Karhunen-Loève expansion for a generalization of Wiener bridge

We derive a Karhunen--Lo\`{e}ve expansion of the Gauss process Bₜ - g(t)∫₀¹ g'(u) dBᵤ, t∈[0,1], where (Bₜ)t∈[0,1] is a standard Wiener process and g:[0,1]→ R is a twice continuously differentiable function with $g(0) = 0$ and ∫₀¹ (g'(u))² du =1. This process is an important limit process in the theory of goodness-of-fit tests.We formulate two special cases with the function g(t)=√2πsin(π t), t∈[0,1], and $g(t)=t$, t∈[0,1], respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.