Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Part One Fourier Series and Periodic Distributions
- 1 Preliminaries
- 2 Fourier Series: Basic Theory
- 3 Periodic Distributions and Sobolev Spaces
- Part Two Applications to Partial Differential Equations
- Part Three Some Nonperiodic Problems
- Appendix A Tools from the Theory of ODEs
- Appendix B Commutator Estimates
- Bibliography
- Index
3 - Periodic Distributions and Sobolev Spaces
Published online by Cambridge University Press: 05 July 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Part One Fourier Series and Periodic Distributions
- 1 Preliminaries
- 2 Fourier Series: Basic Theory
- 3 Periodic Distributions and Sobolev Spaces
- Part Two Applications to Partial Differential Equations
- Part Three Some Nonperiodic Problems
- Appendix A Tools from the Theory of ODEs
- Appendix B Commutator Estimates
- Bibliography
- Index
Summary
In this chapter we introduce a class of generalized functions specially suited for the study of Fourier series and differential equations provided with periodic boundary conditions. The concept of generalized function, as the name itself indicates, is used to generalize the notion of function and the usual calculus, and can be employed to construct scenarios adapted to the study of various problems of mathematical physics and their generalizations. In fact, the theory of generalized functions is intimately connected to the development of applied mathematics and theoretical physics during the first half of the twentieth century. Objects like the Dirac δ function were used in the formulation of quantum mechanics long before they were rigorously defined (see [65], [145] and [146]). Generally speaking, a generalized function is a certain type of linear functional defined on a space of test functions. The reason for this terminology will become clear as we proceed. At this point it is worth while to note that the properties of the generalized functions reflect the properties of the test functions on which they are defined. For example, a generalized function is as differentiable (in a generalized sense) as the corresponding test functions (in the usual sense). Distributions are special classes of generalized functions introduced by L.
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- Fourier Analysis and Partial Differential Equations , pp. 132 - 210Publisher: Cambridge University PressPrint publication year: 2001