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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.
We introduce new moduli of smoothness for functions $fin L_p[-1,1]cap C^{r-1}(-1,1)$, $1le pleinfty$, $rge1$, that have an $(r-1)$st locally absolutely continuous derivative in $(-1,1)$, and such that $varphi^rf^{(r)}$ is in $L_p[-1,1]$, where $varph
We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as [ omega_{k,r}^varphi(f^{(r)},t)_{alpha,beta,p} :=sup_{0leq hleq t} left| {mathcal{W}}_{kh}^{r/2+alpha,r/2+beta}(cdot)
In this article, the authors introduce Besov-type spaces with variable smoothness and integrability. The authors then establish their characterizations, respectively, in terms of $varphi$-transforms in the sense of Frazier and Jawerth, smooth atoms o
The present paper is concerned with new Besov-type space of variable smoothness. Nonlinear spline-approximation approach is used to give atomic decomposition of such space. Characterization of the trace space on hyperplane is also obtained.
In this paper we study approximation theorems for $L^2$-space on Damek-Ricci spaces. We prove direct Jackson theorem of approximations for the modulus of smoothness defined using spherical mean operator on Damek-Ricci spaces. We also prove Nikolskii-