What does it mean for a function to be polynomially bounded

Question

There is a definition in my notes and says,

When functions are polynomially bounded, the initial conditions (the value on small inputs) do not make a difference for the solution in asymptotic notations. The initial conditions can make a difference, when the function is not polynomially bounded.

Can somebody explain what that means? Thank you

Answer 1

If a function $f$ is polynomially bounded it means there exists polynomials $g$ and $h$ such that for all $x$, $$g(x)\le f(x)\le h(x).$$

Answer 2

It depends on the context. In general, $f$ is said polinomially bounded if $|f(x)|\leqslant p(x)$ for some real polynomial $p$. But for example, in the context of the theory of distributions, $f$ is said polinomially bounded if it is smooth and there are real polynomials $p_m$ such that $|d^mf/dx^m|\leqslant p_m(x)$, $m=0,1,2,\dotsc$.