Approximation of functions by partial sums of Fourier series in polynomials orthogonal on an interval

For certain weight functions $p(t)$ and $q(t)$, upper bounds are obtained for the difference between partial sums of Fourier series of a function $f$ with respect to the systems $\sigma_p$ and $\sigma_q$ of polynomials orthogonal on $[-1, 1]$ (a comparison theorem is incidentally proved for the systems $\sigma_p$ and $\sigma_q$). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (for $f\in W^rH^\omega$) of the remainder in a Fourier–Chebyshev series, we establish corresponding results for Fourier series with respect to a system $\sigma_p$.