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Two central objects in constructive approximation, the Christoffel-Darbouxkernel and the Christoffel function, are encoding ample information about theassociated moment data and ultimately about the possible generating measures.We develop a multivariate theory of the Christoffel-Darboux kernel in C^d, withemphasis on the perturbation of Christoffel functions and their level sets withrespect to perturbations of small norm or low rank. The statistical notion ofleverage score provides a quantitative criterion for the detection of outliersin large data. Using the refined theory of Bergman orthogonal polynomials, weillustrate the main results, including some numerical simulations, in the caseof finite atomic perturbations of area measure of a 2D region. Methods offunction theory of a complex variable and (pluri)potential theory are widelyused in the derivation of our perturbation formulas.

Perturbations of Christoffel-Darboux kernels. I: detection of outliers