Hölder classes of vector-valued functions and convergence of the best predictor

Abstract

The Hölder classes Λ a of vector-valued functions are defined. The functions in each space Λ a are completely characterized by conditions concerning the decay of their Fourier coefficients, their smoothness, and their approximability by polynomials. It is shown that, in some sense, Λ a is closed under multiplication, inversion, and factorization. These ideas are applied to a prediction problem for multivariate stationary processes. Specifically, spectral criteria are derived for the convergence rate of the series representation for the best linear predictor.