Theorem 9 (Existence and uniqueness)

If [Maple Math] , with [Maple Math] are [Maple Math] distinct data points (where the [Maple Math] might or might not be generated by a function [Maple Math] ) , then there existes a unique polynomial [Maple Math] of degree at most [Maple Math] with the property that [Maple Math] for each value of [Maple Math] . This polynomial is given by [Maple Math] or [Maple Math]

where [Maple Math]

for each [Maple Math] .

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It is easy to verify that [Maple Math] for [Maple Math] and [Maple Math] .