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]

• Classical orthogonal polynomials are Sobolev orthogonal polynomials, since their derivatives are also orthogonal polynomials.
• Sobolev orthogonal polynomials in general are no longer commutative in the multiplication operator with respect to the inner product, i.e.

${\displaystyle \langle xp_{n},p_{s}\rangle _{W^{n,2}}\neq \langle p_{n},xp_{s}\rangle _{W^{n,2}}}$

Consequently neither Favard's theorem, the three term recurrence or the Christoffel-Darboux formula hold. There exist however other recursion formulas for certain types of measures.

• There exist a lot of literature for the case ${\displaystyle n=1}$.

• Marcellán, Francisco; Xu, Yuan (2015). "On Sobolev orthogonal polynomials". Expositiones Mathematicae. 33 (3): 308–352. arXiv:1403.6249.
• Marcellán, Francisco; Moreno-Balcázar, Juan (2017). "WHAT IS... a Sobolev Orthogonal Polynomial?". Notices of the American Mathematical Society. 64: 873–875. doi:10.1090/noti1562.