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.)

We consider general linear parabolic equations in a given time dependent domain and we describe a general class of Galerkin-type approximations which are continuous with respect to the space variables, but which admit discontinuities with respect to time at each step. Unconditional stability is proved and a general error estimate is established. These results are applied to certain finite element methods based on space-time finite elements.

Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain

Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (x?y?)?=f(x,y), y(0)=A, y(1)=B, 0<?<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all ??(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM1 based on just one evaluation off. For uniform mesh we obtain two methodsM2 andM3 each based on three evaluations off. For ?=0,M1 andM2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h2)-convergence established and illustrated by numerical examples.

Finite difference methods and their convergence for a class of singular two point boundary value problems

Composite overlapping grid methods were studied by Volkov [ Proceedings of the Steklov Institute of Mathematics, 96 (1968), pp. 145--185] and Starius [ Numer. Math., 28 (1977), pp. 243--258] more than 20 years ago. More recently several authors have used such methods for a variety of problems (not necessarily elliptic BVPs). This work is motivated by the recent work of Chesshire and Henshaw [ J. Comp. Phys., 90 (1990), pp. 1--64], who developed general software and have carried out extensive computational studies. This report is limited to the case where the finite difference operators are of positive type. However, within that class of problems we present a general theory.

On the Overlapping Grid Method for Elliptic Boundary Value Problems

I. Introduction. In a previous paper [6], S. V. Parter and the author have studied finite-difference methods for elliptic differential equations of the second order whose coefficients are singular on a portion of the boundary; the uniform convergence of the approximations and the existence of a solution of the Dirichlet problem were proved for a class of such equations. The present work is an extension of those results to parabolic initial boundary-value problems. The class of problems that we consider includes the cases of nonhomogeneous differential equations, of time-dependent coefficients, of time-dependent domains and of over-determined Dirichlet problems.

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Numerical methods and existence theorems for parabolic differential equations whose coefficients are singular on the boundary

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Pierre Jamet

Papers

Numerical methods and existence theorems for parabolic differential equations whose coefficients are singular on the boundary

Finite Difference Methods for the Computation of the "Poisson Kernel" of Elliptic Operators