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Abstract.Bounds for the error of the approximation of the value of a bounded linear functional

Abstract.Optimal numerical approximation of bounded linear funetionals by weighted sums in Hilbert spaces of functions analytic in a circle KF, in a circular annulus Krl,r~ and in an ellipse Er is investigated by Davis' method on the common algebraic background for diagonalising the normal equation matrix. The weights and error functional norms for optimal rules with nodes located angle-equidistant on the concentric circle ~Ks or on the eonfoeal ellipse ~Es and in the interval [-1,1] for an arbitrary bounded linear functional are given expllci~Iy. They are expressed in terms of a complete orthonormal system in the Hilbert space.

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Abstract-ZusammenfassungOn the Determination of the Weights in Multi-Dimensional Numerical Quadrature. Using the concept of the generalized inverse of a bounded linear transformation between R n and 12, a method is given for constructing quadrature rules for integration over an arbitrary bounded m-dimensional region B c R m with the property that the average error over the prescribed family F of the functions continuous in B as well as the variance of the rounding errors according to Sard 18] are minimal. Then we specialize F to the weighted monomials and treat as an example integration on the surface of the m-sphere. Zur Bestlmmung der Gewichte bei mehrdimensionaler numerischer Integration. MitHilfe der Pseudoinversen eines beschr~inkten linearen Operators von R n in 12 wird eine Methode zur Konstruktion von Quadraturformeln fiir die Integration fiber einem beliebigen beschrankten m-dimensionalen Gebiet B c R" hergeleitet, mit der Eigenschaft, dag der mittlere Fehler in einer vorgeschdebenen Familie F sowie die Varianz der Rundungsfehler gem~ig Sard I-8] minimal werden. Sodann w~ihlen wir F als gewichtete Monome und behandeln als Beispiel Integration auf der Oberlt~iche der m-Kugel.