In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.
Dunham, Charles B. Unisolvence on Multidimensional Spaces. Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 469-474. doi: 10.4153/CMB-1968-056-6
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author = {Dunham, Charles B.},
title = {Unisolvence on {Multidimensional} {Spaces}},
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pages = {469--474},
year = {1968},
volume = {11},
number = {3},
doi = {10.4153/CMB-1968-056-6},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-056-6/}
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TI - Unisolvence on Multidimensional Spaces
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