Speaker: Aurelian Gheondea (Institute of Mathematics of the Romanian Academy, Bucharest, and Bilkent University, Ankara)

Date / Time: Tuesday, December 10, 2024 / 15:30 (Ankara, Turkey)

Place: Follow the Link for Online Participation (please self-register first, if necessary)

Abstract: We find probability error bounds for approximations of functions \(f\) in a separable reproducing kernel Hilbert space \(\mathcal{H}\) with reproducing kernel \(K\) on a base space \(X\), firstly in terms of finite linear combinations of functions of type \(K_{x_i}\) and then in terms of the projection \(\pi_x^n\) on \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\), or random sequences of points \(x = (x_i)_i\) in \(X\). Given a probability measure \(P\), letting \(P_K\) be the measure defined by \(\text{d}P_K(x) = K(x, x)\text{d}P(x), \ x\in X\), our approach is based on the nonexpansive operator

\[L^2(X; P_K) \ni \lambda \mapsto L_{P,K} \lambda := \int_X \lambda(x)\, K_x\, \text{d}P(x) \in \mathcal{H},\]

where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by \(\mathcal{H}_P\), that is the operator range of \(L_{P,K}\). Our main result establishes bounds, in terms of the operator \(L_{P,K}\), on the probability that the Hilbert space distance between an arbitrary function \(f\) in \(\mathcal{H}\) and linear combinations of functions of type \(K_{x_i}\), for \((x_i)_i\), sampled independently from \(P\), falls below a given threshold. For sequences of points \((x_i)_i^\infty\) constituting a so-called uniqueness set, the orthogonal projections \(\pi_x^n\) to \(\text{span}\left\{ K_{x_i} \right\}_{i=1}^{n}\) converge in the strong operator topology to the identity operator. We prove that, under the assumption that \(\mathcal{H}_P\) is dense in \(\mathcal{H}\), any sequence of points sampled independently from \(P\) yields a uniqueness set with probability \(1\). This result improves on previous error bounds in weaker norms, such as uniform or \(L^p\) norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space \(H^2(\mathbb{D})\) are presented as well.

Joint work with Ata Deniz Aydın, ETH Zürich, Switzerland.

    Additional Resources

    Biography

    Aurelian Gheondea studied mathematics and got his PhD at the University of Bucharest, Romania, with a thesis “Spectral Theory of Selfadjoint Operators in Krein Spaces” under the supervision of Ion Colojoară. He worked mainly at the Institute of Mathematics of the Romanian Academy in Bucharest and for twenty one years at the Bilkent University in Ankara. His fields of interest spans between Functional Analysis, Operator Theory and Operator Algebras, Matrix and Numerical Analysis, Hermitian Kernels, Quantum Operations, and Mathematical Modelling. He supervised eleven MSc students and one PhD student. He is the author of about eighty articles and three books. Currently he is a Senior Researcher at the Institute of Mathematics of the Romanian Academy in Bucharest and Professor Emeritus at the Bilkent University in Ankara.