A theorem on the expansion of the derivative $f^{(r_1,r_2,\dots,r_n)}$, where $f\in L_p$, and the derivatives of singular integrals into the series of band-limited functions (entire functions of exponential type), which converges in $L_p$ for $1\le p\le q<\infty$, is proved. The norms of their items are estimated by best approximations by ``an angle''.