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Preface | |

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p-adic numbers | |

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Introduction | |

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Archimedean and non-Archimedean normed fields | |

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Metrics and norms on the field of rational numbers | |

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Construction of the completion of a normed field | |

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Construction of the field of p-adic numbers Q<sub>p</sub> | |

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Canonical expansion of p-adic numbers | |

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The ring of p-adic integers Z<sub>p</sub> | |

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Non-Archimedean topology of the field Q<sub>p</sub> | |

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Q<sub>p</sub> in connection with R | |

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The space Q<sup>n</sup><sub>p</sub> | |

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p-adic functions | |

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Introduction | |

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p-adic power series | |

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Additive and multiplicative characters of the field Q<sub>p</sub> | |

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p-adic integration theory | |

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Introduction | |

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The Haar measure and integrals | |

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Some simple integrals | |

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Change of variables | |

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p-adic distributions | |

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Introduction | |

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Locally constant functions | |

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The Bruhat-Schwartz test functions | |

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The Bruhat-Schwartz distributions (generalized functions) | |

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The direct product of distributions | |

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The Schwartz "kernel" theorem | |

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The convolution of distributions | |

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The Fourier transform of test functions | |

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The Fourier transform of distributions | |

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Some results from p-adic L<sup>1</sup> - and L<sup>2</sup>-theories | |

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Introduction | |

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L<sup>1</sup>-theory | |

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L<sup>2</sup>-theory | |

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The theory of associated and quasi associated homogeneous p-adic distributions | |

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Introduction | |

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p-adic homogeneous distributions | |

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p-adic quasi associated homogeneous distributions | |

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The Fourier transform of p-adic quasi associated homogeneous distributions | |

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New type of p-adic ï¿½-functions | |

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p-adic Lizorkin spaces of test functions and distributions | |

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Introduction | |

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The real case of Lizorkin spaces | |

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p-adic Lizorkin spaces | |

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Density of the Lizorkin spaces of test functions in L<sup>p</sup>(Q<sub>n</sub><sub>p</sub>) | |

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The theory of p-adic wavelets | |

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Introduction | |

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p-adic Haar type wavelet basis via the real Haar wavelet basis | |

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p-adic multiresolution analysis (one-dimensional case) | |

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Construction of the p-adic Haar multiresolution analysis | |

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Description of one-dimensional 2-adic Haar wavelet bases | |

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Description of one-dimensional p-adic Haar wavelet bases | |

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p-adic refinable functions and multiresolution analysis | |

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p-adic separable multidimensional MRA | |

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Multidimensional p-adic Haar wavelet bases | |

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One non-Haar wavelet basis in L<sup>2</sup>(Q<sub>p</sub>) | |

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One infinite family of non-Haar wavelet bases in L<sup>2</sup>(Q<sub>p</sub>) | |

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Multidimensional non-Haar p-adic wavelets | |

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The p-adic Shannon-Kotelnikov theorem | |

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p-adic Lizorkin spaces and wavelets | |

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Pseudo-differential operators on the p-adic Lizorldn spaces | |

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Introduction | |

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p-adic multidimensional fractional operators | |

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A class of pseudo-differential operators | |

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Spectral theory of pseudo-differential operators | |

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Pseudo-differential equations | |

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Introduction | |

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Simplest pseudo-differential equations | |

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Linear evolutionary pseudo-differential equations of the first order in time | |

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Linear evolutionary pseudo-differential equations of the second order in time | |

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Semi-linear evolutionary pseudo-differential equations | |

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A p-adic Schrï¿½dinger-type operator with point interactions | |

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Introduction | |

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The equation D<sup>ï¿½</sup> - ï¿½I = ï¿½<sub>x</sub> | |

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Definition of operator realizations of D<sup>ï¿½</sup> + V in L<sub>2</sub>(Q<sub>p</sub>) | |

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Description of operator realizations | |

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Spectral properties | |

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The case of ï¿½-self-adjoint operator realizations | |

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The Friedrichs extension | |

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Two points interaction | |

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One point interaction | |

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Distributional asymptotics and p-adic Tauberian theorems | |

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Introduction | |

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Distributional asymptotics | |

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p-adic distributional quasi-asymptotics | |

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Tauberian theorems with respect to asymptotics | |

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Tauberian theorems with respect to quasi-asymptotics | |

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Asymptotics of the p-adic singular Fourier integrals | |

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Introduction | |

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Asymptotics of singular Fourier integrals for the real case | |

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p-adic distributional asymptotic expansions | |

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Asymptotics of singular Fourier integrals (ï¿½<sub>1</sub>(x) ≡ 1) | |

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Asymptotics of singular Fourier integrals (ï¿½<sub>1</sub>(x) ≠ 1) | |

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p-adic version of the Erdï¿½lyi lemma | |

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Nonlinear theories of p-adic generalized functions | |

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Introduction | |

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Nonlinear theories of distributions (the real case) | |

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Construction of the p-adic Colombeau-Egorov algebra | |

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Properties of Colombeau-Egorov generalized functions | |

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Fractional operators in the Colombeau-Egorov algebra | |

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The algebra A* of p-adic asymptotic distributions | |

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A* as a subalgebra of the Colombeau-Egorov algebra | |

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The theory of associated and quasi associated homogeneous real distributions | |

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Introduction | |

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Definitions of associated homogeneous distributions and their analysis | |

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Symmetry of the class of distributions AH<sub>0</sub>(R) | |

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Real quasi associated homogeneous distributions | |

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Real multidimensional quasi associated homogeneous distributions | |

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The Fourier transform of real quasi associated homogeneous distributions | |

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New type of real ï¿½-functions | |

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Two identities | |

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Proof of a theorem on weak asymptotic expansions | |

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One "natural" way to introduce a measure on Q<sub>p</sup> | |

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References | |

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Index | |