Estimates of rate of convergence of best approximations by local functions

Questions are considered on the rate of convergence (in some abstract space of functions) of approximations that are the best in another space. Under specific conditions it is shown that the best approximations by local functions in a weighted Sobolev space $W^r_{p,B}$ yield almost-best approximation $W^r_{q,B}$ with $q\in[p,+\infty)$.