Inequalities between best polynomial approximations and some smoothness characteristics in the space L2 and widths of classes of functions

被引：0
作者：
S. B. Vakarchuk
V. I. Zabutnaya
机构：
[1] Alfred Nobel Dnepropetrovsk University,
[2] Oles Gonchar Dnepropetrovsk National University,undefined
来源：
Mathematical Notes | 2016年 / 99卷
关键词：
best polynomial approximation; smoothness characteristics; Jackson-type inequality; modulus of continuity; Bernstein ; -width of a function class; Rolle’s theorem;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λk(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L2. On the basis of this, for differentiable functions in the classes L2r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ωk. For classes of functions defined by their smoothness characteristics Λk(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.
引用
收藏
页码：222 / 242
页数：20
相关论文
共 30 条
  • [1] Trigub R. M.(1980)Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus Izv. Akad. Nauk SSSR Ser. Mat. 44 1378-1409
  • [2] Runovskii K. V.(1994)On approximation by families of linear polynomial operators in Lp, 0 < p < 1 Mat. Sb. 185 81-102
  • [3] Pustovoitov N. P.(1997)Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson’s theorem Mat. Sb. 188 95-108
  • [4] Vakarchuk S. B.(2005)Exact constants in Jackson-type inequalities and exact values of widths Mat. Zametki 78 792-796
  • [5] Vakarchuk S. B.(2008)Widths of the function classes from L2 and exact constants in Jackson type inequalities East J. Approx. 14 411-421
  • [6] Zabutna V. I.(2013)Sharp Jackson–Stechkin type inequalities for periodic functions in L2 and widths of function classes Dokl. Ross. Akad. Nauk Russian Academy of Sciences 451 625-628
  • [7] Shabozov M. Sh.(2004)Problems in the approximation of 2p-periodic functions by Fourier sums in the space L2(2p) Mat. Zametki 76 803-811
  • [8] Vakarchuk S. B.(2007)On the approximation in weighted Lebesgue space Proc. A. Razmadze Math. Inst. 143 103-113
  • [9] Zabutnaya V. I.(2009)A sharp inequality of Jackson–Stechkin type in L2 and the widths of functional classes Mat. Zametki 86 328-336
  • [10] Abilov V. A.(2011)Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in L2 Sibirsk. Mat. Zh. 52 1414-1427
123