Existence of the best possible uniform approximation of a function of several variables by a sum of functions of fewer variables

On the proximinality of ridge functions

“…It should be remarked that the results of [7] tell nothing about necessary conditions for proximinality of the spaces considered there. To fill this gap in our case, we want to give a necessary condition for proximinality of R c (X) in C(X).…”

“…For i = 1, ..., m, let ϕ i be the continuous mappings from T onto T i . In [7], the authors obtained sufficient conditions for proximinality of the set…”

“…We are also interested in necessary conditions for proximinality. It follows from one result of Garkavi, Medvedev and Khavinson (see theorem1 [7]) that R a (X) is proximinal in B(X) for all subsets X of R n . There is also an answer (see theorem 2 [7]) for proximinality of R b (X) in B(X).…”

“…In a similar way as above, one can define the sets R a (X), R b (X) and R c (X). Using the results of [7], one can obtain sufficient (but not necessary) conditions for proximinality of these sets. This needs, besides paths (see Section1), the consideration of some additional and more complicated relations between points of X.…”

On the approximation by compositions of fixed multivariate functions with univariate functions

“…The question of existence of a best approximation from the set C h 1 h 2 to a function f in C(Q) (or, in other words, the proximinality of this set in the space of all continuous functions) is far from trivial. Some geometrical conditions on Q sufficient for the existence of a best approximation may be found in [6]. These conditions a priori require that the mapping h = (h 1 , h 2 ) : Q → h 1 (Q) × h 2 (Q) should separate points of Q.…”