Let $$T_n$$ be the linear Hadamard convolution operator acting over Hardy space $$H^q$$ , $$1\le q\le \infty $$ . We call a best approximation-preserving (BAP operator) if $$T_n(e_n)=e_n$$ where $$e_n(z):=z^n,$$ and $$\Vert T_n(f)\Vert _q\le E_n(f)_q$$ for all $$f\in H^q$$ $$E_n(f)_q$$ is approximation by algebraic polynomials of degree most $$n-1$$ in space. give necessary sufficient condition...