Let $P$ be a polynomial function not identically zero with real coefficients. Show that the restriction of $P$ to $\mathbb { R } _ { + } ^ { * }$ belongs to $E$ if and only if $P ( 0 ) = 0$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.