properties of orthogonal polynomials
A countable system of orthogonal polynomials
One also requires that
Such a system (1) may be used as basis for the vector space of functions defined on , i.e. certain such functions may be expanded as a series (http://planetmath.org/FunctionSeries)
where the coefficients have the expression
Every member of (1) is orthogonal to any polynomial of degree less than .
There is a recurrence relation
enabling to determine a .
The zeros of are all real and belong to the open interval ; between two of those zeros there are always zeros of .
The Sturm–Liouville differential equation
where is a polynomial of at most degree 2 and a linear polynomial, gives under certain conditions, as http://planetmath.org/node/8719solutions a system of orthogonal polynomials corresponding suitable values (eigenvalues) of the parametre . Those satisfy the Rodrigues formula
where is a constant and
The classical Chebyshev (http://planetmath.org/ChebyshevPolynomial), Hermite (http://planetmath.org/HermitePolynomials), Laguerre (http://planetmath.org/LaguerrePolynomial), and Legendre polynomials all satisfy an equation (2).
[Not ready . . .]
|Title||properties of orthogonal polynomials|
|Date of creation||2013-03-22 19:05:34|
|Last modified on||2013-03-22 19:05:34|
|Last modified by||pahio (2872)|