On the maximum of Gaussian Fourier series emerging in the analysis of random vibrations

This paper studies the maximum of Gaussian Fourier series emerging in the analysis of random vibrations of a finite string. Evaluating the distribution of the maximal displacement corresponds to the analysis of the maximum for the related Gaussian Fourier series with independent coefficients. When vibrations are triggered by an initial white noise disturbance (where the intantaneous form of the vibrating string is composed of three processes pieced together) the paper gives upper bounds for the maximal displacement. In the last section it considers forced vibrations at special instants where the instantaneous form of the vibrating string has the structure of a Brownian bridge. This enables the paper to give the exact distribution of the maximum for the related sine series.