New moduli of smoothness

Abstract: 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.