Orthogonal polynomials
Under orthogonal polynomials is understood in mathematics an infinite sequence of polynomials
In one unknown, so that the degree has that are orthogonal with respect to a scalar product.
Definition
Be a Borel measure on and consider the Hilbert space of square integrable functions with respect to the scalar product
Next is for everyone. This is the case for example when the carrier has a compact dimension. In particular, the measure is finite and one can without loss of generality call. In the simplest case, the measure is given by a non-negative weighting function.
A sequence of polynomials, orthogonal polynomials is called a result, if the degrees and has different polynomials are orthogonal in pairs:
Construction
If the dimension is given, the corresponding polynomials can be uniquely constructed using the Gram- Schmidt's orthogonalization of the monomials. For this, it suffices obviously the moments
To know. The reverse is known as Stieltjes'sches moment problem.
Standardization
There are different ways of normalization in use. To describe this we introduce the following constants:
And
Then is called the polynomials as orthonormal if necessary and as harmoniously.
Recursion
Orthogonal polynomials satisfy a three-step recursion relation
(where to put the case) with
And the constants in the previous section.
The recursion can also be equivalently in the form
With
Be written.
Especially in the case of orthonormal Poynomen, one obtains a symmetric recursion and the orthonormal polynomials satisfy exactly the generalized eigenvector equation of the associated Jacobi operator. The measure is the spectral measure of the Jacobi operator for the first basis vector.
Christoffel - Darboux formula
It is
And in the case obtained by thresholding
Zeros
The polynomial has exactly zeros, which are all simple and lie in the support of the measure. The zeros of lie strictly between the zeros of.
List of orthogonal polynomials
- Gegenbauer polynomial
- Hahn polynomial
- Hermitian polynomial
- Jacobi polynomial
- Legendre polynomial
- Laguerre polynomials
- Chebyshev polynomial
- Zernike polynomial