Kolmogorov diameters in solution spaces of systems of partial differential equations
The functional dimension of the space of smooth solutions on \({\mathbb{R}}^ n\) of partial differential equations is a weak linear topological invariant, which in general does not have entirely adequate separation properties. In this paper the author considers the stronger linear topological invariant of diametral dimension of solution spaces of partial differential equations. Specifically if \(E_ p(0),0\subset {\mathbb{R}}^ n\) denotes the solutions of P(D) a system of partial differential equations with constant coefficients in a localisable analytically uniform space E then the relative Kolmogorov diameters of the neighbourhoods of 0 in \(E_ p(0)\) are estimated from above and below using a principle due to Ehrenpreis. The diametrial dimension of \(E_ p(0)\) is calculated and it is shown that in special cases \(E_ p({\mathbb{R}}^ n)\) and \(E_ p(0)\) are not isomorphic for 0 bounded.
scientific article; zbMATH DE number 3998311
scientific article