Piecewise Chebyshev-Schoenberg operators: shape preservation, approximation and space embedding
The aim of the paper is to discuss the possibility of the construction of an analogue of the Schoenberg operator and to study its properties in the following very general framework. Consider the points \(a=t_0<t_1<\dotsb<t_q<t_{q+1}=b\) and associate with the interior points the multiplicities \(m_1,\dotsc,m_q\), with \(0\leq m_k\leq n\), for \(1\leq k\leq q\). Let there be given a sequence of spaces \(\mathbb{E}_k\subset C^n([t_k,t_{k+1}])\), \(0\leq k\leq q\), containing constants such that \(D\mathbb{E}_k:=\{DF:=F'\mid F\in \mathbb{E}_k\}\) are \(n\)-dimensional extended Chebyshev spaces. Suppose also that there is a sequence \(M_k\), \(1\leq k\leq q\) of connection matrices, each \(M_k\) being a lower triangular matrix of order \(n-m_k\) with positive diagonal entries. In this context consider the spline space \(\mathbb{S}\), of continuous functions \(S:[a,b]\to\mathbb{R}\) for which the restrictions to \([t_k,t_{k+1}]\) belong to \(\mathbb{E}_k\) for \(0\leq k\leq q\) and which satisfy the connection conditions \[ (DS(t_k^+),\dotsc,D^{n-m_k}S(t_k^+))= M_k(DS(t_k^-),\dotsc,D^{n-m_k}S(t_k^-)),\quad\text{ for }1\leq k\leq q. \] The construction of Schoenberg operators in the particular case where the section-spaces \(\mathbb{E}_k\) are obtained by restricting to \([t_k,t_{k+1}]\) of some space \(\mathbb{E}\subset C^n([a,b])\) was analysed in a previous paper by the author [Constr. Approx. 34, No. 2, 181--208 (2011; Zbl 1229.41010)]. In the generalized framework, the requirements imposed on the sequence \(\mathbb{E}_k\), \(0\leq k\leq q\), assure the existence of the blossom function, which plays a crucial role in the existence of \(B\)-spline basis, [the author, Constr. Approx. 20, No. 4, 603--624 (2004; Zbl 1063.65017)]. A piecewise Chebyshev-Schoenberg operator \(\mathcal{S}\) on \(C([a,b])\) is of the form \(\mathcal{S}F=\sum_{l=-n}^{m}F(\eta_l)N_l\), \(F\in C([a,b])\), where \(N_l\) is a \(B\)-spline basis on the knots \(\eta_l\), \(-n\leq l\leq m\) and \(\mathcal{S}\) preserves the function \({\mathbf 1}\) and another strictly increasing spline \(U\in\mathbb{S}\). The existence of such a type of operators, under certain conditions, and some shape preserving properties are proved. Also, a sequence of such operators which satisfies certain simple conditions possesses the property of uniform approximation of functions and of their derivatives. In the last part of the paper, the problem of embedding a piecewise Chebyshev spline space into another greater space of the same type is discussed. The paper contains a great number of remarks, connections and examples which are very important for clarifying the theory.
scientific article; zbMATH DE number 6137310