Approximation of the continuous functions on \(l_p\) spaces with \(p\) an even natural number (Q2194079)

Let \(p\) be an even natural number, \(K\subset l_p\) a compact nonempty set and \(L_n:C(K)\to C(K)\) a sequence of positive linear operators. The author proves that, under suitable assumptions, for every \(f\in C(K)\), \(\lim_{n\to\infty}L_n(f)=f\) uniformly on \(K\). The results are applied to infinite dimensional versions of the Bernstein operators, compared with the operators introduced and studied by \textit{L. D'Ambrosio} [J. Approx. Theory 140, No. 2, 191--202 (2006; Zbl 1098.41016)], and to infinite dimensional versions of the Kantorovich operators, compared with those introduced and studied by \textit{F. Altomare} et al. [Banach J. Math. Anal. 11, No. 3, 591--614 (2017; Zbl 1383.41003)].