[1] Bakhvalov N S. On the approximate computation of multiple integrals. Vestnik Moskov Univ Ser Mat Mekh Astr Fiz Khim, 1959, 4: 3–18

[2] Traub J F, Wasilkowski G W, Wo′zniakowski H. Information-based Complexity. New York: Academic Press, 1988

[3] Fang G S, Ye P X. Integration error for multivariate functions from anisotropic classes. J Complexity, 2003, 19: 610–627

[4] Fang G S, Ye P X. Complexity of deterministic and randomized methods for multivariate integration problem for the class H (I ). IMA Journal of Numerical Analysis, 2005, 25: 473–485

[5] Heinrich S. Lower bounds for the complexity of Monte Carlo function approximation. J Complexity, 1992,
8: 277–300

[6] Heinrich S. Random approximation in numerical analysis//Bierstedt K D, Bonet J, Horvath J, et al, eds.
Functional Analysis: Proceedings of the Essen Conference. Lect Notes in Pure and Appl Math, Vol 150. Boca Raton: Chapman and Hall/CRC, 1994: 123–171

[7] Heinrich S. Monte Carlo approximation of weakly singular integral operators. J Complexity, 2006, 22: 192–219

[8] Math′e P. Random approximation of Sobolev embedding. J Complexity, 1991, 7: 261–281

[9] Math′e P. Approximation Theory of Stochastic Numerical Methods. Habilitationsschrift, Fachbereich
Mathematik. Berlin: Freie Universit¨at, 1994

[10] Novak E. Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathemat-
ics, Vol 1349. Berlin: Springer, 1988

[11] Temlyakov V N. Approximation of functions with bounded mixed derivative. Tr Mat Inst Akad Nauk SSSR, 1986, 178: 1–112

[12] Temlyakov V N. Approximation of Periodic Functions. New York: Nova Science, 1993

[13] Fang G S, Duan L Q. The complexity of function approximation on Sobolev spaces with bounded mixed
derivative by the linear Monte Carlo methods. J Complexity, 2008, 24: 398–409

[14] Fang G S, Duan L Q. The information-based complexity of approximation problem by adaptive Monte
Carlo methods. Sci China Ser A-Math, 2008, 51(9): 1679–1689

[15] Nikolskii S M. Approximation of Functions of Several Variables and Imbeddings Theorems. Berlin:
Springer-Verlag, 1975

[16] Romanyuk A S. On estimate of the Kolmogorov widths of the classes B in the space L . Ukr Math J, p,q q
2001, 53(7): 1189–1196

[17] Romanyuk A S. Linear widths of the Besov classes of periodic functions of many variables, II. Ukr Math
J, 2001, 53(6): 965–977

[18] Romanyuk A S. Approximation of classes B by linear methods and best approximations. Ukr Math J, p,q 2002, 54(5): 825–838

[19] Pustovoitov N N. Representation and approximation of multivariate periodic functions with a given mixed
modulus of smoothness. Analysis Math, 1994, 20: 35–48

[20] Sun Y S, Wang H P. Representation and approximation of multivariate periodic functions with bounded
mixed moduli of smoothness. Proc Steklov Inst Math, 1997, 219: 350–371

[21] Amanov T I. Representation and imbedding theorems for the function spaces S B(Rn),S B(0 ≤ xj ≤
p,θ p,θ 2π, j = 1,···,n). Trudy Mat Inst Akad Nauk SSSR, 1965, 77: 5–34

[22] Stasyuk S A. Best approximations and Kolmogorov and trigonometric widths of the classes B of periodic
p,θfunctions of many variables. Ukr Math J, 2004, 56(11): 1849–1863

[23] Fedunyk O V. Linear widths of the classes B of periodic functions of many variables in the space Lq. p,θ
Ukr Math J, 2006, 58(1): 103–117

[24] Math′e P. s-Numbers in information-based complexity. J Complexity, 1990, 6: 41–66

[25] Pietsch G. Operator Ideals. Berlin: Deut Verlag Wissenschaften, 1978

[26] Maiorov V E. Discretization of the problem of diameters. Uspekhi Mat Nauk, 1975, 30(6): 179–180

[27] Galeev E M. Kolmogorov widths of classes of periodic functions of many variables W^α _p and H^α_p in the space L_q. Izv Akad Nauk SSSR Ser Mat, 1985, 49: 916–934

[28] Pinkus A. N-widths in Approximation Theory. New York: Springer, 1985