. Proof of Theorem 2.1.2 -- 7.3. Proofs of Theorems 2.1.3 and 2.1.4 -- Chapter 8. Applications -- 8.1. Approximation of Individual Functions and Proof of Theorem 2.3.3 -- 8.2. An Asymptotically Sharp Constant of Weighted Approximation on the Class W[sup(r)]H[sup(λ)][I] -- 8.3. Convergence of Polynomials and a Mehler-Heine Formula for Orthonormal Polynomials.

Chapter 9. Multidimensional Limit Theorems of Polynomial Approximation with Exponential Weights -- 9.1. Multidimensional Limit Theorems with Exponential Weights -- 9.2. Proof of Theorem 9.1.3 -- 9.3. Proofs of Theorems 9.1.1 and 9.1.4 -- 9.4. Approximation of λ-Homogeneous Functions -- Chapter 10. Examples -- 10.1. W(x) = exp(…|x|α), α &gt -- 1 -- 10.2. W(x) = exp(-|x|) -- 10.3. W(x) = exp(…|x|α), 0&lt -- α &lt -- 1 -- 10.4. W(x) = exp(…|x|α), α → ∞ -- 10.5. Examples of Erdös Weights -- Appendix A. Appendix. Negativity of a Kernel -- A. l. Statement of the Main Result -- A. 2. Some Technical Results -- A. 3. Proof of Proposition A.1.1 -- Bibliography -- Index.

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