Riemann problem for p-cone in Vladimirov algebras

Kakichev, V. A.; Shelkovich, V. M.

doi:10.1007/bf01145431

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“…For example, the analytic representation of a distribution f (x) ∈ S (R n ) can be constructed either by the Bochner-Vladimirov type integral representation [24] and [25] or by n-dimensional Cauchy integral representation [7]. …”

Section: Analytic Representations Of Distributionsmentioning

confidence: 99%

“…In the papers by V. K. Ivanov and T. A. Verjbalovich [23], T. A. Verjbalovich [53], V. A. Kakichev and V. M. Shelkovich [24] (see Remark 6.3) and A. M. Kytmanov [30] multidimensional theories were constructed where the Itano's counterexamples [20] do not arise. In [23] and [53] analytic regularizations of distributions are constructed by the Cauchy integral representation, in [24] analytic regularizations of distributions are constructed by the Bochner-Vladimirov integral representation (see also [25]), in [30] harmonic regularizations of distributions are constructed by the Poisson integral representation. Later, a theory of binary multiplication of distributions using harmonic representation of distribution was constructed by V. Boie [5] and by Li Bang-He and Li Ya-Qing [3].…”

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New versions of the Colombeau algebras

Shelkovich

2005

Mathematische Nachrichten

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We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half-space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions. The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one-dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]- [16], [49] and [50].

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Section: Analytic Representations Of Distributionsmentioning

confidence: 99%

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confidence: 99%