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 $L^2((0,1))$?