On functions of bounded $p$-variation

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.