Functions of one complex variable II

Introduction.- 1.Definitions and Basic Results.- 2.Regular Power Series.- 3.Zeros.- 4.Infinite Products.- 5.Singularities.- 6.Integral Representations.- 7.Maximum Modulus Theorem and Applications.- 8.Spherical Series and Differential.- 9.Fractional Transformations and the Unit Ball.- 10.Generalizations and Applications.- Bibliography.- Index.

Regular Functions of a Quaternionic Variable

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 Borel 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 L1 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 Phragmen-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 L 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 Muntz-Szas 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

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

In the plethora of conceptual and algorithmic developments supporting data analytics and system modeling, humancentric pursuits assume a particular position owing to ways they emphasize and realize interaction between users and the data. We advocate that the level of abstraction, which can be flexibly adjusted, is conveniently realized through Granular Computing. Granular Computing is concerned with the development and processing information granules – formal entities which facilitate a way of organizing knowledge about the available data and relationships existing there. This study identifies the principles of Granular Computing, shows how information granules are constructed and subsequently used in describing relationships present among the data.

Granular computing for data analytics: a manifesto of human-centric computing

We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.

Periodic points of birational maps on projective surfaces

In the space ℍ of quaternions, we investigate the natural, invariant geometry of the open, unit disc Δ ℍ and of the open half-space ℍ + . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Mobius transformations of Δ ℍ and ℍ + and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Mobius transformations, and use it to define the analog of the Poincare distances and differential metrics on Δ ℍ and ℍ + .

/pdf/moebius-transformations-and-the-poincare-distance-in-the-3y9hdz1dpj.pdf

Moebius transformations and the Poincare distance in the quaternionic setting

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

/pdf/the-schwarz-pick-lemma-for-slice-regular-functions-xik4z7718o.pdf

The Schwarz-Pick lemma for slice regular functions

https://www.mat.uniroma2.it/~fbracci/download/lfm.pdf

Linear Fractional Maps of the Unit Ball: A Geometric Study

Micro and macro models of granular computing induced by the indiscernibility relation

The aim of this paper is to build a new family of lattices related to some combinatorial extremal sum problems, in particular to a conjecture of Manickam, Miklos and Singhi. We study the fundamental properties of such lattices and of a particular class of boolean functions defined on them.

/pdf/a-class-of-lattices-and-boolean-functions-related-to-the-kvznk9olbj.pdf

Cinzia Bisi

Papers

Moebius transformations and the Poincare distance in the quaternionic setting

The Schwarz-Pick lemma for slice regular functions

Linear Fractional Maps of the Unit Ball: A Geometric Study

Micro and macro models of granular computing induced by the indiscernibility relation

A class of lattices and boolean functions related to the Manickam-Miklös-Singhi conjecture