Some Asymptotic Theory for Functional Regression and Classification

Certain functions of the covariance operator such as the square root of a regularized inverse are important components of many statistics employed for functional data analysis. If Σ is a covariances operator on a Hilbert space, ̂ Σ a sample analogue of this operator, and φ a function on the complex plane, which is analytic on a domain containing a contour around the spectrum of Σ, a tool of generic importance is the comparison of φ ̂ Σ and φ Σ by means of a Taylor expansion: