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Fixed point and interpolation point set of a positive linear operator on \(C(\overline{D})\). (English) Zbl 1240.41069

The aim of this paper is to establish for a bounded open convex subset \(D\subset \mathbb{R}^p\) and a positive linear operator \(A:\mathbb{C}(\overline{D})\to\mathbb{C}(\overline{D})\) relations between the mixed-extremal point set of \(D\), the fixed point set and the interpolation point set of \(A\).
Let \(D\) have nonempty interior, then a point \(x^0=(x_1^0,\dots,x_p^0)\in\partial D\) is called mixed-extremal if, for each \(i\in\{1,\dots,p\}\), \(x_i^0\) is an extremal point of the ordered set \((\{x_i\mid (x_1,\dots, x_p)\in D\},\leq_{\mathbb{R}})\). The set of all mid-extremal points of \(D\) is denoted by \((ME)_D\).
Moreover, a point \(x\in\overline{D}\) is an interpolation point of \(A\) if \(A(f)(x)=f(x)\) for all \(f\in\mathbb{C}(\overline{D})\); a subset \(E\subset\overline{D}\) is an interpolation set of \(A\) if \(A(f)|_E=f|_E\).
The main results are as follows.
{(1)} Suppose that \(A\) is an increasing operator and \(\Pi_1(\overline{D})\subset F_A\), then \((ME)_D\) is an interpolation set of \(A\).
(\(\Pi_1\) denotes the set of all polynomials of degree at most \(1\); \(F_A\) is not defined in the paper).
{(2)} If \(A\) is an increasing operator and \(\Pi_0(\overline{D})\subset F_A\), then \(E:=\{x\in (ME)_D\mid A(q_i)(x)=x_i\}\) is an interpolation set of \(A\).
{(3)} If \(A\) is an increasing operator and \(q_1,\dots,q_p\in F_A\), then \(E:=\{x\in (ME)_D\mid A(\tilde{1}(x)=1\}\) is an interpolation set of \(A\).
{(4)} Suppose \(A:C[a,b]\to C[a,b]\) is an increasing linear operator and \(e_0,e_2\in F_A\), then: if \(A(e_1)(a)=a\), then \(a\) is an interpolation point of \(A\); if \(A(e_1)(b)=b\), then \(b\) is an interpolation point of \(A\).

MSC:

41A36 Approximation by positive operators
41A05 Interpolation in approximation theory
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
46B70 Interpolation between normed linear spaces
30E05 Moment problems and interpolation problems in the complex plane
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