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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 �-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 Schr�dinger-type operator with point interactions | |
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| Introduction | |
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| The equation D<sup>�</sup> - �I = �<sub>x</sub> | |
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| Definition of operator realizations of D<sup>�</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 �-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 (�<sub>1</sub>(x) ≡ 1) | |
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| Asymptotics of singular Fourier integrals (�<sub>1</sub>(x) ≠ 1) | |
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| p-adic version of the Erd�lyi 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 �-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 | |