Definition For 1-variable Case For A Real Measure
Given any non-decreasing function α on the real numbers, we can define the Lebesgue–Stieltjes integral
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, obtained from the sequence of monomials 1, x, x2, ... by the Gram–Schmidt process.
Usually the sequence is required to be orthonormal, namely,
however, other normalisations are sometimes used.
Read more about this topic: Orthogonal Polynomials
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