Could you explain me the "geometric meaning" of the following definition (it's taken from a book on spline functions theory)?
Definition: Given $1 \leq p \leq \infty$ a positive integer $r$, and $0 < t \leq (b - a)/r$, the function $$\omega(f;t)_p = \omega_r(f;t)_{L^p[a,b]} = \sup_{0 \leq h \leq t} ||\Delta^r_h f ||_{L^p[I_{rh}]} $$ is called the $r$-th modulus of smoothness of f in $L_p$. Here $\Delta_h^rf$ is the $r$-th forward difference and $I_{rh} = [a,b-rh]$.
I can understand the idea, but I tried to draw a simple function and tried to relate the moduli of smoothness to the derivative and I can't figure out the relationship. Probably for some cases is the max of the derivative, otherwise is related to the $L_p$ norm.