New moduli of smoothness

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 $\mathcal W_δ(x) = \bigl((1-x-δφ(x)/2) (1+x-δφ(x)/2)\bigr)^{1/2}.$ Related moduli with more general weights are also considered.

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