Classical measures of smoothness

There exist many different ways of measuring the smoothness of a function $f$ . The most natural one is certainly the order of differentiability, i.e. the maximal index $m$ such that $f^{\left(m\right)}={\left(\frac{d}{dx}\right)}^{m}f$ is continuous. To this particular measure of smoothness, we can associate a class of function spaces : if $I$ is an interval of $\mathbb{R}$ , we denote by ${\mathcal{C}}^m\left(I\right)$ the space of continuous functions which have bounded and continuous derivatives, up to the order $m$ . This space can beequipped with the norm

for which it is a Banach space. (That is, the space is a vector space; the norm satisfies the triangleinequality; $\parallel f\parallel =0$ is possible only if $f=0$ ; finally, all Cauchy sequences converge: if wehave a sequence with entries $f_n\in {\mathcal{C}}^m\left(I\right)$ for which $\parallel f_n-f_{n^{\text{'}}}\parallel$ can be made arbitrarily small simply by choosing $n,n^{\text{'}}$ sufficiently large, then the $f_n$ (and all their derivatives up to the $m$ th) converge uniformly to some function $f$ in ${\mathcal{C}}^m$ (and its derivatives).

In the case of a multivariate domain $\Omega \in {\mathbb{R}}^d$ , we define ${\mathcal{C}}^m\left(\Omega \right)$ to be the space of continuous functions which have bounded and continuous partial derivatives ${\partial }^{\alpha }f:=\frac{{\partial }^{\left|\alpha \right|}f}{\partial x_1^{{\alpha }_1}\cdots \partial x_d^{{\alpha }_d}}$ , for $\left|\alpha \right|:={\alpha }_1+\cdots +{\alpha }_d=0,\cdots ,m$ . This space canalso be equipped with the norm

for which it is a Banach space.

In many instances, one is somehow interested in measuring smoothness in an average sense: for this purpose it is natural to introduce the Sobolev spaces $W^{m,p}\left(\Omega \right)$ consisting of all functions $f\in L^p$ with partial derivatives up to order $m$ in $L^p$ . Here $p$ is a fixed index in $[1,+\infty ]$ . (Recall that ${\parallel f\parallel }_{L^p}=[{\int }_{\Omega }{\left|f\left(x\right)\right|}^p{]}^{\frac{1}{p}}$ if $p<+\infty$ and ${\parallel f\parallel }_{L^{\infty }}={sup}_{x\in \Omega }\left|f\left(x\right)\right|$ .) This space is also a Banach space, when equipped with the norm

Note that the norm for ${\mathcal{C}}^m$ spaces coincides with the $W^{m,\infty }$ norm.