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The Christoffel function is the reciprocal of the diagonal Christoffel-Darboux kernel,

where are orthonormal polynomials for a finite positive measure on the real line with finite moments of all orders and infinite support. Here is real. This convention uses polynomials of polynomial degree at most ; another common convention indexes the same function by , the number of terms.

The function has the variational characterization

where the minimum is over real polynomials. Its unique minimizer is

Expanding in the orthonormal polynomials and applying the Schwarz's inequality to proves the characterization (Simon 2008).

The Christoffel function is positive and nonincreasing as increases. At the zeros of , the weights in Gaussian quadrature are the Christoffel numbers

Scaling the measure by multiplies the Christoffel function by .

For on , the Legendre polynomials give

The figure compares , 1, 2, 4, and 8. The marked points on the blue curve are at , 0, and , with heights , , and . These are the weights of three-point Gaussian quadrature.

See also

Christoffel-Darboux Kernel, Christoffel Number, Gaussian Quadrature, Orthogonal Polynomials

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References

Simon, B. "The Christoffel-Darboux Kernel." In Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (Ed. D. Mitrea and M. Mitrea). Providence, RI: Amer. Math. Soc., pp. 295-335, 2008. https://doi.org/10.1090/pspum/079/2500498. https://arxiv.org/abs/0806.1528.

Cite this as:

Weisstein, Eric W. "Christoffel Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ChristoffelFunction.html

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