Lipschitz classes of periodic stochastic processes and Fourier series (Q1077079)

Different problems of approximation of complex valued integrable periodic stochastic processes by trigonometric polynomials with stochastic coefficients are discussed in the way of ordinary approximation theory. The relationships between the best approximation errors and continuity modulus of processes, between the magnitude of errors (or Fourier coefficients) and the orders of Lipschitz classes to which processes belong are established. Sufficient conditions, such as the smoothness in the mean of processes, for the absolute convergence of Fourier series are obtained. From these results the conditions for the smoothness of sample functions of processes can be derived as well. The author has made a series of works on this topic published in Japanese journals.