Sobolev orthogonal polynomials

Let ${\displaystyle \mu _{0},\mu _{1},\dots ,\mu _{n}}$ be positive Borel measures on ${\displaystyle \mathbb {R} }$ with finite moments. Consider the inner product

${\displaystyle \langle p_{r},p_{s}\rangle _{W^{n,2}}=\int _{\mathbb {R} }p_{r}(x)p_{s}(x)\;\mathrm {d} \mu _{0}+\sum \limits _{k=1}^{n}\int _{\mathbb {R} }p_{r}^{(k)}(x)p_{s}^{(k)}(x)\;\mathrm {d} \mu _{k}}$

and let ${\displaystyle W^{n,2}}$ be the corresponding Sobolev space. The Sobolev orthogonal polynomials ${\displaystyle \{p_{n}\}_{n\geq 0}}$ are defined as

${\displaystyle \langle p_{n},p_{s}\rangle _{W^{n,2}}=c_{n}\delta _{n,s}}$

where ${\displaystyle \delta _{n,s}}$ denotes the Kronecker delta. One says that these polynomials are sobolev orthogonal.^[1]