Approximation of periodic functions by interpolation polynomials in \(L_ 1\) (Q2639288)

In the paper the author proves an \(L^ 1\)-error estimate for the trigonometric interpolatory polynomial \(L_ nf\in T_ n\) based on the equidistant nodes \(x_ k=2k\pi /(2n+1),\quad k=0,...,2n.\) Therefore let \[ \omega_ n(f)=\sum^{2n}_{k=0}\{\sup_{x_ k\leq x\leq x_{k+1}}f(x)-\inf_{x_ k\leq x\leq x_{k+1}}f(x)\}. \] Then for \(2\pi\)-periodic Riemann integrable f the following inequality holds \[ \int^{2\pi}_{0}| f(x)-L_ n(x)| dx\leq \frac{2\omega_ n(f) \ln n}{\pi -n}+\frac{C\cdot \omega_ n(f)}{n}. \] The constant 2/\(\pi\) in the first term of the right hand side is best possible. As simple corollaries one has convergence results for functions f of bounded variation and functions from \(W^ r_ 1\), \(r\in {\mathbb{N}}\).