Eine Ungleichung von van der Corput und Kemperman

Abstract

An Inequality of van der Corput and Kemperman. The tool of van der Corput's difference theorem in the theory of uniform distribution is his so-called fundamental inequality. KEMPERMAN showed that even the non-constructive proofs of the difference theorem by BASS, BERTRANDIAS and CIGLER implicitly use a more general form of van der Corput's fundamental inequality. In this article, the inequality which constitutes the basis of the difference theorem will be proved under a very general setting, applications will be demonstrated in connection with the uniform distribution of products of linear forms and a quantitative version of the difference theorem, i. e. an estimation of discrepancies, will be derived.10" 0026--9255/81/0091/0139/$02.80

Topics in Mathematical Analysis and Applications

“…x; y/f .x/g.y/dxdy < M kD ṅ f k L p .R C ;'.x// kD ṅ gk L q .R C ; .y// (84) and " Z R C y .p 1/.1Cq˛2/ Z R C K .…”

“…Considering dual inequalities (84) and (85) accompanied with the kernel K .x; y/ D .x C y/ , > 0, the constants M and m become, respectively, For some other applications of Theorems 20 and 21, the reader is referred to [7].…”

“…Then, there exists a unique quadratic mapping Q W X ! Y which satisfies (1) kf .x/ Q.x/k Ä 1 .k C h/.1 L/ '.x; x; x; : : : ; x; x/ (83) for all x 2 X , where .k C h/ is an even integer; (2) kf .x/ Q.x/k Ä 1 .k C h 1/.1 L/ '.x; x; x; : : : ; x; x/ (84) for all x 2 X , where .k C h/ is an odd integer.…”

“…This is neither curious nor tragic since Weyl was discussing this type of questions with a certain gingerliness. As Taschner [84] showed, Weyl's reasoning can be made constructive.…”

A simple constructive proof of Kronecker's Density Theorem

“…-From today's perspective, Kronecker's theorem stood at the beginning of the theory of uniform distribution continued by Hermann Weyl, and brought to some perfection by the Vienna school around Edmund Hlawka. Kronecker's theorem is a by-product of Weyl's approach to uniform distribution, which can be constructivised relatively easily[4].…”