On random Fourier-Hermite transform associated with stochastic process

Abstract

Liu and Liu in 2007 introduced the Fourier - Hermite transform $\sum a_{n}\lambda_{n}^{R}\psi_{n}(t)$ which is a random Fourier - Hermite series with random variables $\lambda_{n}^{R}$ choosen randomly from the unit circle of $\mathbb{C}$, where $\psi_{n}(t)$ are Hermite functions and $a_{n}$ are Fourier - Hermite coefficients of an $L^{2}(\mathbb{R})$ function. They used it in image encryption and decryption and expected its application in general signal and image processing. This motivated us to investigate more on random Fourier - Hermite transform by replacing the random variables $\lambda_{n}^{R}$ by some other random variables. It leads to address two problems. First to focus on convergence of random Fourier - Hermite series. Secondly to investigate on finding Fourier transform of the sum function of these random Fourier - Hermite series. The random variables those has been choosen are Fourier - Hermite coefficients of stochastic process. They are independent if associated with Wiener process and dependent if associated with symmetric stable process. The scalars $a_{n}$ are Fourier - Hermite coefficients of functions of suitable $L^{p}$ spaces. The Fourier transform of the sum functions are found out which is possible in case of $p = 2$ only.