Let $m\in \mathbb{N}$, $p\in [1,\infty]$, $W^{m,p}([0,1])$ the space of all functions $[0,1]\rightarrow \mathbb{R}$ which are $m$ times weakly differentiable and weak derivatives in $L^p$, 
$$|u|\_{W^{m,p}([0,1])}:=\|u^{(m)}\|\_{L^p([0,1])}$$
for all $u\in W^{m,p}$ the Sobolev seminorm where $u^{(m)}$ is the $m$-th derivative of $u$, $[n]=\{0,\dots,n\}$, $\ell([n],\mathbb{R})$ the set of all functions $[n]\rightarrow \mathbb{R}$, $\nabla \colon \ell([n],\mathbb{R})\rightarrow \ell([n-1],\mathbb{R})$ defined by 
$$(\nabla x)\_i:=x\_{i+1}-x\_i,$$
$$\|f\|\_{\ell^p}:=\left(\frac{1}{n}\sum\_{i\in [n]} |f\_i|^p\right)^{\frac{1}{p}}$$ the discrete $p$-norm and 
$$|f|\_{\ell^{m,p}}:=\|n^m\nabla^mf\|\_{\ell^p}$$
the discrete Sobolev norm. I want to find a linear operator $E\colon \ell([n],\mathbb{R})\rightarrow W^{m,p}([0,1])$ which is interpolatory, i.e. $Ef(k/n)=f\_k$ for all $k\in [n]$, and satisfies 
$$C\_1|f|\_{\ell^{m,p}}\leq |Ef|\_{W^{m,p}([0,1]}\leq C\_2|f|\_{\ell^{m,p}}$$for some $C\_1,C\_2$ independent of $n$ and $f$. I tried spline-interpolation of degree $m$. However I don't know how to prove the statement. Such a theorem would be very useful for me and maybe others to get some high-order approximation estimates.



Interpolation Operator Bounded in Sobolev Norm
I prove here that the upper bound for your inequality is true for $m=1$. However, this approach should help you prove what you want in general. (EDIT: This is true in general after checking with Charles Fefferman--see edit at bottom.)


What you're trying to do is similar to recent work by Charles Fefferman, Arie Israel, and Garving Luli. Using your notation, define $X\_m \,\colon= W^{m,p}(\mathbb{R})$ equipped with the homogeneous seminorm $|F|\_{X\_m} := ||F^{(m)}||\_{L^p(\mathbb{R})}$, and define $X\_m([n])\, \colon= \ell([n],\mathbb{R})$ equipped with the seminorm $|f|\_{X\_m([n])} \,\colon= \inf\{|F|\_{X\_m} \, \colon\, F \in X\_m,\, F=f \, \text{on}\, [n]\}$. Let $p \in (1,\infty)$. In [Sobolev Extension by Linear Operators](http://arxiv.org/abs/1205.2525), one of the theorems Fefferman, Israel, and Luli prove is that there exists a linear map $T \colon X\_m([n]) \rightarrow X\_m$ such that $Tf = f$ on $[n]$ and $|Tf|\_{X\_m} \leq C|f|\_{X\_m([n])}$, where $C$ depends only on $m,n,p$. Note that we also have the trivial fact $|f|\_{X\_m([n])} \leq |Tf|\_{X\_m}$. Their result is in fact true for extensions of any subset in $\mathbb{R}^n$ with $p \in (n,\infty)$, not just finite subsets of $\mathbb{R}$ (the restriction on $p$ comes from the Sobolev Embedding Theorem). Recently, they proved in [Fitting a Sobolev function to data](http://arxiv.org/abs/1411.1786), that when the subset is finite, $T$ additionally has the important but technical condition of "$\Omega$-assisted bounded depth," where $\Omega$ is a certain set of linear functionals on $X\_m([n])$. 


Applying this to your problem, define $Y\_m\,\colon= W^{m,p}([0,n])$ with $|F|\_{Y\_m} := ||F^{(m)}||\_{L^p(\mathbb{[0,n]})}$, and define $Y\_m([n])\, \colon= \ell([n],\mathbb{R})$ equipped with seminorm $|f|\_{Y\_m([n])} \, \colon= \inf\{|F|\_{Y\_m} \, \colon\, F \in Y\_m,\, F=f \, \text{on}\, [n]\}$. Note that $|f|\_{Y\_m([n])} = |f|\_{X\_m([n])}$. Suppose we have that $|f|\_{Y\_m([n])} = \big(\sum\limits\_{i \in [n-m]} |\nabla^m f|^p\big)^{\frac{1}{p}}$. Then, $$\big(\sum\limits\_{i \in [n-m]} |\nabla^m f|^p\big)^{\frac{1}{p}} = |f|\_{Y\_m([n])} \leq |Tf|\_{Y\_m} \leq |Tf|\_{X\_m} \leq C|f|\_{X\_m([n])} = C\big(\sum\limits\_{i \in [n-m]} |\nabla^m f|^p\big)^{\frac{1}{p}}$$ 
Multiplying in by $n^{m -\frac{1}{p}}$, this implies your desired inequality, using that $$n^{m -\frac{1}{p}}\big(\int\limits\_{0}^n |u^{(m)}(x)|^p \,dx\big)^{\frac{1}{p}} = \big(\int\limits\_{0}^1 |u^{(m)}(nx)|^p \,dx\big)^{\frac{1}{p}}$$
for $u \in Y\_m$ by scaling and the chain rule.


So, your problem comes down to comparing $|f|\_{Y\_m([n])}$ and $\big(\sum\_{i \in [n-m]} |\nabla^m f|^p\big)^{\frac{1}{p}}$. It is easy to see that $|f|\_{Y\_1([n])} \leq \big(\sum\_{i \in [n-1]} |\nabla f|^p\big)^{\frac{1}{p}}$, which gives the upper bound in your desired inequality for $m=1$. Let $f \in \ell([n],\mathbb{R})$, and let $G\_1 \colon [0,n] \rightarrow \mathbb{R}$ be the function whose graph consists of the straight lines connecting $f(i)$ to $f(i+1)$ for $i=0,\dots,n-1$. Then, $G\_1 \in Y\_1$ and for $z \in (i,i+1)$, $\frac{dG\_1}{dx}(z) = (\nabla f)\_i$. Then, $|f|\_{Y\_1([n])} \leq ||G\_1^{(1)}||\_{L^p(\mathbb{R})} = \big(\sum\_{i \in [n-1]} |\nabla f|^p\big)^{\frac{1}{p}}$.


EDIT: I asked Charles Fefferman today, just to make sure the direction I'm going is correct, if your discrete Sobolev norm and the norm induced on $[n]$ by restrictions of extensions are equivalent, and he said that they certainly are. The lower bound follows from considering a polynomial extension, and the upper bound should follow from an argument where you prove it for small subsets and use a partitions of unity to prove it for all of $[n]$. Since the norms on $[n]$ are equivalent, this means that using my little argument above, your problem is true, since it is equivalent to a specific case of Fefferman, Israel, and Luli's result.


