Theorem (Recursive Polynomial Construction).  Given the function    that is to be approximated, and the set of distinct nodes  

    .  

For any pair of nodes  ,  suppose that we have constructed the polynomials:

      which agrees with at the nodes  ,  

      which agrees with at the nodes  .  

Then      is formed by making a combination of the above two polynomials

    ,
    or
    ,

and it agrees with at all the nodes  .

Remark. Other equivalent ways to write   are

    ,
    or
    .

Proof.

For    we evaluate    as follows:  

    

    

    
    
    

    

For    we evaluate    as follows:  

    

    

    

    

For    we evaluate    as follows:  

    

    

    

    

Therefore, we have

        for      

(c) John H. Mathews 2005