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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.
41A05 Interpolation in approximation theory
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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