Sharp estimates for the deviation of the mean value of a periodic function in terms of moduli of continuity of higher order (Q2773594)

The authors obtain exact estimates for the deviation of the mean value of a periodic function. The main result of the article provides the following estimate: NEWLINE\[NEWLINE \sup_{f\in C}\frac{\|f - A_0(f)\|}{\omega_{2r}(f,\pi)} = \sup_{f\in C^*}\frac{\|f - A_0(f)\|}{\omega_{2r}(f,\pi)} = \frac{1}{C_{2r}^r}, NEWLINE\]NEWLINE where \(C\) denotes the space of continuous \(2\pi\)-periodic functions with uniform norm, \(C^*\subset C \) is the set of even functions with nonnegative Fourier coefficients, \(\omega_s(f,h)\) denotes the \(s\)-order modulus of continuity of a function \(f\in C\) with step size \(h\), \(A_0(f) = (2\pi)^{-1}\int_{-\pi}^{\pi}f\). The results obtained are also extended to \(L_2\)-spaces. As an example, the authors expose exact estimates for the error term of the compound rectangle formula.