Polynomial function

A polynomial function is the function obtained by evaluating a polynomial on a specified ring or field. Distinct polynomials can induce the same polynomial function over a finite field; for example, $x^p$ and $x$ agree on ${\mathbb{F}}_p$.

Over ${\mathbb{F}}_2$, the nonzero formal polynomial $X^2-X$ evaluates to zero at both possible inputs, zero and one. It therefore induces the same polynomial function as the zero polynomial, despite having different coefficients and polynomial degree. Over an infinite field, equality at every input forces equality of polynomials by the root bound for a polynomial. The distinction matters whenever a representation is supposed to identify the function uniquely.

A piecewise polynomial function agrees with a polynomial on each member of a partition. Approximation claims usually require finitely many pieces with controlled interfaces. In one dimension, finitely many partition points and localized wavelets with more vanishing moments than the polynomial degree leave only a bounded number of nonzero coefficients per scale, yielding exponential squared best N-term approximation error.

For integer $r\ge 1$, the truncated power is zero for $x\le t$ and equals $(x-t{)}^r$ for $x>t$. It is globally $C^{r-1}$ and its $r$th derivative has a jump at $t$. Order zero is the step-function convention $1_{\{x>t\}}$. Knot divided differences of these functions produce B-splines.