Some theorems on function smoothing (Q1358606)

Here is part of a sample theorem (Theorem 4): Let \(X\) be a real Banach space and \(B\) its unit ball. Then \(X\) is superreflexive if and only if, for each uniformly continuous real-valued function \(f\) on \(B\), \[ \inf|f-\varphi|_B= 0, \] where the infimum is taken over all \(\varphi: B\to\mathbb{R}\) which are Fréchet differentiable with uniformly continuous Fréchet derivative and \(|\cdot|_B\) denotes the sup-norm. The paper contains several other results which are too technical to be described here in more detail. Unfortunately, it is marred by typographical errors, some unexplained notation (even in the Russian original), and by the bad translation.