Best approximation of sums of elements and a theorem of Newman and Shapiro

D. J. Newman and H. S. Shapiro, “Some theorems on Chebyshev approximation,” Duke Math. J.,30, No. 4, 673–681 (1963). V. M. Tikhomirov, Some Questions in Approximation Theory [in Russian], Moscow State Univ. (1976). W. Rudin, Functional Analysis, McGraw-Hill, New York (1973). S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1977). S. N. Bernshtein, “On the best approximation of continuous functions of a given degree. V,” in: Collected Works, Vol. II, Constructive Function Theory [in Russian], Izd. Akad. Nauk SSSR, Moscow (1954), pp. 390–395. M. I. Ganzburg, “Multidimensional limit theorems of the theory of best polynomial approximations,” Sib. Mat. Zh.,23, No. 3, 30–47 (1982). S. N. Bernshtein, “On certain elementary extremal properties of polynomials of several variables,” in: Collected Works, Vol. II, Constructive Function Theory [in Russian], Izd. Akad. Nauk SSSR, Moscow (1954), pp. 433–436.