Theory of p-adic distributions : linear and nonlinear models /
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| Author / Creator: | Albeverio, Sergio. |
|---|---|
| Imprint: | Cambridge, UK ; New York : Cambridge University Press, 2010. |
| Description: | xvi, 351 p. ; 23 cm. |
| Language: | English |
| Series: | London Mathematical Society lecture note series ; 370 London Mathematical Society lecture note series ; 370. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/8052713 |
Table of Contents:
- Preface
- 1. p-adic numbers
- 1.1. Introduction
- 1.2. Archimedean and non-Archimedean normed fields
- 1.3. Metrics and norms on the field of rational numbers
- 1.4. Construction of the completion of a normed field
- 1.5. Construction of the field of p-adic numbers Q p
- 1.6. Canonical expansion of p-adic numbers
- 1.7. The ring of p-adic integers Z p
- 1.8. Non-Archimedean topology of the field Q p
- 1.9. Q p in connection with R
- 1.10. The space Q n p
- 2. p-adic functions
- 2.1. Introduction
- 2.2. p-adic power series
- 2.3. Additive and multiplicative characters of the field Q p
- 3. p-adic integration theory
- 3.1. Introduction
- 3.2. The Haar measure and integrals
- 3.3. Some simple integrals
- 3.4. Change of variables
- 4. p-adic distributions
- 4.1. Introduction
- 4.2. Locally constant functions
- 4.3. The Bruhat-Schwartz test functions
- 4.4. The Bruhat-Schwartz distributions (generalized functions)
- 4.5. The direct product of distributions
- 4.6. The Schwartz "kernel" theorem
- 4.7. The convolution of distributions
- 4.8. The Fourier transform of test functions
- 4.9. The Fourier transform of distributions
- 5. Some results from p-adic L 1 - and L 2 -theories
- 5.1. Introduction
- 5.2. L 1 -theory
- 5.3. L 2 -theory
- 6. The theory of associated and quasi associated homogeneous p-adic distributions
- 6.1. Introduction
- 6.2. p-adic homogeneous distributions
- 6.3. p-adic quasi associated homogeneous distributions
- 6.4. The Fourier transform of p-adic quasi associated homogeneous distributions
- 6.5. New type of p-adic ¿-functions
- 7. p-adic Lizorkin spaces of test functions and distributions
- 7.1. Introduction
- 7.2. The real case of Lizorkin spaces
- 7.3. p-adic Lizorkin spaces
- 7.4. Density of the Lizorkin spaces of test functions in L p (Q n p )
- 8. The theory of p-adic wavelets
- 8.1. Introduction
- 8.2. p-adic Haar type wavelet basis via the real Haar wavelet basis
- 8.3. p-adic multiresolution analysis (one-dimensional case)
- 8.4. Construction of the p-adic Haar multiresolution analysis
- 8.5. Description of one-dimensional 2-adic Haar wavelet bases
- 8.6. Description of one-dimensional p-adic Haar wavelet bases
- 8.7. p-adic refinable functions and multiresolution analysis
- 8.8. p-adic separable multidimensional MRA
- 8.9. Multidimensional p-adic Haar wavelet bases
- 8.10. One non-Haar wavelet basis in L 2 (Q p )
- 8.11. One infinite family of non-Haar wavelet bases in L 2 (Q p )
- 8.12. Multidimensional non-Haar p-adic wavelets
- 8.13. The p-adic Shannon-Kotelnikov theorem
- 8.14. p-adic Lizorkin spaces and wavelets
- 9. Pseudo-differential operators on the p-adic Lizorldn spaces
- 9.1. Introduction
- 9.2. p-adic multidimensional fractional operators
- 9.3. A class of pseudo-differential operators
- 9.4. Spectral theory of pseudo-differential operators
- 10. Pseudo-differential equations
- 10.1. Introduction
- 10.2. Simplest pseudo-differential equations
- 10.3. Linear evolutionary pseudo-differential equations of the first order in time
- 10.4. Linear evolutionary pseudo-differential equations of the second order in time
- 10.5. Semi-linear evolutionary pseudo-differential equations
- 11. A p-adic Schrödinger-type operator with point interactions
- 11.1. Introduction
- 11.2. The equation D ¿ - ¿I = ¿ x
- 11.3. Definition of operator realizations of D ¿ + V in L 2 (Q p )
- 11.4. Description of operator realizations
- 11.5. Spectral properties
- 11.6. The case of ¿-self-adjoint operator realizations
- 11.7. The Friedrichs extension
- 11.8. Two points interaction
- 11.9. One point interaction
- 12. Distributional asymptotics and p-adic Tauberian theorems
- 12.1. Introduction
- 12.2. Distributional asymptotics
- 12.3. p-adic distributional quasi-asymptotics
- 12.4. Tauberian theorems with respect to asymptotics
- 12.5. Tauberian theorems with respect to quasi-asymptotics
- 13. Asymptotics of the p-adic singular Fourier integrals
- 13.1. Introduction
- 13.2. Asymptotics of singular Fourier integrals for the real case
- 13.3. p-adic distributional asymptotic expansions
- 13.4. Asymptotics of singular Fourier integrals (¿ 1 (x) ≡ 1)
- 13.5. Asymptotics of singular Fourier integrals (¿ 1 (x) ≠ 1)
- 13.6. p-adic version of the Erdélyi lemma
- 14. Nonlinear theories of p-adic generalized functions
- 14.1. Introduction
- 14.2. Nonlinear theories of distributions (the real case)
- 14.3. Construction of the p-adic Colombeau-Egorov algebra
- 14.4. Properties of Colombeau-Egorov generalized functions
- 14.5. Fractional operators in the Colombeau-Egorov algebra
- 14.6. The algebra A* of p-adic asymptotic distributions
- 14.7. A* as a subalgebra of the Colombeau-Egorov algebra
- A. The theory of associated and quasi associated homogeneous real distributions
- A.1. Introduction
- A.2. Definitions of associated homogeneous distributions and their analysis
- A.3. Symmetry of the class of distributions AH 0 (R)
- A.4. Real quasi associated homogeneous distributions
- A.5. Real multidimensional quasi associated homogeneous distributions
- A.6. The Fourier transform of real quasi associated homogeneous distributions
- A.7. New type of real ¿-functions
- B. Two identities
- C. Proof of a theorem on weak asymptotic expansions
- D. One "natural" way to introduce a measure on Q p
- References
- Index