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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)
##### Keywords:
abstract interpolation polynomial
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