New moduli of smoothness

Abstract

In this paper, we discuss various properties of the new modulus of smoothness \[ ω^φ_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) Δ_{hφ(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, \] where ${\mathcal{W}}_{\delta }(x)=((1-x-\delta \phi (x)/2)(1+x-\delta \phi (x)/2){)}^{1/2}.$ Related moduli with more general weights are also considered.