Having a white random process $s(t)$, is it possible to write $$s(t)=\sum\_{i=0}^\infty\alpha\_i\phi\_i(t)$$ where the $\alpha\_i$ are random variables and the $\phi\_i$ orthogonal polynomials (Jacobi Polynomials, Legendre Polynomials...)?



Expansion of white noise into infinite series using orthogonal polynomials
Yes, if $\{f\_j\}$ is an orthonormal basis of $L^2[0,1]$ and $Z\_j$ are i.i.d. standard normal variables, then $S(t)=\sum\_j Z\_j f\_j(t)$ is a white noise. Note that this sum does not converge pointwise and should be understood as a distribution. Alternatively,
if $F\_j(t)=\int\_0^t f\_j(r) \,dr$ then $B(t)=\sum\_j Z\_j F\_j(t)$ is a standard Brownian motion. When $f\_j$ is the trigonometric basis, this is Wiener's original construction of Brownian motion, see [1]. When $f\_j$ is the Haar basis, this yields Paul Levy's interpolation construction, see [2], [3]. When $L\_j$ are the Legendre polynomials and $f\_j(t)$ equals $L\_j(2t-1)$ normalized in $L^2$, one gets the desired polynomial representation. See [4] for a refined study.


[1] Jean-Pierre Kahane. Some random series of functions. Vol. 5. Cambridge University Press, 1993.


[2] <https://sites.stat.washington.edu/jaw/COURSES/491-2/HO-492/HaarFcnConstr-HO-492.pdf>


[3] [http://math.stanford.edu/~ryzhik/STANFORD/STANF227-10/notes-bm.pdf](http://math.stanford.edu/%7Eryzhik/STANFORD/STANF227-10/notes-bm.pdf)


[4] An Optimal Polynomial Approximation of Brownian Motion
James Foster, Terry Lyons, and Harald Oberhauser
SIAM Journal on Numerical Analysis 2020 58:3, 1393-1421
<https://arxiv.org/abs/1904.06998>


