Polynomial interpolation answers a basic reconstruction question: if a function is known only at finitely many points, when can those values be matched by a single [polynomial](/page/Polynomial), and what does that polynomial tell us about the original function? The question is algebraic because finitely many coefficients must satisfy finitely many equations, but it is also analytic because the resulting polynomial may be used to approximate a function between the data points. This dual nature is why interpolation appears early in [numerical analysis](/page/Numerical%20Analysis), [approximation theory](/page/Approximation%20Theory), and the study of Taylor polynomial approximations.

The central tension is that exact agreement at selected points does not automatically mean good approximation elsewhere. A polynomial may pass through every measured value and still oscillate badly between nodes. Polynomial interpolation is therefore not just the act of fitting a curve; it is the study of which data determine a polynomial, how that polynomial can be represented, and how its error depends on the underlying function and the placement of the nodes.

## Definition

The page topic is not merely a formula for drawing a curve through points. Polynomial interpolation is the finite-dimensional reconstruction problem in which point samples are treated as exact constraints and the unknown object is restricted to a polynomial space of matching dimension. Here $\mathbb{R}[x]$ denotes the ring, or equivalently the [vector space](/page/Vector%20Space), of polynomials in one variable $x$ with real coefficients.

[definition: Polynomial Interpolation] Let $n \in \mathbb{N}$, let $x_0, x_1, \ldots, x_n \in \mathbb{R}$ be distinct, and let $y_0, y_1, \ldots, y_n \in \mathbb{R}$. Polynomial interpolation at the nodes $x_0, x_1, \ldots, x_n$ is the construction of a polynomial $p \in \mathbb{R}[x]$ such that $\deg p \le n$ and \begin{align*} p(x_i) = y_i, \qquad i=0,1,\ldots,n. \end{align*} [/definition]

The input data for interpolation must specify where the function values are sampled. These sampling locations need to be separated, because asking for two different function values at the same point is not an ordinary pointwise interpolation problem.

[definition: Interpolation Nodes] Let $n \in \mathbb{N}$. Interpolation nodes are distinct points $x_0, x_1, \ldots, x_n \in \mathbb{R}$. [/definition]

The distinctness condition is not cosmetic. If the same node is listed twice while only function values are prescribed, the data may demand two incompatible values at one input.

[example: Repeated Node Failure] Suppose a polynomial $p \in \mathbb{R}[x]$ is required to satisfy both $p(0)=1$ and $p(0)=2$. Since a polynomial defines a function $\mathbb{R}\to\mathbb{R}$, evaluating $p$ at the single input $0$ produces one real number. Thus the two requirements force the equality \begin{align*} 1=p(0)=2. \end{align*} Subtracting $1$ from both sides gives \begin{align*} 0=1, \end{align*} which is impossible in $\mathbb{R}$. Therefore no such polynomial exists. The obstruction is not special to polynomials: the data assign two different values to the same node, so they do not define ordinary pointwise interpolation data. If repeated points are intended to encode derivative conditions, the appropriate framework is Hermite interpolation rather than ordinary polynomial interpolation. [/example]

Given nodes, the basic problem is to replace a list of data values by one polynomial of controlled degree. The degree bound is part of the definition: without it, many polynomials can be forced through the same finite data by adding a polynomial that vanishes at every node.

[definition: Polynomial Interpolation Problem] Let $x_0, x_1, \ldots, x_n \in \mathbb{R}$ be interpolation nodes, and let $y_0, y_1, \ldots, y_n \in \mathbb{R}$. The polynomial interpolation problem is the problem of finding a polynomial $p \in \mathbb{R}[x]$ such that $\deg p \le n$ and \begin{align*} p(x_i) = y_i, \qquad i = 0,1,\ldots,n. \end{align*} [/definition]

When the data values come from an actual function, interpolation should preserve the distinction between the sampled function and the polynomial constructed from the samples. This matters because the polynomial may be used as a proxy away from the nodes, where it need not equal the original function.

[definition: Interpolating Polynomial] Let $x_0, x_1, \ldots, x_n \in \mathbb{R}$ be interpolation nodes, and let $f: A \to \mathbb{R}$ be a function with $\{x_0, x_1, \ldots, x_n\} \subset A \subset \mathbb{R}$. An interpolating polynomial for $f$ at the nodes $x_0, x_1, \ldots, x_n$ is a polynomial $p \in \mathbb{R}[x]$ such that $\deg p \le n$ and \begin{align*} p(x_i) = f(x_i), \qquad i = 0,1,\ldots,n. \end{align*} [/definition]

Many applications fix the same nodes and vary the sampled function. Quadrature rules, finite element constructions, and spectral collocation methods all need a reusable operation that turns function values into a polynomial. Packaging interpolation as an operator records this reusable construction and makes algebraic questions, such as linearity, meaningful.

[definition: Interpolation Operator] Let $x_0, x_1, \ldots, x_n \in \mathbb{R}$ be interpolation nodes, and let $A \subset \mathbb{R}$ contain all the nodes. The interpolation operator associated to these nodes is the map $I_n: \{f: A \to \mathbb{R}\} \to \mathbb{R}_n[x]$ where $\mathbb{R}_n[x] = \{p \in \mathbb{R}[x] : \deg p \le n\}$ and $I_n f$ is the interpolating polynomial for $f$ at $x_0, x_1, \ldots, x_n$. [/definition]

The definition of $I_n$ depends on the fact that the interpolating polynomial exists and is unique. That result is stated below in the properties section; the definition records the intended object once existence and uniqueness have been established.

## Equivalent Characterisations

An explicit interpolation formula should avoid solving a fresh system of equations every time the data values change. The natural strategy is to build polynomials that act like coordinate vectors with respect to evaluation at the nodes: one basis polynomial detects one node and vanishes at all the others.

[definition: Lagrange Basis Polynomial] Let $x_0, x_1, \ldots, x_n \in \mathbb{R}$ be interpolation nodes. For each $i \in \{0,1,\ldots,n\}$, the $i$-th Lagrange basis polynomial is the polynomial \begin{align*} \ell_i(x) = \prod_{0 \le j \le n,\ j \ne i} \frac{x - x_j}{x_i - x_j}. \end{align*} [/definition]

The Lagrange basis is designed so that data values become coefficients. This gives a direct formula for the interpolant and makes the dependence on the prescribed values transparent. The resulting theorem is the standard closed-form solution of the interpolation problem.