On \(j\)-convex preserving interpolation operators
The author researches the existence of \(j\)-convex preserving interpolation operators. The author includes an application in which we made use of a sufficient set of test-functions to characterize when every degree of convexity can be preserved among particular families of polynomial interpolation operators, which include the Bernstein operators. Let \((X,\|\cdot \|)= (C[a,b],\|\cdot\|_\infty)\) be a linear operator \(P:X\to X\) that can be written \(P=\sum^m_{i=0} \delta_{t_i} \otimes v_i\), where \(t_i\in [a,b]\), \(t_i< t_{i+1}\) and \(v_i\in X\), \(i=0,\dots,m\), is said to be an interpolation operator. We denote the set of interpolation operators by \({\mathcal P}\). By \(S\) we denote the set of \(j\)-convex functions of \(X\). Definition 2.1. Fix \(a=t_0<t_1< \cdots< t_m=b\) and let \(\Omega\subset X\). Let \({\mathcal P}_j\) denote the operators of \({\mathcal P}\) that leave \(\prod_{j-1}\) invariant. We say \(\Omega\) is a sufficient test-function set if, for all \(P\in{\mathcal P}_j\), \(P\Omega\subset S\) implies \(P\) preserves \(j\)-convexity. Similarly, we say \(\Omega\) is a necessary test-function set if \(P\Omega\subset S\) whenever \(P\in{\mathcal P}_j\) preserves \(j\)-convexity. Definition 2.2. For the integer \(j\geq 2\) and for \(k=0,\dots,m-j\), define \[ \omega^+_k (x):= \begin{cases} 0\quad & \text{for } a\leq x\leq t_{k+j-1}\\ \omega_k (x) \quad & \text{for } t_{k+j-1}\leq x\leq b\end{cases} \] where \(\omega_k(x):=(x-t_{k+1}) \cdots(x-t_{k+j}-1)\). Let \(\Omega=\{\omega^+_k\}^{m-j}_{k=0}\). Theorem 2.1: \(\Omega\) is a sufficient test-function set. Theorem 2.2: If \(j=1,2\) then \(\Omega\) is a necessarry and sufficient test -- function set. Theorem 2.3: If \(m=j\), \(j+1\) then \(\Omega\) is a necessary and sufficient test -- function set. Theorem 3.1: Let \(\Omega= \{\omega_1, \cdots,\omega_n\} \subset X\). Then \(\Omega\) is not a necessary and sufficient test-function set. Definition 4.1: An interpolation operator \(P=\sum^n_{i=0} \delta_{t_i} \otimes v_i\) is said to be an \(n\)-th degree Bernstein-type operator if \(t_i=i/n\) and, in addition, each \(v_i\) contains a \(t^i\) term but contains no \(t^j\) term, \(j<i\). Let \({\mathcal B}_n\) denote the set of all \(n\)th degree Bernstein-type operators. Theorem 4.1: Let \(P\) be an \(n\)-th degree Bernstein-type operator. Then \(P\) preserves (simultaneously \(j\)-convexity for \(j=0,1,\dots n\) iff \(P\) preserves \(p\)=convexity or positivity.
scientific article; zbMATH DE number 1455758
scientific article