Volterra operator ($(Af)(x)={\int }_0^{x}f(t)dt$)

The Volterra integration operator on $L^2([0,1])$ is compact and injective, but its inverse is differentiation and is unbounded. Its singular values $2/((2n-1)\pi )$ tend to zero, making recovery of a function from noisy indefinite-integral data an ill-posed inverse problem.