vi Contents
Chapter 4. Summability of Fourier Series 91
4.1. Ces` aro summability of Fourier Series 91
4.2. Special coefficients 111
4.3. Summability 120
4.4. Summability kernels 130
Notes 134
Chapter 5. Fourier Series in Mean Square 135
5.1. Vector spaces of functions 135
5.2. Parseval’s Identity 138
Notes 148
Chapter 6. Trigonometric Polynomials 149
6.1. Sampling and interpolation 149
6.2. Bernstein’s Inequality 158
6.3. Real-valued and nonnegative trigonometric polynomials 162
6.4. Littlewood polynomials 165
6.5. Quantitative approximation of continuous functions 175
Notes 182
Chapter 7. Absolutely Convergent Fourier Series 183
7.1. Convergence 183
7.2. Wiener’s theorem 191
Notes 194
Chapter 8. Convergence of Fourier Series 195
8.1. Conditions ensuring convergence 195
8.2. Functions of bounded variation 198
8.3. Examples of divergence 205
Notes 209
Chapter 9. Applications of Fourier Series 211
9.1. The heat equation 211
9.2. The wave equation 213
9.3. Continuous, nowhere differentiable functions 215
9.4. Inequalities 217
9.5. Bernoulli polynomials 220
9.6. Uniform distribution 229
9.7. Positive definite kernels 239
9.8. Norms of polynomials 241
Notes 246
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