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arXiv2007

The space of distributions with discontinuous test functions and a family of zero-sum games with discontinuous payoffs

In the present paper we consider one class of zero-sum games with discontinuous payoffs which may have no solutions in the sets of pure or mixed strategies. We show that, however, the solution always exists in the set of so-called $\mathcal R'$-mixed strategies which are the elements of the space $\mathcal R'$ of distributions with discontinuous test functions. The advantages of the new space of distributions (in comparison with the classical space $\mathcal D'$ of distributions with continuous or smooth test functions), that is, the possibility to define in $\mathcal R'$ the operations of integrations of distributions and multiplication of distributions by discontinuous functions, which are correct in the sense that they are everywhere defined, continuous and coincide with the ordinary operations for regular distributions, are crucial for our proof of existence of solution.

math.FA↗

arXiv2006

Distributions with dynamic test functions and multiplication by discontinuous functions

As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the present paper we construct a space of distributions where the general operation of multiplication by a discontinuous function is defined, continuous, commutative, associative and for which the Leibniz product rule holds. In the new space of distributions, the classical delta-function $δ_τ$ extends to a family of delta-functions $δ_τ^α$, dependent on the \textit{shape} $α$. We show that the various known definitions of the product of the Heaviside function and the delta-function in the classical space of distributions $\mathcal D'$ become particular cases of the multiplication in the new space of distributions, and provide the applications of the new space of distributions to the ordinary differential equations which arise in optimal control theory. Also, we compare our approach of the Schwartz distribution theory with the approach of the Colombeau generalized functions algebra, where the general operation of multiplication of two distributions is defined.

math.CA↗