An estimate for the rate of approximation of functions by Chebyshev polynomials (Q2711412)

Let \(f:[-1,1]\rightarrow \mathbb R\) be a function of bounded variation on [-1,1] and let \((U_n)_{n\geq 0}\) be the sequence of Chebyshev polynomials of the second kind. The author obtains estimates for \(|C_n(f;x)-\frac 12(f(x+0)+f(x-0))|,\) where \(C_n(f;x)\) is the \(n\)'th partial sum of the expansion of \(f\) in the series of polynomials \((U_n)_{n\geq 0},\) in terms of the total variation of a function \(g_x\) attached to \(f.\)