Definition:Polynomial Function/Real

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Definition

Let $S \subset \R$ be a subset of the real numbers.


Definition 1

A real polynomial function on $S$ is a function $f : S \to \R$ for which there exist:

such that for all $x\in S$:

$f(x) = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where $\sum$ denotes indexed summation.


Definition 2

Let $\R \sqbrk X$ be the polynomial ring in one variable over $\R$.

Let $\R^S$ be the ring of mappings from $S$ to $\R$.

Let $\iota \in \R^S$ denote the inclusion $S \hookrightarrow \R$.


A real polynomial function on $S$ is a function $f: S \to \R$ which is in the image of the evaluation homomorphism $\R \sqbrk X \to \R^S$ at $\iota$.


Coefficients

The parameters $a_0, \ldots a_n \in \R$ are known as the coefficients of the polynomial $P$.


Also see