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Journal of Complexity, ISSN 0885-064X, 04/2014, Volume 30, Issue 2, pp. 72 - 94
We prove new optimal bounds for the error of numerical integration in bivariate Besov spaces with dominating mixed order r. The results essentially improve on... 
Faber basis | Mixed Besov spaces | Cubature formula | Hammersley point sets | MATHEMATICS | MATHEMATICS, APPLIED | INTEGRATION | DISCREPANCY | Cuba | Monte Carlo methods | Tensors | Periodic functions | Tools | Estimates | Smoothness | Optimization
Journal Article
Constructive Approximation, ISSN 0176-4276, 10/2017, Volume 46, Issue 2, pp. 227 - 242
In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic... 
46B15 | 46E35 | Sobolev space | 42C40 | Numerical Analysis | Analysis | Unconditional bases | Triebel–Lizorkin space | Mathematics | Haar system | Besov space | MATHEMATICS | Triebel-Lizorkin space
Journal Article
SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2016, Volume 54, Issue 2, pp. 969 - 993
We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on... 
Frolov cubature formula | Numerical integration | Sobolev space | Besov-Triebel-Lizorkin space | QUADRATURE-FORMULAS | DIMENSIONALITY | MATHEMATICS, APPLIED | numerical integration | SMOOTH FUNCTIONS | SQUARE | APPROXIMATION | BESOV-SPACES | LIPSCHITZ | NUMERICAL-INTEGRATION | DISCREPANCY | TRIEBEL-LIZORKIN SPACES | Mathematics - Numerical Analysis
Journal Article
Journal of Fourier Analysis and Applications, ISSN 1069-5869, 10/2018, Volume 24, Issue 5, pp. 1319 - 1339
Journal Article
Constructive Approximation, ISSN 0176-4276, 8/2017, Volume 46, Issue 1, pp. 69 - 108
In a recent article, two of the present authors studied Frolov’s cubature formulae and their optimality in Besov–Triebel–Lizorkin spaces of functions with... 
68Q25 | Numerical integration | Periodization | Mathematics | 65Y20 | Change of variable | 42B35 | 47B38 | 41A25 | Besov–Triebel–Lizorkin spaces | 41A63 | Numerical Analysis | Analysis | Dominating mixed smoothness | MATHEMATICS | Besov-Triebel-Lizorkin spaces | CUBATURE FORMULAS | INEQUALITIES | BOUNDS | DISCREPANCY | ERROR
Journal Article
Numerische Mathematik, ISSN 0029-599X, 8/2017, Volume 136, Issue 4, pp. 993 - 1034
Journal Article
Foundations of Computational Mathematics, ISSN 1615-3375, 12/2013, Volume 13, Issue 6, p. 965
  In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths d ^sub n^(W,H ^sup γ^), and [straight epsilon]-dimensions n... 
Approximations | Polynomials
Journal Article
Foundations of Computational Mathematics, ISSN 1615-3375, 12/2013, Volume 13, Issue 6, p. 965
In this paper, we study linear trigonometric hyperbolic cross approximations, Kolmogorov n-widths [d.sub.n](W, [H.sup.[gamma]]), and [epsilon]-dimensions... 
Approximation theory | Metric spaces | Research
Journal Article
Monatshefte für Mathematik, ISSN 0026-9255, 11/2017, Volume 184, Issue 3, pp. 425 - 441
We deal with lattices that are generated by the Vandermonde matrices associated to the roots of Chebyshev polynomials. If the dimension d of the lattice is a... 
Numerical integration | 06D50 | Mathematics | 11J71 | 52C07 | Admissible lattices | 65D30 | Frolov cubature formula | 65D32 | Orthogonal lattices | 11K06 | Mathematics, general | Lattice points in convex bodies | MATHEMATICS | INTEGRATION | BOUNDS | ERROR | FORMULAS | CUBATURE | Numerical analysis
Journal Article
Mathematische Nachrichten, ISSN 0025-584X, 05/2015, Volume 288, Issue 7, pp. 743 - 762
We prove lower bounds for the error of optimal cubature formulae for d‐variate functions from Besov spaces of mixed smoothness Bp,θα(Gd) in the case 1≤p≤∞,... 
41A55 | 41A25 | 41A58 | 41A63 | Quasi‐Monte‐Carlo integration | 65D32 | Fibonacci lattice | Besov spaces of mixed smoothness | B‐spline representations | Smolyak grids | Quasi-Monte-Carlo integration | B-spline representations | MATHEMATICS | SPACES | DISCREPANCY | FORMULAS
Journal Article
2018, Advanced Courses in Mathematics - CRM Barcelona, ISBN 9783319922393
eBook
Journal of Complexity, ISSN 0885-064X, 10/2018, Volume 48, pp. 69 - 102
We consider the problem of determining the asymptotic order of the Gelfand numbers of mixed-(quasi-)norm embeddings ℓpb(ℓqd)↪ℓrb(ℓud) given that p≤r and q≤u,... 
Besov spaces with small mixed smoothness | Gelfand numbers | Block sparsity | [formula omitted]-spaces | Compressed sensing | Sparsity-in-levels | spaces | MATHEMATICS, APPLIED | APPROXIMATION | l(p)(l(q))-spaces | SIGNAL RECOVERY | INEQUALITY | HOLDER-NIKOLSKII CLASSES | MATHEMATICS | WIDTHS | ENTROPY
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 2011, Volume 260, Issue 11, pp. 3299 - 3362
Journal Article
SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2016, Volume 54, Issue 6, pp. 3625 - 3647
In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier... 
Sobolev and Gevrey type spaces | Tractability | Approximation numbers | D-dependence | Rate of convergence | Preasymptotics | MATHEMATICS, APPLIED | approximation numbers | preasymptotics | d-dependence | BANACH-SPACES | rate of convergence | tractability
Journal Article
Numerische Mathematik, ISSN 0029-599X, 09/2016, Volume 134, Issue 1, pp. 163 - 196
We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces with dominating mixed smoothness . We... 
42C10 | 65D30 | 46E35 | 91G60 | 65D32 | 11K06 | 65C05 | 11J71 | DOMINATING MIXED SMOOTHNESS | DISCREPANCY BOUNDS | MATHEMATICS, APPLIED | CUBATURE FORMULAS | SEQUENCES | BESOV-SPACES | Analysis | Numerical analysis | Mathematics - Numerical Analysis
Journal Article
Journal of Complexity, ISSN 0885-064X, 2010, Volume 26, Issue 6, pp. 629 - 640
We provide sharp lower and upper bounds for the Gelfand widths of ℓ -balls in the N-dimensional ℓ -space for 0 Compressive sensing | Sparse recovery | Gelfand widths | ℓ1-minimization
Journal Article
Constructive Approximation, ISSN 0176-4276, 12/2015, Volume 42, Issue 3, pp. 353 - 398
We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $$s>0$$ s > 0 on the... 
Quasi-polynomial tractability | Mathematics | Sobolev spaces of mixed smoothness | Approximation numbers | 41A25 | 46E35 | 41A63 | Numerical Analysis | Analysis | 42A10 | 65D15 | Rate of convergence | Preasymptotics | d -Dependence | d-Dependence | INTERPOLATION | MATHEMATICS | SPACES | SPARSE GRIDS | TRACTABILITY
Journal Article
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