On approximation by integral Müntz polynomials
Let \(C_ 0[0,\alpha]\) be the set of all continuous functions f on the interval [0,\(\alpha\) ] such that f(m) is integer for any integer m in [0,\(\alpha\) ]. Let \(0=\lambda_ 0<\lambda_ 1<\lambda_ 2<..\). be a fixed sequence of real numbers and \(P_{\Lambda}\) the set of Müntz polynomials \(p(x)=\sum^{n}_{k=0}a_ k x^{\lambda_ k}\) with integer coefficients \(a_ k\). If the Müntz condition \(\sum 1/\lambda_ k=\infty\) is satisfied, then \(P_{\Lambda}\) is dense in \(C_ 0[0,1]\). On the other hand, it was shown by \textit{Le Baron O. Ferguson} [Fourier analysis and approximation theory, Vol. I, Budapest 1976, Colloq. Math. Soc. Janos Bolyai 19, 359-370 (1978; Zbl 0421.41005) that \(P_{\Lambda}\) is closed in \(C_ 0[0,\alpha]\) for \(\alpha >1\) if the sequence \(\Lambda\) is such that (1) \(\lambda_{k+1}-\lambda_ k\geq c>0\) and (2) lim sup N/\(\lambda\) \({}_ N=0\). Ferguson derived this result by means of a criterion which states that \(P_{\Lambda}\) is closed in \(C_ 0[0,\alpha]\) if (3) lim inf \(M_ n>0\), where \(M_ n\) are the norms of certain Chebyshev polynomials. The author of the paper under review cites this criterion but with an unjustified replacement of the lim inf by lim sup in (3). He then uses this modified criterion to replace in (2) the lim sup by a lim inf.
scientific article; zbMATH DE number 3925454
scientific article