### Applications of P-adic generalized functions and approximations by a system of P-adic translations of a function.

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For various ${L}^{p}$-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 ${L}^{p}\left(\mu \right)$. For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...

For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and ${C}^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of ${C}^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. 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-$\sigma $-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma $-porous Suslin subset of a topologically complete metric space contains a non-$\sigma $-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma $-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology...