Brownian half-integer sine expansion
= Brownian half-integer sine expansion
{c}
{title2=$W=\sum_n\frac{\xi_nh_n}{(n-1/2)\pi}\quad\text{in }L^2[0,1]$}
The functions $h_n(t)=\sqrt2\sin((n-1/2)\pi t)$ give the Brownian covariance eigenbasis on $[0,1]$, with eigenvalues $\lambda_n=((n-1/2)\pi)^{-2}$. Integrating by parts against $g_n=\sqrt2\cos((n-1/2)\pi t)$ gives independent coefficients $\xi_n=\int g_n\,dW$. Pathwise L2 completeness yields $W=\sum_n\sqrt{\lambda_n}\xi_nh_n$ in L2 and the squared-norm series.