Linear methods in some smoothing problems (Q1905333)

In this very technical paper the author continues the investigations of \textit{N. J. Kalton}, \textit{J. W. Roberts} [Trans. Am. Math. Soc. 278, 803- 816 (1983; Zbl 0524.28008)], and \textit{Yu. A. Brudnyj} [Mat. Sb., N. Ser. 82(124), 175-191 (1970; Zbl 0204.135)] on extensions of Whitney's theorem on the approximation of functions by polynomials (vector valued case). The author shows, e.g., that for an arbitrary real Banach space \(Y\) there exists a continuous linear projection \(A : C(B_n, Y) \to {\mathcal P}_{k - 1} (B_n, Y)\) such that \(|f - Af |\leq C_k n^{(k - 1)/2} \omega_k (f,1)\) where \(\omega_k\) is a higher order module of continuity defined on the space of \(Y\)-valued continuous functions on the unit ball \(B_n\) of \(\mathbb{R}^n\). Moreover, uniqueness, existence and stability of best approximations of \(Y\)-valued continuous functions by \(Y\)-valued polynomials are given.