Abstract
Optimal rules of numerical approximation of bounded linear functionals in a general Hilbert space of periodic functions are investigated by Davis' method. The weights and error norms for optimal rules with equidistant nodes for an arbitrary bounded linear functional are given explicitly in terms of a total orthonormal system in the Hilbert space. Further the optimal approximation rules are compared with the numerical approximations obtained by means of trigonometric interpolation. As examples the numerical integration and the numerical evaluation of conjugate harmonic functions are considered in two distinct Hardy spaces and also in Sobolev spaces.
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