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Discrepancy and integration in function spaces with dominating mixed smoothness

100%

Markhasin L.

Instytut Matematyczny Polskiej Akademii Nauk

Optimal lower bounds for discrepancy in Besov spaces with dominating mixed smoothness are known from the work of Triebel. Hinrichs proved upper bounds in the plane. In this work we systematically analyse the problem, starting with a survey of discrepancy results and the calculation of the best known constant in Roth's Theorem. We give a larger class of point sets satisfying the optimal upper bounds than already known from Hinrichs for the plane and solve the problem in arbitrary dimension for certain parameters considering celebrated constructions by Chen and Skriganov which are known to achieve the optimal L₂-norm of the discrepancy function. Since those constructions are b-adic, we give b-adic characterizations of the spaces. Finally results for Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness and for the integration error are concluded.

$L_p$- and $S_{p,q}^rB$-discrepancy of (order 2) digital nets

96%

Markhasin L.

Acta Arithmetica

2015

tom 168

nr 2

139-159

Dick proved that all dyadic order 2 digital nets satisfy optimal upper bounds on the $L_p$-discrepancy. We prove this for arbitrary prime base b with an alternative technique using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds on the discrepancy function in Besov spaces with dominating mixed smoothness for a certain parameter range, and enlarge that range for order 2 digital nets. The discrepancy function in Triebel-Lizorkin and Sobolev spaces with dominating mixed smoothness is considered as well.

Ograniczanie wyników

1 Acta Arithmetica

2 Markhasin L.

1 2015

1 2013

Wyniki wyszukiwania

Discrepancy and integration in function spaces with dominating mixed smoothness

$L_p$- and $S_{p,q}^rB$-discrepancy of (order 2) digital nets