  We consider a methodology based in B-splines scaling functions to numerically
invert Fourier or Laplace transforms of functions in the space
$L^2(\mathbb{R})$. The original function is approximated by a finite
combination of $j^{th}$ order B-splines basis functions and we provide
analytical expressions for the recovered coefficients. The methodology is
particularly well suited when the original function or its derivatives present
peaks or jumps due to discontinuities in the domain. We will show in the
numerical experiments the robustness and accuracy of the method.
