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MATH
[1251]

Theorems from Data fitting and polynomial interpolation

All theorems from Data fitting and polynomial interpolation.

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Distinct inputs give a unique interpolant

If t1,…,tnt_1,\ldots,t_n are distinct, the columns of the Vandermonde matrix are linearly independent and span Rn\mathbb R^n. Hence Ax=yAx=y has a unique solution for every data vector yy, so there is exactly one polynomial in Pn−1P_{n-1} satisfying

f(ti)=yi,i=1,…,n.f(t_i)=y_i,\qquad i=1,\ldots,n.
  1. [1]:Interpolate in the standard polynomial basis
  2. [2]:Polynomial interpolation
  3. [3]:Proof

The Lagrange polynomials form a basis

The set

L=(ℓ1,…,ℓn)L=(\ell_1,\ldots,\ell_n)

is a basis for Pn−1P_{n-1}.

  1. [1]:Lagrange polynomials
  2. [2]:Basis
  3. [3]:Proof
  4. [4]:Interpolate in the Lagrange basis

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Distinct inputs give a unique interpolantThe Lagrange polynomials form a basis