In this article, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a=$\{a_j\}_{j\geq1}$ and b=$\{b_j\}_{j\geq1}$ of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on a, b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.
Jidong HAO
,
Heping WANG
. STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1765
-1782
.
DOI: 10.1007/s10473-020-0611-x
[1] Chen J, Wang H P. Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings. J Complexity, 2017, 39:94-110
[2] Chen J, Wang H P. Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball-Preasymptotics, asymptotics, and tractability. J Complexity, 2019, 50:1-24
[3] Cobos F, Kühn T, Sickel W. Optimal approximation of multivariate periodic Sobolev functions in the sup-norm. J Funct Anal, 2016, 270(11):4196-4212
[4] Dick J, Kritzer P, Pillichshammer F, Woźniakowski H. Approximation of analytic functions in Korobov spaces. J Complexity, 2014, 30:2-28
[5] Dick J, Larcher G, Pillichshammer F, Woźniakowski H. Exponential convergence and tractability of multivariate integration for Korobov spaces. Math Comp, 2011, 80:905-930
[6] Irrgeher C, Kritzer P, Pillichshammer F, Woźniakowski H. Tractability of multivariate approximation defined over Hilbert spaces with exponential weights. J Approx Theory, 2016, 207:301-338
[7] Kritzer P, Pillichshammer F, Woźniakowski H. Multivariate integration of infinitely many times differentiable functions in weighted Korobov spaces. Math Comp, 2014, 83:1189-1206
[8] Kritzer P, Pillichshammer F, Woźniakowski H. Tractability of multivariate analytic problems//Kritzer P, Niederreiter H, Pillichshammer F, Winterhof A, eds. Uniform Distribution and Quasi-Monte Carlo Methods, Discrepancy, Integration and Applications. Berlin:De Gruyter, 2014:147-170
[9] König H. Eigenvalue Distribution of Compact Operators. Basel:Birkhäuser, 1986
[10] Kühn T, Mayer S, Ullrich T. Counting via entropy:new preasymptotics for the approximation numbers of Sobolev embeddings. SIAM J Numer Anal, 201654(6):3625-3647
[11] Kühn T, Sickel W, Ullrich T. Approximation numbers of Sobolev embeddings-Sharp constants and tractability. J Complexity, 2014, 30:95-116
[12] Kühn T, Sickel W, Ullrich T. Approximation of mixed order Sobolev functions on the d-torus:asymptotics, preasymptotics, and d-dependence. Constr Approx, 2015, 42(3):353-398
[13] Liu Y P, Xu G Q. A note on tractability of multivariate analytic problems. J Comlexity 2016, 34:42-49
[14] Liu Y P, Xu G Q. (s, t)-Weak tractability of multivariate linear problems in the average case setting. Acta Math Sci, 2019, 39B(4):1033-1052
[15] Novak E, Woźniakowski H. Tractablity of Multivariate Problems, Volume I:Linear Information. Zürich:European Math Soc, 2008
[16] Novak E, Woźniakowski H. Tractablity of Multivariate Problems, Volume II:Standard Information for Functionals. Zürich:European Math Soc, 2010
[17] Novak E, Woźniakowski H. Tractablity of Multivariate Problems, Volume III:Standard Information for Operators. Zürich:European Math Soc, 2012
[18] Papageorgiou A, Petras I. A new criterion for tractability of multivariate problems. J Complexity 2014, 30:604-619
[19] Pietsch A. Operator Ideals. Amsterdam:North-Holland, 1980
[20] Pietsch A. Eigenvalues and s-Numbers. Cambridge:Cambridge University Press, 1987
[21] Pinkus A. n-Widths in Approximation Theory. Berlin:Springer, 1985
[22] Xu G Q. Exponential convergence-tractability of general linear problems in the average case setting. J Complexity, 2015, 31:617-636
[23] Wang H P. A note about EC-(s, t)-weak tractability of multivariate approximation with analytic Korobov kernels. J Complexity, 2019, 55, 101412, 19 pp
[24] Wang X. Volumes of generalized unit balls. Math Mag, 2005, 78(5):390-395
[25] Werschulz A, Woźniakowski H. Tractability of multivariate approximation over weighted standard Sobolev spaces. J Complexity, 2019, 53:95-112
