اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn some properties of moduli of smoothness with Jacobi weights
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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) Delta_{hvarphi(cdot)}^k (f^{(r)},cdot)right|_p ] where $varphi(x) = sqrt{1-x^2}$, $Delta_h^k(f,x)$ is the $k$th symmetric difference of $f$ on $[-1,1]$, [ {mathcal{W}}_delta^{xi,zeta} (x):= (1-x-deltavarphi(x)/2)^xi (1+x-deltavarphi(x)/2)^zeta , ] and $alpha,beta > -1/p$ if $0<p<infty$, and $alpha,beta geq 0$ if $p=infty$. We show, among other things, that for all $m, nin N$, $0<ple infty$, polynomials $P_n$ of degree $<n$ and sufficiently small $t$, begin{align*} omega_{m,0}^varphi(P_n, t)_{alpha,beta,p} & sim t omega_{m-1,1}^varphi(P_n, t)_{alpha,beta,p} sim dots sim t^{m-1}omega_{1,m-1}^varphi(P_n^{(m-1)}, t)_{alpha,beta,p} & sim t^m left| w_{alpha,beta} varphi^{m} P_n^{(m)}right|_{p} , end{align*} where $w_{alpha,beta}(x) = (1-x)^alpha (1+x)^beta$ is the usual Jacobi weight. In the spirit of Yingkang Hus work, we apply this to characterize the behavior of the polynomials of best approximation of a function in a Jacobi weighted $L_p$ space, $0<pleinfty$. Finally we discuss sharp Marchaud and Jackson type inequalities in the case $1<p<infty$.
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