On Tractability of Weighted Integration for Certain Banach Spaces of Functions

There are a number of results on tractability and strong tractability for the integration problem over bounded domains. However, the majority of them assume functions with dominating mixed partial derivatives bounded in the L 2 norm. Much less is known when the derivatives are bounded in the L 1 norm, and almost nothing when a finite L p norm is assumed for an arbitrary p. The focus of this paper is to extend known tractability results to weighted spaces of functions with the derivatives bounded in L p norms and the norms of the derivatives then combined via weighted L q norms. Moreover, we consider weighted integration with the domain of integration being not necessarily bounded. It turns out that tractability and strong tractability depend only on q for bounded D, whereas they depend on both parameters p and q for unbounded D.

[1]  Xiaoqun Wang.  A Constructive Approach to Strong Tractability Using Quasi-Monte Carlo Algorithms , 2002, J. Complex..

[2]  E. Novak.  Deterministic and Stochastic Error Bounds in Numerical Analysis , 1988 .

[3]  E. Novak,et al.  The inverse of the star-discrepancy depends linearly on the dimension , 2001 .

[4]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[5]  Henryk Wozniakowski,et al.  On the Power of Standard Information for Weighted Approximation , 2001, Found. Comput. Math..

[6]  Fred J. Hickernell,et al.  On strong tractability of weighted multivariate integration , 2004, Math. Comput..

[7]  Grzegorz W. Wasilkowski,et al.  Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains , 2002 .

[8]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[9]  Grzegorz W. Wasilkowski,et al.  Complexity of Weighted Approximation over R , 2000 .

[10]  H. Wozniakowski.  Efficiency of Quasi-Monte Carlo Algorithms for High Dimensional Integrals , 2000 .

[11]  Henryk Wozniakowski,et al.  Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces , 2002, J. Complex..

[12]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[13]  Fred J. Hickernell,et al.  On tractability of weighted integration over bounded and unbounded regions in Reals , 2004, Math. Comput..

[14]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[15]  Henryk Wozniakowski,et al.  Weighted Tensor Product Algorithms for Linear Multivariate Problems , 1999, J. Complex..

[16]  E. Novak,et al.  Foundations of Computational Mathematics: When are integration and discrepancy tractable? , 2001 .

[17]  Henryk Wozniakowski,et al.  Intractability Results for Integration and Discrepancy , 2001, J. Complex..

[18]  Fred J. Hickernell,et al.  The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension , 2002, Math. Comput..