Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces $B_{σ}^{p}$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from $B_{σ}^{p}$ . The difficult situation of derivative-free error estimates is also covered.

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A Representation for the Remainder of the Maclaurin Quadrature Formula.

Error Estimates for Interpolatory Quadrature Formulae.

Characterization of the Speed of Convergence of the Trapezoidal Rule.

Basic relations valid for the Bernstein spaces B ² σ and their extensions to larger function spaces via a unified distance concept

Basic relations valid for the Bernstein spaces $B ²_{σ}$ and their extensions to larger function spaces via a unified distance concept