We investigate the conditions under which global smoothness of a function / (as measured by its modulus of continuity) is retained by the elements of approximating sequences (£"/). As one consequence we obtain statements concerning the invariance of Lipschitz classes under operators of several types. A crucial tool in our approach is the least concave majorant of a modulus of continuity.

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In the center of our paper are two counterexamples showing the independence of the concepts of global smoothness preservation and variation diminution for sequences of approximation operators. Under certain additional assumptions it is shown that the variation-diminishing property is the stronger one. It is also demonstrated, however, that there are positive linear operators giving an optimal pointwise degree of approximation, and which preserve global smoothness, monotonicity and convexity, but are not variationdiminishing.

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Die Studienbücher Wirtschaftsmathematik behandeln anschaulich, systematisch und fachlich fundiert Themen aus der Wirtschafts-, Finanz-und Versicherungsmathematik entsprechend dem aktuellen Stand der Wissenschaft. Die Bände der Reihe wenden sich sowohl an Studierende der Wirtschaftsmathematik, der Wirtschaftswissenschaften, der Wirtschaftsinformatik und des Wirtschaftsingenieurwesens an Universitäten, Fachhochschulen und Berufsakademien als auch an Lehrende und Praktiker in den Bereichen Wirtschaft, Finanz-und Versicherungswesen.