Set of polynomials where any two are orthogonal to each other
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
of a function f. If this integral is finite for all polynomials f, we can define an inner product on pairs of polynomials f and g by
This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero.
Then the sequence (Pn)∞ n=0 of orthogonal polynomials is defined by the relations
In other words, the sequence is obtained from the sequence of monomials 1, x, x2, … by the Gram–Schmidt process with respect to this inner product.
Usually the sequence is required to be orthonormal, namely,
however, other normalisations are sometimes used.
Absolutely continuous case
Sometimes we have
where
is a non-negative function with support on some interval [x1, x2] in the real line (where x1 = −∞ and x2 = ∞ are allowed). Such a W is called a weight function.[1] Then the inner product is given by
However, there are many examples of orthogonal polynomials where the measure dα(x) has points with non-zero measure where the function α is discontinuous, so cannot be given by a weight function W as above.
Examples of orthogonal polynomials
The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
Meixner classified all the orthogonal Sheffer sequences: there are only Hermite, Laguerre, Charlier, Meixner, and Meixner–Pollaczek. In some sense Krawtchouk should be on this list too, but they are a finite sequence. These six families correspond to the NEF-QVFs and are martingale polynomials for certain Lévy processes.
There are some families of orthogonal polynomials that are orthogonal on plane regions such as triangles or disks. They can sometimes be written in terms of Jacobi polynomials. For example, Zernike polynomials are orthogonal on the unit disk.
The advantage of orthogonality between different orders of Hermite polynomials is applied to Generalized frequency division multiplexing (GFDM) structure. More than one symbol can be carried in each grid of time-frequency lattice.[2]
Properties
Orthogonal polynomials of one variable defined by a non-negative measure on the real line have the following properties.
Relation to moments
The orthogonal polynomials Pn can be expressed in terms of the moments
as follows:
where the constants cn are arbitrary (depend on the normalization of Pn).
This comes directly from applying the Gram–Schmidt process to the monomials, imposing each polynomial to be orthogonal with respect to the previous ones. For example, orthogonality with prescribes that must have the form
which can be seen to be consistent with the previously given expression with the determinant.
Recurrence relation
The polynomials Pn satisfy a recurrence relation of the form
where An is not 0. The converse is also true; see Favard's theorem.
If the measure dα is supported on an interval [a, b], all the zeros of Pn lie in [a, b]. Moreover, the zeros have the following interlacing property: if m < n, there is a zero of Pn between any two zeros of Pm. Electrostatic interpretations of the zeros can be given.[citation needed]
Combinatorial interpretation
From the 1980s, with the work of X. G. Viennot, J. Labelle, Y.-N. Yeh, D. Foata, and others, combinatorial interpretations were found for all the classical orthogonal polynomials. [3]
These are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. Including derivatives has big consequences for the polynomials, in general they no longer share some of the nice features of the classical orthogonal polynomials.
Orthogonal polynomials with matrices
Orthogonal polynomials with matrices have either coefficients that are matrices or the indeterminate is a matrix.
There are two popular examples: either the coefficients are matrices or :
Variante 1: , where are matrices.
Variante 2: where is a -matrix and is the identiy matrix.
Quantum polynomials
Quantum polynomials or q-polynomials are the q-analogs of orthogonal polynomials.
^Catak, E.; Durak-Ata, L. (2017). "An efficient transceiver design for superimposed waveforms with orthogonal polynomials". 2017 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom). pp. 1–5. doi:10.1109/BlackSeaCom.2017.8277657. ISBN978-1-5090-5049-9. S2CID22592277.