Convergence Problems of Orthogonal Series

PrefaceChapter I Fundamental Ideas. Examples of Series of Orthogonal Functions 1. Orthogonality, Orthogonalization, Series of Orthogonal Functions 2. The Space L2μ. The Riesz-Fischer Theorem. Complete Orthogonal Systems 3. Orthogonal Polynomials 4. The Jacobi Polynomials 5. Bounds for General Orthonormal Systems and Orthonormal Systems of Polynomials 6. Haar's Orthogonal System 7. Rademacher's and Walsh's Orthogonal Systems. Relations to the Theory of ProbabilityChapter II Investigation of the Convergence Behaviour of Orthogonal Series by Methods Belonging to the General Theory of Series 1. General Summation Processes 2. The Abel Transform. Some Tauberian Theorems 3. The Fundamental Theorem Concerning the Convergence of Orthogonal Series 4. Everywhere Divergent Orthogonal Series 5. Convergence with Every Arrangement of the Terms 6. Generalities on the Cesàro Summation of Orthogonal Series 7. Convergence of Subsequences and Cesàro Summability 8. Coefficient Tests for the Cesàro Summability of Orthogonal Series 9. Abel Non-Summable Orthogonal Series with Monotone Coefficients 10. Menchoff's Summation Theorem 11. Orthogonal Universal SeriesChapter III The Lebesgue Functions 1. The Significance of the Lebesgue Functions for Convergence Problems 2. Multiplicatively Orthogonal Systems. Generalization of the Walsh Series 3. The Lebesgue Functions of the Cesàro Summation 4. Summability of Orthogonal Series Arising in Terms of the Functions of a Polynomial-Like System 5. the Order of Magnitude of the Lebesgue Functions 6. Impossibility of the Improvement of the Coefficient TestsChapter IV Classical Convergence Problems 1. Banach Spaces. Functionals 2. The Singular Integrals 3. Convergence of the Polynomial-Like Orthogonal Expansions at Points of Continuity 4. The Convergence Features of the Singular Integrals at the Lebesgue Points 5. Generalities on the Degree of Approximation 6. Approximation Properties of Some Orthogonal Systems 7. Structural Convergence Conditions 8. Absolute Convergence of the Orthogonal SeriesBibliographyGeneral Index