On random Fourier-Hermite transform

16 Dec 2019  ·  Sahoo Sabita, Mangaraj Bharatee ·

Motivated by the work of Z. Liu and S. Liu, we have introduced random Fourier-Hermite transform $\sum_{n=0}^{\infty} c_{n}A_n(\omega)\lambda_{n}\phi_{n}(t)$, where $c_n$ are Fourier-Hermite coefficient of a function $f$ in $L^{p}(-\infty, \infty)$, $A_n(\omega)$ are specific type of random variables associated with symmetric stable process $X(s,\omega)$ of index $\gamma$, $\frac{4}{3} < \gamma \leq 2$, $\lambda_{n}$ are the eigen values of the conventional Fourier transform and $\phi_{n}(t)$ are $n^{th}$ order normalized Hermite-Gaussian function. It is shown that this series converges to a stochastic integral, whose existence is also proved ... Random fractional Fourier transform $\sum_{n=0}^{\infty} c_{n}A_n(\omega)\lambda_{n}^{\alpha}\phi_{n}(t)$ of rational order $\alpha$ is also introduced. read more

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Probability Classical Analysis and ODEs