Weakly compact and Deutsch compact sets in approximation theory. Application to exponential sums

Approximative compactness type properties in problems of min- and max-approximation are studied, in this way, CLUR, Day–Oshman, Anderson–Megginson, CMLUR and AT classes of spaces arise naturally. These spaces are characterized in approximative terms. In particular, we characterize the spaces with max-approximatively compact unit ball and the spaces with approximatively compact unit ball, we also obtain results on min- and max-approximative  compactness for suns and max-suns.
In addition, the following problems of min- and max-approximation are solved:

• characterization of the spaces where each closed neighborhood of each convex existence set is approximatively compact;
• characterization of the spaces where each closed neighborhood of each convex existence  set  is an approximatively weakly compact Chebyshev set;
• characterization of the spaces where each closed neighborhood of each is approximatively weakly compact set is approximatively weakly compact;
• characterization of the spaces where the classes of  strongly and weakly approximatively compact sets coincide;
• characterization of the spaces where  where the classes of strongly and weakly max-approximatively compact sets coincide;
• characterization of the spaces where each sun (max-sun) is approximatively weakly compact (max-approximatively weakly compact);
• characterization of the spaces such that each point of the space, except the origin, is a point  of weak max-approximative compactness of the unit ball.

Various classes of exponential sums are proved to be monotone path-connected / Menger-connected. The set of extended exponential sums is shown to be a sun in $C[a,b]$. 
The results were obtained by the author in collaboration with Prof. I.G. Tsar'kov.