Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

CONTENTS Introduction Story One. Historical notes Story Two. Some methods for deriving the cardinal and allied series Story Three. L 2 and L p theory Story Four. The cardinal series and LCA groups Story Five. Extensions to higher dimensions Conclusion INTRODUCTION

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Five short stories about the cardinal series

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.)

T s Ts = o ( l o g l o g I~]), t~]-+ 0% for almost eve ry x in T,. I n Sections 1 to 3 in this pape r we prove among o ther things the following theorem, which generalizes the L p es t imate of the opera tor M* in [7]. TEEORE~. Assume that k is a Calderdn -Zygmund kernel defined in R', s ~_ 2, which has continuous derivatives of order ~ s 41 outside the origin. Let the operator M be defined by

/pdf/convergence-almost-everywhere-of-certain-singular-integrals-vsfm97sw9m.pdf

Convergence almost everywhere of certain singular integrals and multiple Fourier series

0. Preface. This article is a survey of certain aspects of the theory of multiple Fourier and trigonometric series. I t is by no means meant to be a complete survey; for example, it is practically disjoint with the material covered on the subject in Zygmund's book [38, Chapter 17.]. There are eight sections to this survey. §1 is the introduction. §2, §3, and §4 are expository in the sense that the main theorems in each section are proved. §5, §6, and §7 are descriptive. §8 consists of two bibliographies, a bibliography for the survey itself, and a general bibliography. §2 deals with the now classical theory of the Bochner-Riesz summability of multiple Fourier series and the Abel summability of multiple Fourier series. §3 presents Bochner's counter-example for the critical index in summability theory in considerable detail. §4 is concerned with the uniqueness of multiple trigonometric series and proves the main theorem in the subject so far, i.e. uniqueness under Abel summability (due to the present author). §5 describes some results in conjugate multiple Fourier series defined by means of the Calderón-Zygmund kernel and related topics, i.e. analyticity in several variables. §6 deals with the Riemannian theory of multiple trigonometric series. §7 describes some applications to geometric integration theory and potential theory.

https://projecteuclid.org/download/pdf_1/euclid.bams/1183525779

Fourier series in several variables

The Divergence Theorem for Discontinuous Vector Fields

Uniqueness of Multiple Trigonometric Series

Quasilinear ellipticity and the ist eigenvalue

Summability of Multiple Fourier Series Introduction Iterated Fejer Summability of Fourier Series Bochner-Riesz Summability of Fourier Series Abel Summability of Fourier Series Gauss-Weierstrass Summability of Fourier Series Further Results and Comments Conjugate Multiple Fourier Series Introduction Abel Summability of Conjugate Series Spherical Convergence of Conjugate Series The Calpha-Condition An Application of the Calpha-Condition An Application of the Lp-Condition Further Results and Comments Uniqueness of Multiple Trigonometric Series Uniqueness for Abel Summability Uniqueness for Circular Convergence Uniqueness, Number Theory, and Fractals Further Results and Comments Positive Definite Functions Positive Definite Functions on SN-l Positive Definite Functions on TN Positive Definite Functions on SN1-l x TN Further Results and Comments Nonlinear Partial Differential Equations Reaction-Diffusion Equations on the N-Torus Quasilinear Ellipticity on the N-Torus Further Results and Comments The Stationary Navier-Stokes Equations Distribution Solutions Classical Solutions Further Results and Comments Appendix A: Integrals and Identities Appendix B: Real Analysis Appendix C: Harmonic and Subharmonic Functions Bibliography Index

Victor L. Shapiro

Papers

Fourier series in several variables

The Divergence Theorem for Discontinuous Vector Fields

Uniqueness of Multiple Trigonometric Series

Quasilinear ellipticity and the ist eigenvalue

Fourier Series in Several Variables with Applications to Partial Differential Equations