In this article we obtain an asymptotic formula for the approximations to functions in the class $V_p^\alpha$ ( $0\leqslant\alpha<\infty$ , $1\leqslant p<\infty$ ) by Fourier sums in the metric of $L^p $ ( $1\leqslant p<q\leqslant\infty$ ). We find sufficient conditions and also criteria for the continuity of the derivative $f^\alpha(t)$ of a function in the class $V_p^\alpha$ . We also give some results on the Fourier coefficients of functions in the above class.