Inequalities from complex analysis [[electronic resource] /] / John P. D'Angelo
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| Autore: |
D'Angelo John P
|
| Titolo: |
Inequalities from complex analysis [[electronic resource] /] / John P. D'Angelo
|
| Pubblicazione: | [Washington, D.C.], : Mathematical Association of America, c2002 |
| Descrizione fisica: | 1 online resource (280 p.) |
| Disciplina: | 515/.9 |
| Soggetto topico: | Functions of complex variables |
| Inequalities (Mathematics) | |
| Mathematical analysis | |
| Soggetto genere / forma: | Electronic books. |
| Note generali: | Description based upon print version of record. |
| Nota di bibliografia: | Includes bibliographical references (p. 257-259) and index. |
| Nota di contenuto: | ""Cover ""; ""Title ""; ""Contents""; ""Preface""; ""I. Complex Numbers""; ""I.1 The real number system""; ""I.2 Definition of the complex number field""; ""I.3 Elementary complex geometry""; ""I.4 Alternative definitions of the complex numbers""; ""I.4.1 Using matrices""; ""I.4.2 Using polynomials""; ""I.5 Completeness""; ""I.6 Convergence for power series""; ""I.7 Trigonometry""; ""I.8 Roots of unity""; ""I.9 Summary""; ""II. Complex Euclidean Spaces and Hilbert Spaces""; ""II.1 Hermitian inner products""; ""II.2 Orthogonality, projections and closed subspaces"" |
| ""II.3 Orthonormal expansion""""II.4 The polarization identity""; ""II.5 Generating functions and orthonormal systems""; ""III. Complex Analysis in Several Variables""; ""III.1 Holomorphic functions""; ""III.2 Some calculus""; ""III.3 The Bergman kernel function""; ""IV. Linear Transformations and Positivity Conditions""; ""IV.1 Adjoints and Hermitian forms""; ""IV.2 Solving linear equations""; ""IV.3 Linearization""; ""IV.4 Eigenvalues and the spectral theorem in finite dimensions""; ""IV.5 Positive definite linear transformations in finite dimensions""; ""IV.6 Hilbert's inequality"" | |
| ""IV.7 Additional inequalities from Fourier analysis""""V. Compact and Integral Operators""; ""V.1 Convergence properties for bounded linear transformations""; ""V.2 Compact operators on Hilbert space""; ""V.3 The spectral theorem for compact Hermitian operators""; ""V.4 Integral operators""; ""V.5 A glimpse at singular integral operators""; ""VI. Positivity Conditions for Real-valued Functions""; ""VI.1 Real variables analogues""; ""VI.2 Real-valued polynomials on C^n""; ""VI.3 Squared norms and quotients of squared norms""; ""VI.4 Plurisubharmonic functions"" | |
| ""VI.5 Positivity conditions for polynomials""""VII. Stabilization and Applications""; ""VII.1 Stabilization for positive bihomogeneous polynomials""; ""VII.2 Positivity everywhere""; ""VII.3 Positivity on the unit sphere""; ""VII.4 Applications to proper holomorphic mappings between balls""; ""VII.5 Positivity on zero sets""; ""VII.6 Proof of stabilization""; ""VIII. Afterword""; ""APPENDIX A""; ""A.1 Algebra""; ""A.2 Analysis""; ""Bibliography""; ""Index"" | |
| Sommario/riassunto: | Inequalities from Complex Analysis is a careful, friendly exposition of inequalities and positivity conditions for various mathematical objects arising in complex analysis. The author begins by defining the complex number field, and then discusses enough mathematical analysis to reach recently published research on positivity conditions for functions of several complex variables. The development culminates in complete proofs of a stabilization theorem relating two natural positivity conditions for real-valued polynomials of several complex variables. The reader will also encounter the Bergman kernel function, Fourier series, Hermitian linear algebra, the spectral theorem for compact Hermitian operators, plurisubharmonic functions, and some delightful inequalities. Numerous examples, exercises, and discussions of geometric reasoning appear along the way. Undergraduate mathematics majors who have seen elementary real analysis can easily read the first five chapters of this book, and second year graduate students in mathematics can read the entire text. Some physicists and engineers may also find the topics and discussions useful. The inequalities and positivity conditions herein form the foundation for a small but beautiful part of complex analysis. John P. D'Angelo was the 1999 winner of the Bergman Prize; he was cited for several important contributions to complex analysis, including his work on degenerate Levi forms and points of finite type, as well as work, some joint with David Catlin, on positivity conditions in complex analysis |
| Titolo autorizzato: | Inequalities from complex analysis |
| ISBN: | 0-88385-970-X |
| Formato: | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione: | Inglese |
| Record Nr.: | 9910463061403321 |
| Lo trovi qui: | Univ. Federico II |
| Opac: | Controlla la disponibilità qui |
Serie:
Carus mathematical monographs ; ; no. 28.