Fetching the paper…
Reading the bibliography…
In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness.
Approximate evaluation of repeated integrals
N. M. Korobov, · 1959
Earlier work this paper cites.
Zur angenäherten Berechnung mehrfacher Integrale
E. Hlawka, · 1962
Earlier work this paper cites.
Optimal convergence bounds for quadrature processes and integration methods of Monte Carlo type for classes of functions
N. S. Bakhvalov, · 1963
Earlier work this paper cites.
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93, 107–115, 1971
1971
Earlier work this paper cites.
Spaces of Differentiable Functions with Dominating Mixed Derivatives
T. I. Amanov, · 1976
Earlier work this paper cites.
K. K. Frolov, Upper error bounds for quadrature formulas on function classes, Dokl. Akad. Nauk SSSR 231, 818–821, 1976
1976
Earlier work this paper cites.
Approximation of Functions of Several Variables and Embedding Theorems
S. M. Nikol’skij, · 1977
Earlier work this paper cites.
B. Stöckert, Ungleichungen vom Plancherel-Polya-Nikolskij-Typ in gewichteten L p Ω L_{p}^{\Omega} -Räumen mit gemischten Normen, Math. Nachr
1978
Earlier work this paper cites.
H. Triebel, Theory of Function Spaces . Birkhäuser, Basel, 1983
1983
Earlier work this paper cites.
V. A. Bykovskii, On the correct order of the error of optimal cubature formulas in spaces with dominant derivative, and on quadratic deviations of grids info Preprint, Computing Center Far-Eastern Scientific Center, Akad. Sci. USSR, Vladivostok, 1985
1985
Earlier work this paper cites.
On reconstruction of multivariate periodic functions based on their values at the knots of number-theoretical nets
V. N. Temlyakov, · 1986
Earlier work this paper cites.
H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces
1987
Earlier work this paper cites.
On a way of obtaining lower estimates for the errors of quadrature formulas
V. N. Temlyakov, · 1990
Earlier work this paper cites.
Error estimates for Fibonacci quadrature formulas for classes of functions with a bounded mixed derivative
V. N. Temlyakov, · 1991
Earlier work this paper cites.
H. Triebel, Theory of Function Spaces II . Birkhäuser, Basel, 1992
1992
Cited alongside, same era.
R. A. DeVore and G. G. Lorentz, Constructive Approximation
1993
Cited alongside, same era.
V. V. Dubinin, Cubature formulas for classes of functions with bounded mixed difference, Matem. Sbornik, 183 (1992), issue 7; English transl. in Math. USSR Sbornik 76 (1993), 283–292
1993
Cited alongside, same era.
V. N. Temlyakov, Approximation of Periodic Functions . Computational Mathematics and Analysis Series, Nova Science Publishers, Inc., Commack, NY, 1993
1993
Cited alongside, same era.
M. M. Skriganov, Constructions of uniform distributions in terms of geometry of numbers, Algebra i Analiz 6, 200–230, 1994
1994
Cited alongside, same era.
Smolyak’s algorithm, sparse grid approximation and periodic function spaces with dominating mixed smoothness, Thesis, Friedrich-Schiller-Universität Jena, Jena, 2007
T. Ullrich, · 2007
Later among the works it cites.
H. Triebel, Function spaces and wavelets on domains. EMS Publ. House, Zürich, 2008
2008
Later among the works it cites.
T. Ullrich, Local mean characterization of Besov-Triebel-Lizorkin type spaces with dominating mixed smoothness on rectangular domains, Preprint, Uni Bonn, 2008
2008
Later among the works it cites.
Discrepancy of Hammersley points in Besov spaces of dominating mixed smoothness
A. Hinrichs, · 2010
Later among the works it cites.
E. Novak and H. Woźniakowski, Tractability of Multivariate Problems , Volume II: Standard Information for Functionals, European Math. Soc. Publ. House, Zürich, 2010
2010
Later among the works it cites.
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations
T. Runst and W. Sickel, · 1996
Cited alongside, same era.
V. V. Dubinin, Cubature formulae for Besov classes, Izvestiya Math 61(2) (1997), 259-283
1997
Cited alongside, same era.
W. Sickel, On pointwise multipliers for F p , q s F^{s}_{p,q} . The case σ p , q < s < n / p \sigma_{p,q}<s<n/p , Annali di Matematica pura ed applicata
1999
Cited alongside, same era.
V. N. Temlyakov, Cubature formulas, discrepancy, and nonlinear approximation, J. Complexity 19, 352–391, 2003
2003
Cited alongside, same era.
Monte Carlo Methods in Financial Engineering
P. Glasserman, · 2004
Cited alongside, same era.
M. Christ and A. Seeger, Necessary conditions for vector-valued operator inequalities in harmonic analysis. Proc. London Math. Soc. (3) 93, no. 2, 447–473, 2006
2006
Cited alongside, same era.
Function spaces with dominating mixed smoothness; characterization by differences
T. Ullrich, · 2006
Cited alongside, same era.
D. Nuyens and R. Cools, Higher Order Quasi‐Monte Carlo Methods: A Comparison , AIP Conf. Proc. 1281, 553–557, 2010
2010
Later among the works it cites.
H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration . EMS Tracts in Mathematics, European Mathematical Society (EMS), Zürich, 2010
2010
Later among the works it cites.
H. Yserentant, Regularity and Approximability of Electronic Wave Functions
2010
Later among the works it cites.
J. Dick and F. Pillichshammer, Discrepancy Theory and Quasi-Monte Carlo Integration
2013
Later among the works it cites.
L. Markhasin, Discrepancy and integration in function spaces with dominating mixed smoothness, Dissertationes Math. 494, 1–81, 2013
2013
Later among the works it cites.
D. Dũng and T. Ullrich, Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square, Math. Nachrichten , DOI 10.1002/mana.201400048, 2014
2014
Later among the works it cites.
M. Ullrich, On ”Upper error bounds for quadrature formulas on function classes” by K. K. Frolov, ArXiv e-prints 1404.5457, 2014
2014
Later among the works it cites.
2015
Closest in time.
M. Ullrich and T. Ullrich, The role of Frolov’s cubature formula for functions with bounded mixed derivative, Preprint, 2015
2015
Closest in time.