This book is based on two premises. First, the ideas and methods of the theory of distributions lead to formulations of classical theories which are satisfying and complete mathematically, and which at the same time provide the most useful viewpoint for applications. Second, mathematics and science students alike can profit from an approach which treats· the particular in a careful, complete, and modern way, and which treats the general as obtained by abstraction for the purpose of illuminating the basic structure exemplified in the particular. As an example, the basic L2 theory of Fourier series can be established quickly and with no mention of measure theory once L2(0, 277) is known to be complete. Here L2(0, 21r) is viewed as a subspace of the space of periodic distributions and is shown to be a Hilbert space. This leads to a discussion of abstract Hilbert space and orthogonal expansions. It is easy to derive necessary and sufficient conditions that a formal trigonometric series be the Fourier series of a distribution, an L2 distribution, or a smooth function. This in turn facilitates a discussion of smooth solutions and distribution solutions of the wave and heat equations. The book is organized as follows. The first two chapters provide background material which many readers may profitably skim or skip. Chapters 3, 4, and 5 treat periodic functions and distributions, Fourier series, and applications. Included are convolution and approximation (including the Weierstrass theorems), characterization of periodic distributions, elements of Hilbert space theory, and the classical problems of mathematical physics. The basic theory of functions of a complex variable is taken up in Chapter 6. Chapter 7 treats the Laplace transform from a distribution-theoretic point of view and includes applications to ordinary differential equations. Chapters 6 and 7 are virtually independent of the preceding three chapters; a quick reading of sections 2, 3, and 5 of Chapter 3 may help motivate the procedure of Chapter 7.
This book is based on two premises. First, the ideas and methods of the theory of distributions lead to formulations of classical theories which are satisfying and complete mathematically, and which at the same time provide the most useful viewpoint for applications. Second, mathematics and science students alike can profit from an approach which treats· the particular in a careful, complete, and modern way, and which treats the general as obtained by abstraction for the purpose of illuminating the basic structure exemplified in the particular. As an example, the basic L2 theory of Fourier series can be established quickly and with no mention of measure theory once L2(0, 277) is known to be complete. Here L2(0, 21r) is viewed as a subspace of the space of periodic distributions and is shown to be a Hilbert space. This leads to a discussion of abstract Hilbert space and orthogonal expansions. It is easy to derive necessary and sufficient conditions that a formal trigonometric series be the Fourier series of a distribution, an L2 distribution, or a smooth function. This in turn facilitates a discussion of smooth solutions and distribution solutions of the wave and heat equations. The book is organized as follows. The first two chapters provide background material which many readers may profitably skim or skip. Chapters 3, 4, and 5 treat periodic functions and distributions, Fourier series, and applications. Included are convolution and approximation (including the Weierstrass theorems), characterization of periodic distributions, elements of Hilbert space theory, and the classical problems of mathematical physics. The basic theory of functions of a complex variable is taken up in Chapter 6. Chapter 7 treats the Laplace transform from a distribution-theoretic point of view and includes applications to ordinary differential equations. Chapters 6 and 7 are virtually independent of the preceding three chapters; a quick reading of sections 2, 3, and 5 of Chapter 3 may help motivate the procedure of Chapter 7.