Some analytic varieties in the polydisc and the Müntz-Szasz problem in several variables

Abstract:

For $1 \leqq {p_1} < {p_2} < \infty$ and $n \geqq 2$ it is shown that there exists a sequence of monomials $\{ \prod _{j = 1}^nS_j^\lambda mj\}$ with ${\lambda _{mj}} \sim m$ for each $j = 1, \ldots ,n$ whose linear span is dense in ${L^{{p_1}}}({I^n})$ but not in ${L^{{p_2}}}({I^n})$ (${I^n}$ is the Cartesian product of $n$ copies of the closed unit interval $[0, 1]$). Construction of the examples is via duality, making use of suitable analytic varieties in the polydisc.