Dense sets in spaces of integrable functions
The author establishes some theorems concerning approximation of \(L^{p}\)-functions by 'polynomials of many variables' \(g\in L(\mathcal F)\) in which the role of variables play functions from a given set \(\mathcal F\) (i.e., \(L(\mathcal F) := \{g=\sum_{\alpha }^{}c_{\alpha }f_{1}^{\alpha _{1}}\cdot f_{2}^{\alpha _{2}}\cdot\dots\cdot f_{m}^{\alpha _{m}}; f_{i}\in \mathcal F\}\), where \(m\) are nonnegative integers, \(\alpha := (\alpha_{1},\dots,\alpha _{m})\) are multiindices, and \(c_{\alpha }\) are complex constants which differ from zero for a finite number of multiindices only). The paper defines the space \(L^{p}\) as a space of functions integrable with \(p-\)th power on a Hausdorff space \(T,\) being the countable union of compact subsets, with respect to a finite and regular Borel measure [and hence it represents a generalization of the paper of the same author, J. Math. Anal. Appl. 181, No. 2, 505-523 (1994; Zbl 0801.46031)]). Various sufficient conditions on \(\mathcal F\) are established which guarantee the density of the above-mentioned 'polynomials': the functions from \(\mathcal F\) are supposed to separate the points, some continuity properties are requested, and the set \(\mathcal F\) contains together with any function \(f\) its conjugate, too. To this aim, the author uses a slightly modified version of the Stone-Weierstrass theorem: he shows that the `Stone-Weierstrass approximation polynomials' from \(\mathcal F\) which tend to the function \(f\) can be chosen in such a way that their absolute values are uniformly bounded, which makes possible -- with the help of the Luzin theorem -- to use Lebesgue's dominated convergence theorem.
scientific article; zbMATH DE number 786116