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In mathematics, a collection of ''n'' functions ''f''1, ''f''2, ..., ''f''''n'' is unisolvent on domain Ω if the vectors : are linearly independent for any choice of ''n'' distinct points ''x''1, ''x''2 ... ''x''''n'' in Ω. Equivalently, the collection is unisolvent if the matrix ''F'' with entries ''f''''i''(''x''''j'') has nonzero determinant: det(''F'') ≠ 0 for any choice of distinct ''x''''j'''s in Ω. Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. Polynomials are unisolvent by the unisolvence theorem Examples: * 1, ''x'', ''x''2 is unisolvent on any interval by the unisolvence theorem * 1, ''x''2 is unisolvent on (), but not unisolvent on () * 1, cos(''x''), cos(2''x''), ..., cos(''nx''), sin(''x''), sin(2''x''), ..., sin(''nx'') is unisolvent on () Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension ''d'' = 2 and higher (Ω ⊂ R''d''), the functions ''f''1, ''f''2, ..., ''f''''n'' cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points ''x''1 and ''x''2 along continuous paths in the open set until they have switched positions, such that ''x''1 and ''x''2 never intersect each other or any of the other ''x''''i''. The determinant of the resulting system (with ''x''1 and ''x''2 swapped) is the negative of the determinant of the initial system. Since the functions ''f''''i'' are continuous, the intermediate value theorem implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent. ==References== * Philip J. Davis: ''Interpolation and Approximation'' pp. 31–32 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Unisolvent functions」の詳細全文を読む スポンサード リンク
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