I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous representative $\bar{f}$ and show a property about it.
My questions (due to a lack of imagination) are now
What is meant by the continuous representative of a $W^{1,2}$ function?
How are $f$ and $\bar{f}$ related?
Any explanation would be very nice, as I don't get the idea behind this... :-)
On
As you probably know, elements of $W^{1,2}$ aren't functions but equivalence classes of functions (up to almost everywhere equality). A representative is then just an element of this equivalence class. The Sobolev embedding theorem asserts that every $f\in W^{1,2}(\mathbb{R})$ has a continuous representative.
Hopefully, this explains both 1. and 2.
A priori, an element $[f] \in W^{1,2}(\mathbb{R}) \subset L^2(\mathbb{R})$ is only an equivalence class of functions in $L^2(\mathbb{R})$ (where two functions are identified if the set on which they differ is of measure zero). However, one can show that if $[f] \in W^{1,2}(\mathbb{R})$, then there is a (unique) $\overline{f} \in [f]$ that is continuous. The functions $f$ and $\overline{f}$ are the same up to a measure zero subset, but $\overline{f}$ is continuous while $f$ is (possibly) not. The function $\overline{f}$ is called the continuous representative of $[f]$ (or $f$).
To summarize, a function in $L^2$ which has a weak derivative in $L^2$ is not necessarily continuous but agrees with a continuous function on a set whose complement is of measure zero.