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By Marchenko V. M.

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B C j /1Œ0;b/ . j / C. 2 j b/1Œb;1 . ti ; tj // is the same as 2 1 j i bj C j bj j j jj i : Here, we suppress the dependence on k by taking Œtj ; a; j ; b D Œtj;k ; ˛k ; j;k ; ˇk . That is, for given tj and a we want to find j and b such that the corresponding equalities hold. Assume first that ˛ D 0, which corresponds to the L2 discrepancy anchored at the origin. b 1 j i C j /1Œ0;b/ . j / bj C j j bj C. 2 j jj j b/1Œb;1 . j /; : Note that this indeed holds if we take j D tj and b D 1, or j D 1 tj and b D 0.

However, the function f may be discontinuous at ˛. x/ dx: 0 Here, the derivatives are meant point-wise almost everywhere. Despite many similarities to a subspace of the Sobolev space anchored at ˛, the property that f may be discontinuous at ˛ makes this space different than the Sobolev space. K1 / and the Sobolev space is also necessary from a different point of view. ˛ tj / mod 1 to get a relation to the Sobolev space. K1 / we used the same sample points for both discrepancy and integration. K1 / to be different than the Sobolev space.

In this section we wish to only illustrate a few tractability results for multivariate integration based on its relations with discrepancy. These tractability results will follow from discrepancy error bounds reported above as well as results obtained in a number of papers cited here. We mainly limit ourselves to multivariate integration for the (weighted) Sobolev space anchored at 0 or 1, which as we now know, is related to the (weighted) L2 discrepancy anchored at 0. Even in this standard case, there are still open questions, which we will present as open problems.

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A Brief Review of the Development of Qualitative Control Theory in Belarus by Marchenko V. M.


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