(美)沃爾特·魯丁
沃爾特·魯丁（Walter Rudin），1953年于杜克大學獲得數學博士學位。曾先後執教於麻省理工學院、羅切斯特大學、威斯康星大學麥迪遜分校、耶魯大學等。他的主要研究興趣集中在調和分析和複變函數上。除本書外，他還著有《Functional Analysis》（泛函分析）和《Principles of Mathematical Analysis》（數學分析原理）等其他名著。這些教材已被翻譯成十幾種語言，在世界各地廣泛使用。
目錄
Preface Prologue: The Exponential Function Chapter 1 Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ∞] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2 Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borei measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3 LP-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4 Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5 Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6 Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7 Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises
Chapter 8 Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9 Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra Lt Exercises Chapter 10 Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11 Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12 The Maximum Modulus Principle Introduction The Schwarz lemma The Phragrnen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13 Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14 Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class y
Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15 Zeros of Holomorphic Functions Infinite products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Miintz-Szasz theorem Exercises Chapter 16 Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17 Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18 Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19 Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20 Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index
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