Integration, Distributions, Holomorphic Functions, Tensor and Harmonic Analysis

Maps. Grassmann Algebra.- 2. Differential Forms.- 3. Cohomology Spaces. Poincaré Lemma.- 4. Integration of Differential Forms.- 5. Elements of Vector Analysis.- 6. The Differentiate Manifold.- 7. Tangent Spaces.- 8. Covariant Tensor Fields. Riemannian Metric and Differential Forms on a Manifold.- 9. Orientation of Manifolds. Examples.- 10. Poincaré-Stokes Theorem for a Manifold with Boundary.- 11. Tensor Densities. Weyl Duality. Homology.- 12. Weyl Duality and Hodge *operator. Generalized Green's formulas on Riemannian Manifold.- 13. Harmonic Forms. Hodge-Kodaira-de Rham Theory.- 14. Application to Electrodynamics.- 15. Invariant Forms (Hurwitz Integral). Cohomology of Compact Lie Groups.- 16. Complements. Exercises.- XV. Elementary Properties of Holomorphic Functions of Several Variables. Harmonic Functions.- 1. Holomorphic Mappings. Cauchy-Riemann Equations.- 2. Differential Forms on Complex Manifolds. Forms of Type (p, q). Operators d? and d??.- 3. Cauchy's Formula and its Applications.- 4. The Topology of the Space of Holomorphic Functions A(?).- 5. Elementary Properties of Harmonic Functions.- 6. Green's Function. Poisson Integral Formula. Harnack Theorems..- 7. Subharmonic Functions. Perron's Solution of the Dirichlet Proble.- XVI. Complex Analysis in One Dimension (Riemann Surfaces.- 1. Zeros of Holomorphic Functions of One Variable.- 2. Functions Holomorphic in an Annulus. The Laurent Expansion. Singularities.- 3. Meromorphic Functions.- 4. Application of the Calculus of Residues to the Evaluation of Integrals.- 5. Applications of the Argument Principle.- 6. Functions and Differential Forms on Riemann Surfaces.- 7. Analytic Continuation. Coverings. Fundamental Group. The Theory of Poincaré.- 8. The Koebe-Riemann Theorem. Non-Euclidean Geometry. Möbius Transformations.- 9. The Perron Method for Riemann Surfaces. The Radó Theorem.- 10. Resolutive Functions. Harmonic Measures. Brelot's Theorem.- 11. The Green's Function of a Riemann Surface.- 12. The Uniformization Theorem.- 13. Runge's Theorem. Theorem of Behnke and Stein. Theorem of Malgrange.- 14. Cousin Problems for Open Riemann Surfaces. Theorems of Mittag-Leffler and Weierstrass.- 15. Examples of Partial Fractions and Factorizations. Functions cos ?z, ?2/sin2 ?z, ?(z). Mellin and Hankel Formulae. Canonical Products.- 16. Elliptic Functions. Eisenstein Series. The Function ?.- 17. Modular Functions and Forms. The Modular Figure, Discontinuous Groups of Automorphisms.- 18. The Multiplicity Formula for Zeros of a Modular Form. Dimension of Vector Spaces M°(k, ?) of Cusp Forms.- 19. Mapping Properties of j. Picard Theorem. Elliptic Curves. Jacobi's Inversion Problem. Abel's Theorem.- 20. Uniformization Principle. Automorphic Forms. Riemann-Roch Theorem and Its Consequences. Historical Sketch.- 21. Appendices. Exercises (Proofs of Theorems of Runge, Florack, Koebe, and Hurwitz. Triangle Groups. Elliptic Integrals and Transcendental Numbers).- XVII. Normal and Paracompact Spaces. Partition of Unity..- 1. Locally Compact Spaces Countable at Infinity.- 2. Normal Spaces. Urysohn's Lemma.- 3. Extendibility of Continuous Functions on Normal Spaces.- 4. Tychonoff Spaces. Uniformizability. Compactification.- 5. The Theory of Maximal Ideals.- 6. The Gel'fand Theory of Maximal Ideals.- 7. Connection with Quantum Mechanics.- 8. Locally Finite Families.- 9. Paracompact Spaces. Partition of Unity. Metric Spaces are Paracompact.- XVIII. Measurable Mappings. The Transport of a Measure. Convolutions of Measures and Functions.- 1. Measurable Mappings.- 2. Topologies Determined by Families of Mappings.- 3. The Transport of a Measure.- 4. The Projective Limits of Hausdorff Spaces. Infinite Tensor Products and the Projective Limits of Measures.- 5. Convolutions of Measures and Functions.- 6. Convolutions of Functions and Measures on Rp.- 7. Convolutions of Integrable Functions.- XIX. The Theory of Distributions. Harmonic Analysis.- 1. The Space C0? (?).- 2. A Differentiate Partition of Unity on Rn.- 3. The Space of Test Functions. Distributions.- 4. Inductive Limits. The Topology of the Space D.- 5. The Pasting Together Principle for Distributions. The Support of a Distribution.- 6. The Space e(?). Distributions with Compact Supports.- 7. Operations on Distributions.- 8. The Convolution Algebra e (Rn).- 9. The Image of a Distribution.- 10. Remarks on the Tensor Products E??F and E??F. The Kernel Theorem.- 11. The Tensor Product E ? ?F of Hilbert Spaces.- 12. Regularization of Distributions.- 13. Examples of Distributions Important for Applications.- 14. The Fourier Transformation. The Space l.- 15. The Fourier Transformation as a Unitary Operator on the Space l2(Rn).- 16. Tempered Distributions. The Fourier Transformation in l'.- 17. The Laplace-Fourier Transformation for Functions and Distributions. The Paley-Wiener-Schwartz Theorem.- 18. Fundamental Solutions of Differential Operators.- 19. Positive-Definite Functions. Positive Distributions. The Theorems of Bochner and Minlos.- 20. Representations of Locally Compact Groups. The Relation between Unitary Representations and Positive-Definite Functions.- 21. The Haar Integral.- Index of Symbols.- Name Index.