Information-based complexity
On the numerical integration of Walsh series by number-theoretic methods
Mathematics of Computation
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms
Mathematics of Computation
The existence of good extensible rank-1 lattices
Journal of Complexity
Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers
Foundations of Computational Mathematics
SIAM Journal on Numerical Analysis
Lattice rule algorithms for multivariate approximation in the average case setting
Journal of Complexity
Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order
SIAM Journal on Numerical Analysis
Multivariate L∞ approximation in the worst case setting over reproducing kernel Hilbert spaces
Journal of Approximation Theory
On the power of standard information for multivariate approximation in the worst case setting
Journal of Approximation Theory
Constructions of (t ,m,s)-nets and (t,s)-sequences
Finite Fields and Their Applications
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In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order @a1 in each variable. The approximation error is studied in the worst case setting in the L"2 norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.