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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 $varphi(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $L_p[-1,1]$ (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.
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-de
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)
We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on
We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of curves that are defined in terms of Weierstrass points.
We discuss the worldvolume description of intersecting D-branes, including the metric on the moduli space of deformations. We impose a choice of static gauge that treats all the branes on an equal footing and describes the intersection of D-branes as