复分析中的不等式（影印版）

本书细致且友好地讲述 了一些相当有趣的数学内容 。作者首先定义了复数域， 并在前几章中对一些标准的 数学分析内容做了新颖的介 绍。作者从最近的研究文献 所引入的一些成果，将本书 推向高潮，对这些成果，本 书给出易于理解的完整证明 和一些令人惊讶的推论。一 个统一主题是希尔伯特第十 七问题的复变量类似。书中 大量的例子、练习以及关于 几何推理的讨论，对读者十 分有益。本书可供数学专业 的本科生和研究生阅读，也 可供物理学家和工程师参考 。

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.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 IⅣ.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 Cn VI.3 Squared norms and quotients of squared norms VI.4 Plurisubharmonic functions VI.5 Positívity 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