Lagrange Polynomial - Proof

Proof

The function L(x) being sought is a polynomial in of the least degree that interpolates the given data set; that is, assumes value at the corresponding for all data points :

Observe that:

  1. In there are k terms in the product and each term contains one x, so L(x) (which is a sum of these k-degree polynomials) must also be a k-degree polynomial.
  2. \ell_j(x_i) = \prod_{m=0,\, m\neq j}^{k} \frac{x_i-x_m}{x_j-x_m}

We consider what happens when this product is expanded. Because the product skips, if then all terms are (except where, but that case is impossible, as pointed out in the definition section—in that term, and since, contrary to ). Also if then since does not preclude it, one term in the product will be for, i.e., zeroing the entire product. So

  1. \ell_j(x_i) = \delta_{ji} = \begin{cases} 1, & \text{if } j=i \\ 0, & \text{if } j \ne i \end{cases}

where is the Kronecker delta. So:

Thus the function L(x) is a polynomial with degree at most k and where .

Additionally, the interpolating polynomial is unique, as shown by the unisolvence theorem at Polynomial interpolation.

Read more about this topic:  Lagrange Polynomial

Famous quotes containing the word proof:

    “Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
    —Johan Huizinga (1872–1945)

    “He who has never failed somewhere, that man can not be great. Failure is the true test of greatness. And if it be said, that continual success is a proof that a man wisely knows his powers,—it is only to be added, that, in that case, he knows them to be small.”
    —Herman Melville (1819–1891)