Steckin-Marchaud-type inequality in connection with \(L^p\) approximation for multivariate Bernstein-Durrmeyer operators (Q2732540)

Let \(T\) be a simplex in \(\mathbb{R}^d\), \(T:=\{x=(x_1, x_2,\dots, x_d)\), \(x_i\geq 0\), \(i=1,2,\dots,d\), \(1-|x|\geq 0\}\), here \(|x|= \sum^d_{i=1} x_i\). \(L^{+\infty} (T):=C(T)\). For \(f\in L'(T)\), \(n\in\mathbb{N}\), the Bernstein-Durrmeyer operators are defined by NEWLINE\[NEWLINEM_{n,d}(f,x)= \sum_{|k |\leq n}P_{n,k} (x){(n+d)!\over n!} \int_TP_{n,k} (u)f(u)du,NEWLINE\]NEWLINE here NEWLINE\[NEWLINEP_{n, k}(x) ={n!\over k!(n- |k|)!} x^k\bigl(1- |x|\bigr)^{n-|k |},\;x\in T,NEWLINE\]NEWLINE \(k=(k_1,k_2, \dots,k_d) \in N_0^d\), \(x^k= x_1^{k_1} x_2^{k_2} \cdots x_d^{k_d}\), \(|k|= \sum^d_{i=1} k_i\), \(k!=k_1!k_2! \cdots k_d!\). For \(x\in T\), the authors define \(\varphi_i (x)=\varphi_{ii} (x): =\sqrt {x_i(1- |x|)}\), \(1\leq i\leq d\), \(\varphi_{ij}: =\sqrt{x_ix_j}\), \(1\leq i<j\leq d\). Let \(e_i:=(0,0,\dots, 0,1,0,\dots,0)\) be the unit vector in \(\mathbb{R}^d\), \(e_{ij}= e_i-e_j\), for an arbitrary vector \(e\) in \(R^d\), by \(\Delta^r_{he} f(z)\) the authors denote the symmetric difference on the direction \(e\). Let \(f\in L^p(T) (1\leq p\leq +\infty)\), its modulus of \(r\)th smoothness \(\omega^r_\Phi (f,t)_p:= \sup_{0<h\leq t} \sum_{1\leq i\leq j\leq d}\|\Delta^r_h \varphi_{ij}e_{ij} f \|_p\). The authors prove an inverse theorem: Theorem 1.1: Let \(f\in L^p(T)\), \(1\leq p\leq+\infty\), then NEWLINE\[NEWLINE\omega^2_\Phi (f,1/ \sqrt n)_p \leq{\text{const} \over n}\sum^n_{k=1} \|M_{k,d} f-f\|_p.NEWLINE\]NEWLINE The authors get an equivalent theorem: Corollary 1.1: For \(f\in L^p(T)\), \(1\leq p\leq+ \infty\), then NEWLINE\[NEWLINE\|M_{n,d} f-f\|_p=O(n^{-\alpha}) \Leftrightarrow \omega^2_\Phi (f,t)=O (t^{2\alpha})(0< \alpha<1).NEWLINE\]