# fa.functional analysis – A density question for the Hilbert transform

Let $$mathscr Hf$$ denote the Hilbert transform of a function $$f$$ defined on the real-line $$mathbb R$$. Are the set of functions
$${(f+mathscr Hf)_{|_{(0,1)}},:, f in C^{infty}(mathbb R)quad text{and}quad textrm{supp} f Subset (0,infty)}$$
dense in $$L^2((0,1))$$?