Perturbations of Christoffel-Darboux kernels: detection of outliers. (English) Zbl 1470.42049

Considering the Lebesgue space \(L^{2}(\mu)\), where \(\mu\) is a positive Borel measure in the complex plane with compact and infinite support supp(\(\mu\)), and scalar product and norm \[\langle f , g \rangle_{2,\mu} = \int f(z) \overline{g(z)} d\mu(z),\quad \| f\|_{2, \mu } = \left( \langle f,f \rangle_{2,\mu} \right)^{1/2}, \] the authors review the definition and main properties of the univariate Christoffel-Darboux kernel \[K_{n}^{\mu} (z,w) = \sum_{j=0}^{n} p_{j}^{\mu}(z) \overline{p_{j}^{\mu}(w)}\] for the associated sequence of orthonormal polynomials \(p_{j}^{\mu}\) of degree \(j\) with positive leading coefficient, as well as the Christoffel-Darboux function. The multivariate setup follows taking into account the graded lexicographical order on the set of indices \[\alpha = \left( \alpha_{1}, \alpha_{2}, \ldots, \alpha_{d} \right) \in \mathbb{N}^{d}, \quad z^{\alpha} = z_{1}^{\alpha_{1}} z_{2}^{\alpha_{2}}\cdots z_{d}^{\alpha_{d}} .\] Several results regarding orthogonal polynomials of several variables are presented, leading to the multivariate Christoffel-Darboux kernel whose properties are discussed, together with connections with statistics notions as the Mahalanobis distance and leverage score. In particular, “up to some additive constant, \(\sqrt{K_{n}^{\mu}(z,z)}\) can be considered as a natural generalization of the Mahalanobis distance between a point \(z \in \mathbb{C}^{d}\) and the probability measure \(\mu\)”.
After studying the approximation of two Christoffel-Darboux kernels \(K_{n}^{\mu} (z,z)\) and \(K_{n}^{\nu} (z,z)\), for a fixed \(n\) and all \(z \in \mathbb{C}^{d}\), where \(\mu\) and \(\nu\) are two positive measures with compact support in \(\mathbb{C}^{d}\), the authors provide upper and lower bounds for the ratio of Christoffel-Darboux kernels \(\frac{K_{n}^{\mu+\sigma} (z,z)}{K_{n}^{\mu} (z,z)}\) for \( z \notin \) supp(\(\mu\)), but possible \(z \in \) supp(\(\sigma\)). The corresponding univariate situation is addressed followed by the asymptotics in Bergman space. The results obtained by the authors allow the identification of outliers as those elements \(z_{j}\) of the support where the leverage score \(t_j K_{n}^{\tilde{\nu}}(z_{j},z_{j})\) is close to \(1\), being \(\tilde{\nu} = \tilde{\mu}+ \sigma \), \(\tilde{\mu}=\sum_{j=l+1}^{N} t_{j}\delta_{z_{j}} \), \(t_{j} >0\), \(z_{l+1}, \ldots,z_{N} \in \) supp(\(\mu\)) distinct, and assuming certain conditions of proximity between \(\tilde{\mu}\) and \(\mu\). We also find several numerical examples in this thorough contribution.


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
30H20 Bergman spaces and Fock spaces
30E05 Moment problems and interpolation problems in the complex plane


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