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\(p\)-adic Colombeau–Egorov type theory of generalized functions. (English) Zbl 1065.46056
Summary: The \(p\)-adic Colombeau–Egorov algebra of generalized functions on \(\mathbb Q^n_p\) is constructed. For generalized functions, the operations of multiplication, Fourier transform, convolution, taking point values are defined. The operations of (fractional) partial differentiation and (fractional) partial integration are introduced in terms of Vladimirov’s pseudodifferential operator. The products of Bruhat–Schwartz distributions are well defined as elements of this algebra. In contrast to the usual Colombeau and Egorov \(\mathbb C\)-theories, where generalized functions on \(\mathbb R^n\) are not determined by their point values on \(\mathbb R^n\), \(p\)-adic Colombeau–Egorov generalized functions are uniquely determined by their point values on \(\mathbb Q^n_p\).

MSC:
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)
22E50 Representations of Lie and linear algebraic groups over local fields
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