L^2-centroids and polynomial-based approximations

We propose a novel class of centroids for real-valued random variables, based on the minimization of a smooth functional derived from the classical quantization problem in the L^2 setting. Unlike traditional representative points, which typically require numerical optimization, the new centroids admit closed-form expressions under mild moment assumptions. We establish their connection with orthogonal polynomials and demonstrate their effectiveness through theoretical analysis and numerical comparisons across both discrete and continuous distributions. Furthermore, we show that these centroids can be successfully employed in clustering tasks using a modified 2-means algorithm, and in function approximation via orthogonal projections. The proposed approach offers a computationally efficient and analytically tractable alternative to classical quantization methods.