Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Lebesgue integration
- 3 Some useful theorems
- 4 Convergence of sequences of functions
- 5 Local averages and convolution kernels
- 6 Some general remarks on Fourier transformation
- 7 Fourier theorems for good functions
- 8 Fourier theorems in Lp
- 9 Fourier theorems for functions outside Lp
- 10 Miscellaneous theorems
- 11 Power spectra and Wiener's theorems
- 12 Generalized functions
- 13 Fourier transformation of generalized functions I
- 14 Fourier transformation of generalized functions II
- 15 Fourier series
- 16 Generalized Fourier series
- Bibliography
- Index
14 - Fourier transformation of generalized functions II
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Lebesgue integration
- 3 Some useful theorems
- 4 Convergence of sequences of functions
- 5 Local averages and convolution kernels
- 6 Some general remarks on Fourier transformation
- 7 Fourier theorems for good functions
- 8 Fourier theorems in Lp
- 9 Fourier theorems for functions outside Lp
- 10 Miscellaneous theorems
- 11 Power spectra and Wiener's theorems
- 12 Generalized functions
- 13 Fourier transformation of generalized functions I
- 14 Fourier transformation of generalized functions II
- 15 Fourier series
- 16 Generalized Fourier series
- Bibliography
- Index
Summary
Functionals of types D′ and Z′
In order to give meaning to the Fourier transform of any locally integrable function it is necessary to generalize beyond the functionals in S′, and we describe now the functionals in classes D′ and Z′ which provide the necessary generalization. We give examples in section 14.2, but start in this section with the basic concepts.
In summary, the functionals of class D′ are based on the use of test functions in class D (the good functions of bounded support), whilst the functionals in class Z′ are based on the use of test functions in class Z (the Fourier transforms of functions in D). Much, but not all, of chapter 12 can simply be adapted by replacing S by D (or Z) and S′ by D′ (or Z′). For instance, functionals of type D′ and Z′ are defined as follows.
Definition An association of exactly one real or complex number with each ϕ∈D is said to be a functional in class D′ if there exists at least one sequence of ordinary functions such that fnϕ∈L for each ϕ∈D and each n, and such that for each ϕ∈D the number associated with ϕ is equal to limn→∞ int; fnϕ.
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- Chapter
- Information
- A Handbook of Fourier Theorems , pp. 145 - 154Publisher: Cambridge University PressPrint publication year: 1987