New moduli of smoothness
  In this paper, we discuss various properties of the new modulus of smoothness \[ \omega^\varphi_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, \] where $\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. 
K. A. Kopotun, D. Leviatan and I. A. Shevchuk
math.CA