Christoffel numbers

Christoffel coefficients

The coefficients of a quadrature formula

which is exact for algebraic polynomials of degrees . The interpolation nodes of such a formula are the zeros of a polynomial of degree which is orthogonal on relative to the distribution to all polynomials of degree ; if , the Christoffel numbers are uniquely determined. One has , and

If the polynomials are orthonormal, the Christoffel numbers may be expressed as

where is the leading coefficient of . In the case , and , the are the Legendre polynomials, and

These expressions are due to E.B. Christoffel [1]. For they were evaluated by C.F. Gauss. See also Gauss quadrature formula.

References

[1]	E.B. Christoffel, "Ueber die Gaussche Quadratur und eine Verallgemeinerung derselben" J. Reine Angew. Math. , 55 (1858) pp. 61–82
[2]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[3]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)

References

[a1]	F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974)