Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)\_{L^{2}([0,1])} + (f',g')\_{L^{2}([0,1])}$. Here $[0,1]$ is not parametrizing the circle; it is just an interval and the functions on it are not assumed to be periodic. My question is: What is a orthonormal basis of $W^{1,2}([0,1])$? Can one write it down explicitely? 



Orthonormal basis in $W^{1,2}([0,1])$
If you change the interval to $[-1,1]$ instead of $[0,1]$, the equivalent inner product would be $$\tag{$\*$} (f,g) = (f,g)\_{L^2([-1,1])} + \lambda (f',g')\_{L^2([-1,1])},$$ where $\lambda = 2$.


One way to get an orthogonal basis is to start with polynomials and apply Gram-Schmidt orthogonalization to them. You get what might be called Sobolev-Legendre polynomials. If you search for *Sobolev orthogonal polynomials* you find lots of references, of which [MX] seems to be a reasonable review. Digging a bit deeper, one finds [M, Thm.3.3] an explicit recurrence relation for Sobolev-Legendre polynomials $S^\lambda\_n(x)$, normalized by $S^\lambda\_n(1) = 1$, with respect to the inner product $(\*)$:


$$
 S^\lambda\_n(x) = S^\lambda\_{n-2} + a\_n (P\_n(x) - P\_{n-2}(x)) ,
$$
where $P\_n$ are the usual [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomials) and
$$
 a\_n = \sum\_{k=0}^{[\frac{n-1}{2}]} \left(\frac{\lambda}{4}\right)^k
 \frac{1}{(2n)!} \frac{(n+2k-1)!}{(n-2k-1)!}
$$


[MX] *Francisco Marcellán and Yuan Xu*, MR 3360352 [**On Sobolev orthogonal polynomials**](http://dx.doi.org/10.1016/j.exmath.2014.10.002), *Expo. Math.* **33** (2015), no. 3, 308--352.


[M] *H. G. Meijer*, [**A short history of orthogonal
polynomials in a Sobolev space I. The non-discrete case**](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.28.5540&rep=rep1&type=pdf), *Niew Arch. Wisk.* **14** (1996), 93--112.


