English

New moduli of smoothness

Classical Analysis and ODEs 2015-05-13 v2

Abstract

In this paper, we discuss various properties of the new modulus of smoothness ωk,rφ(f(r),t)p:=sup0<htWkhr()Δhφ()k(f(r),)Lp[1,1], \omega^\varphi_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, where Wδ(x)=((1xδφ(x)/2)(1+xδφ(x)/2))1/2.\mathcal W_\delta(x) = \bigl((1-x-\delta\varphi(x)/2) (1+x-\delta\varphi(x)/2)\bigr)^{1/2}. Related moduli with more general weights are also considered.

Keywords

Cite

@article{arxiv.1408.2018,
  title  = {New moduli of smoothness},
  author = {K. A. Kopotun and D. Leviatan and I. A. Shevchuk},
  journal= {arXiv preprint arXiv:1408.2018},
  year   = {2015}
}
R2 v1 2026-06-22T05:23:47.111Z