Let $0<\lambda\le1$ and consider $$ \Theta:(\Bbb R[X]_0,||\cdot||_{\lambda})\longrightarrow(\mathcal C^{\lambda}[0,1],||\cdot||_{\lambda}) $$ defined as $$ \Theta(p):=\sup_{u\le\cdot}p(u) $$ (in the sense that the polynomial $p$ is sent by $\Theta$ to the function $t\mapsto\sup_{u\le t}p(u)$) where $\Bbb R[X]_0$ denotes the space of one variable polynomials with real coefficients which vanish at $0$, $\mathcal C^{\lambda}[0,1]$ is the space of $\lambda-$ Holder continuous functions $f:[0,1]\to\Bbb R$ and $||\cdot||_{\lambda}$ denotes the usual $\lambda-$Holder norm.
What kind of function is $\Theta$? What property has? Does $\Theta$ belong to a family of functions which has a name and for which a theory is developed somewhere? Can someone suggest me some reference?
I guess we are near Functional Analysis field, but, beyond this, only fog.
Many thanks