Various extensions of the Müntz-Szász theorem

Abstract

The Müntz-Szász Classical Theorem characterizes increasing sequences $\{\lambda_{j}\}^{+\infty}_{j=0}$ with $0=\lambda{_0}<\lambda{_1}<\lambda{2}<...$ for which the space $\langle1, x^{\lambda{_1}}, x^{\lambda{_2}},...\rangle$ is dense or not in $C([0, 1])$, depending on if the series $\sum^{+\infty}_{j=1}1/\lambda_{j}$ diverges or not respectively. In the book Polynomials and Polynomials Inequalities (see [7]), Tamás Erdélyi and Peter Borwein explain the tools needed in order to show a complete and extended proof of the Müntz-Szász Theorem. To do so, they use some techniques of complex analysis and also the algebraic properties of the zeros of some functions called Chebyshev functions. On these notes we put together all these ideas, beginning with the well known Weierstrass Approximation Theorem, continuing with the development of the complex analysis results needed and giving a complete proof of an extended version of the Müntz-Szász Theorem. Such new version characterizes arbitrary sequences $\{\lambda_{j}\}^{+\infty}_{j=0}$ of different arbitrary positive real numbers (except for $\lambda_{0}=0$ for which the space of continuous functions spanned by the powers $x^{\lambda j}$ is dense or not in $C([0, 1])$. In that case, it depends on if the series $\sum^{+ \infty}_{j=1}\lambda_{j}/(\lambda^{2}_{j}+1)$ diverges or not respectively. Moreover, pursuing in this direction, we also have studied an equivalent result for the Lebesgue spaces that characterizes arbitrary different sequences $\{\lambda_{j}\}^{+\infty}_{j=1}$ of real numbers greater than $-1/p$ for which the space $\langle x^{\lambda 1}, x^{\lambda 2}, x^{\lambda 3}...\rangle$ is dense or not in $L^p([0,1])$ which in that case depends on if the series $\sum^{+\infty}_{j=1}(\lambda_{j}+1/p)/((\lambda_{j}+1/p)^{2}+1)$ diverges or not respectively.