On random Fourier-Hermite transform associated with stochastic process

TL;DR

This paper investigates the convergence and Fourier transform of random Fourier-Hermite series associated with stochastic processes, extending prior work by exploring different types of random variables.

Contribution

It introduces new analysis of the convergence and Fourier transform of random Fourier-Hermite series with stochastic process-based randomness, expanding applications in signal processing.

Findings

Convergence of the series depends on the type of stochastic process involved.

Fourier transform of the sum function is derived for the case p=2.

Results extend the applicability of Fourier-Hermite transforms in stochastic settings.

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.