Sharp estimates for errors of numerical differentiation type formulas on trigonometric polynomials (Q2773588)

The trigonometric approximation technique is used to obtain exact estimates for error terms of numerical differentiation formulas. In particular, the class of expansions includes the Stirling-Bessel formula for numerical differentiation, the Euler-Maclaurin formula, and an expansion of the so-called central difference. For example, the following inequality is proven: NEWLINE\[NEWLINE P(f') \leq \frac{n^2}{1 - \frac{\sin((nh)/2)}{nh/2}} P(F - S_{h,1}(F)) \leq \frac{n}{2\sin((nh)/2)} P(\delta^1_h(f)), NEWLINE\]NEWLINE where \(P\) denotes a seminorm on \(H_n\) (the set of trigonometric polynomials of order \( \leq n\)), \(F\) is the antiderivative for \(f\), and \(S_{h,1}(F)\) is the Steklov function for \(F\).