Let \(f\) be a sufficiently smooth function defined on \([-1,\,1]\). A. Iserles and S. P. Nørsett [IMA J. Numer. Anal. 28, No. 4, 862–887 (2008; Zbl 1221.65348)] have observed that the \(n\)-th partial sums \(f_n\) of the modified Fourier series
\[ \frac{c_0}{2} + \sum_{k=1}^{\infty} [c_k\, \cos (\pi k x) + s_k\,\sin(\pi (k - \frac{1}{2})x)] \]
with
\[ c_k = \int_{-1}^1 f(x) \, \cos (\pi k x) \, dx, \quad s_k = \int_{-1}^1 f(x) \, \sin(\pi (k - \frac{1}{2})x) \, dx \]
converge for \(n\to \infty\) pointwise at a faster rate than the classical Fourier series of \(f\).
In the paper under review, the author shows the following result: If \(f\in C^2[-1,\,1]\) and \(f''\) has bounded variation, then \(f(x) - f_n(x) = {\mathcal O}(n^{-2})\) for \(x \in (-1,\, 1)\) and \(f(\pm 1) - f_n(\pm 1) = {\mathcal O}(n^{-1})\) for \(n\to \infty\). Further, if \(f\in C^{2s}[-1,\, 1]\) \((s \in \mathbb N)\) and \(f^{(2s)}\) has bounded variation, then an asymptotic expansion of \(f(x) - f_n(x)\) of order \({\mathcal O}(n^{-2s})\) contains terms with \(f^{(2j-1)}(\pm 1)\) \((j=1\dots,s)\). Since often these derivatives \(f^{(2j-1)}(\pm 1)\) \((j=1\dots,s)\) are not available, the author uses a Richardson-like extrapolation method of \(f_n\) to increase the convergence rate. Finally, for the computation of \(c_k\) and \(s_k\) an efficient method based on Filon-type quadrature is presented.