The coset

Algebra Level 4

Suppose $x_1,\cdots,x_n,c_1,\cdots,c_n \in \mathbb{R}$ where $x_i$ are distinct from each other.

Let $S$ be the set of all polynomials $p(x) \in \mathbb{R}[x]$ such that $p(x_i)=c_i$ for all $i$ .

Is $\displaystyle \sum_{i=1}^n \frac{\displaystyle \prod_{j=1, j \neq i}^n (x-x_j)}{\displaystyle \prod_{j=1, j\neq i}^n (x_i-x_j)}c_i$ the polynomial of minimal degree for $S$ ?

It depends on the boundary conditions No Yes

1 solution

Daniel Xiang
Feb 15, 2018

for $x_1,\dots, x_n, y_1, \dots, y_n \in \mathbb{R}$

$\displaystyle L(x) := \sum_{i=1}^{n} \left( y_i \prod_{\begin{smallmatrix}1\le j\le n\\ j\neq i\end{smallmatrix}} \frac{x-x_j}{x_i-x_j}\right)$

The above is the definition of a Lagrange polynomial $L(x)$ , which satisfies $L(x_i)=y_i\space\space\forall i$

obviously $L(x)$ is a polynomial of degree $n-1$ .

A polynomial function of degree $k$ can be determined if it passes through $k+1$ known points, because it has $k+1$ coefficients.

therefore $L(x)$ is the polynomial of minimal degree for $S$

Yes, except technically, $L(x)$ can have degree less than $n-1$ in some cases.

Jonathan Dunay - 3 years, 3 months ago

The coset

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