How to Prove the Lagrange Interpolation Formula?

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SUMMARY

The discussion focuses on proving the Lagrange Interpolation Formula, specifically the equation $$\sum_{i=0}^nq(x_i)L_{n,i}(x)= q(x)$$ for any polynomial $q \in P_n$. Participants emphasize the importance of understanding the Lagrange nodal basis $L_{n,i}$, defined for distinct points $x_0 < x_1 < ... < x_n$. The conversation highlights the necessity of grasping polynomial interpolation concepts to effectively approach the proof.

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  • Understanding of polynomial interpolation
  • Familiarity with Lagrange nodal basis functions
  • Knowledge of polynomial spaces, specifically $P_n$
  • Basic principles of mathematical proof techniques
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  • Explore the concept of polynomial basis functions
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Mathematicians, students studying numerical methods, and anyone interested in understanding polynomial interpolation techniques and their proofs.

HMPARTICLE
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$\text{Let } L_{n,i}, i = 0,...,n, \text{be the Lagrange nodal basis at} x_0 < x_1<...<x_n$. Show that, for any polynomial $q \in P_n$

$$\sum_{i=0}^nq(x_i)L_{n,i}(x)= q(x)$$

I don't know how to begin this proof. I know what a lagrange polynomial is, but I am not sure how to begin. If someone could give me a point to start please.
 
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