Applications of P-adic generalized functions and approximations by a system of P-adic translations of a function.
For various -spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in . For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...
For a Tychonoff space , is the lattice-ordered group (-group) of real-valued continuous functions on , and is the sub--group of bounded functions. A property that might have is (AP) whenever is a divisible sub--group of , containing the constant function 1, and separating points from closed sets in , then any function in can be approximated uniformly over by functions which are locally in . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...
It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.
We show a general method of construction of non--porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non--porous Suslin subset of a topologically complete metric space contains a non--porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non--porous element. Namely, if we denote the space of all compact subsets of a compact metric space with the Vietoris topology...