In this paper, we discuss various properties of the new modulus of smoothness [ omega^varphi_{k,r}(f^{(r)},t)_p := sup_{0 < hleq t}|mathcal W^r_{kh}(cdot) Delta_{hvarphi(cdot)}^k (f^{(r)},cdot)|_{L_p[-1,1]}, ] where $mathcal W_delta(x) = bigl((1-x-deltavarphi(x)/2) (1+x-deltavarphi(x)/2)bigr)^{1/2}. $ Related moduli with more general weights are also considered.
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