Bochner's theorem (orthogonal polynomials)
In the theory of orthogonal polynomials, Bochner's theorem is a characterization theorem of certain families of orthogonal polynomials as polynomial solutions to Sturm–Liouville problems with polynomial coefficients.
The theorem is named after Salomon Bochner, who discovered it in 1929.
Setup
Define notations- is the differential operator.
- are linear operators on functions. For example, is one. They are often assumed to be polynomials in.
- are real or complex functions, and are often assumed to be polynomials.
- are real or complex numbers. They are eigenvalues of linear operators.
- are real or complex numbers. They parameterize the solution families.
Statement
Let. Bochner's problem asks the following: consider the SL problemFor what values of, are the eigenpairs such that is a polynomial of degree 0, is a polynomial of degree 1, etc?Observe first that by plugging in the solution, we have, so we may WLOG assume that. Similarly, by plugging in the, we find that must be polynomials of degree at most 1, 2 respectively.
Bochner's theorem states that, up to a complex-affine transform of, there are only 5 families of solutions:
| Polynomial solution | Condition | |||
| Jacobi polynomials | ||||
| Laguerre polynomials | ||||
| Hermite polynomials | ||||
| Bessel polynomials | ||||
| Monomials |
Proof sketch: By the previous observation, there are only 5 parameters in total that characterize the prob/em:Setting, then up to a complex affine transform, it reduces to the formThis is the form of the Jacobi differential equation, and has polynomial solutions precisely when there exists such that. These are the Jacobi polynomials. The other cases are, up to affine scaling, the various limits of the case. The solution families are then obtained by taking the respective limits of the Jacobi polynomials. The conditions on the parameters are necessary to prevent the leading coefficient from going zero.
The original proof by Bochner directly considered the 3 possible cases:
and for each, performed a complex-affine transform of the variable and solved the corresponding equation.
Extensions
The Bochner's theorem allows many extensions, by relaxing various conditions on the setup of Bochner's problem. Other extensions are reviewed in and.Real case
If instead of complex-affine transforms, we only permit real-affine transforms, then there are 2 more families: twisted Hermite, and twisted Jacobi:| Polynomial solution | Condition | |||
| twisted Jacobi polynomials | ||||
| twisted Hermite polynomials |
The 2 extra cases are orthogonal, but not positive-definite. The twisted Hermite satisfies the following complex-orthogonality:They are orthogonal with respect to the real weight functionwhere is any real function supported on, that satisfies, and has all moments zero.
The twisted Jacobi satisfies the following complex-orthogonality:where the average is taken over the function.
They are orthogonal with respect to the real weight functionwhere is any real function supported on, that satisfies, and has all moments zero.