Lagrange Interpolation

 

For a number of unspecified functions that are independent of , we define in the form

 

(1)        

 

To comply with the Existence and Uniqueness Theorem, the interpolation conditions must be met, so the equality has to be satisfied for all , .  This requirement is accomplished by letting

 

,

 

which is achieved by

 

(2)        

 

where each is a polynomial of at most degree .  Therefore, is a polynomial of at most degree .

 

Equations (1) and (2) form what is known as Lagrange’s polynomial interpolation.