The closed linear span of \(\{x^k-c_k\}_1^{\infty}\) (Q1065304)

Several easily verified conditions on a sequence \((c_k)_1^{\infty}\) of real numbers are given which imply that the sequence of functions \((x^k-c_k)_1^{\infty}\) is total in \(C[0,1]\). This problem is equivalent to demanding that the function \(f(x) \equiv 1\) belongs to the closed linear hull of \((x^k-c_k)_1^{\infty}\) in \(C[0,1]\). For instance, if the sequence \((c_k)_1^{\infty}\) is such that for all \(k \geq M\), \(\epsilon(-1)^k(c_k-c)\geq 0,\) where \(c\in {\mathbb{R}}\) and \(\epsilon\in \{-1,1\}\), fixed, and if \(c_k-c\not\equiv 0\), then \((x^k-c_k)_1^{\infty}\) is total in \(C[0,1]\); if, in addition, \(c_k\neq c\) for infinitely many \(k\), with the help of Chebyshev polynomials an effective approximation to \(f(x) \equiv 1\) in \(C[0,1]\) by finite linear combinations of the \(x^k-c_k\) is given. Another condition is: \(|c_{n_k}-c|^{1/n_k}\to 0\) as \(k\to \infty\), where the subsequence \((n_k)_1^{\infty}\) satisfies the Müntz condition \(\sum^{\infty}_{k=1}(n_k)^{-1}=\infty\) and \(c_k\not\equiv c\); in the case when \(|c_k|^{1/k}\to 0\) as \(k\to \infty\), again, a good approximation to f(x)\(\equiv 1\) is explicitly constructed.