Solarity and Proximinality in Generalized Rational Approximation in Spaces $$C(Q)$$ and $$L^p$$
A. R. Alimov, Игорь Германович Царьков
Abstract
The present paper is concerned with problems of solarity, proximinality, approximative compactness, stability, and monotone path-connectedness in generalized rational approximation in spaces $$L^p$$ and $$C(Q)$$ . Efficiency of the new approaches and concepts used in the paper is illustrated by a number of examples. Solar properties of sets of generalized rational functions in the space $$C(Q)$$ are proved with the help of the new concept of $$ \mathring B $$ -complete sets: a closed set $$M$$ is called $$ \mathring B $$ -complete if, for each $$x\in X$$ and $$r>0$$ , the condition $$M_0:= \mathring B (x,r)\cap M)\ne\varnothing$$ implies that $$ {}\kern.2em\overline{\kern-.2em M}{} _0 \supset M\cap B(x,r)$$ . Existence and generalized approximative compactness in problems of generalized rational approximations in spaces $$C(Q)$$ and $$L^p$$ are proved using the new definition of algebraic completeness with the machinery of regular Deutsch convergence.