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On the interpolation in linear spaces. (English) Zbl 1051.41004
Let \(X\) and \(Y\) be two linear spaces and let \(f:X\rightarrow Y\) be a given function. If \(x_{0},x_{1},\dots,x_{n}\in X\) are distinct points, the author defines, in some conditions, an abstract interpolation polynomial of the form \(L( x_{0},x_{1},\dots,x_{n};f) (x) =D_{n}x^{n}+D_{n-1}x^{n-1}+\cdots +D_{1}x+D_{0}\), where \(D_{k}\in L_{k}( X,Y) \) \(( 1\leq k\leq n) \) are \(k\)-linear mappings defined on \(X^{k}\) with values in \(Y\) \(\left( D_{0}\in Y\right)\). The application \(D_{n}\in L_{n}(X,Y) \) is called the generalized divided difference of order \(n\) for the function \(f\) on the nodes \(x_{0},x_{1},\dots,x_{n}\). Some properties of the abstract interpolation polynomials and generalized divided differences are presented in this context.
MSC:
41A05 Interpolation in approximation theory
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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