Interpolation and approximation

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Interpolation and approximation

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[QUICK DESCRIPTION]
Interpolation is the task of finding a function $p$ from a certain class that matches given data $(x_1,y_i)$, $i=1,2,\ldots,N$.  That is,
[maths] p(x_i)=y_i,\qquad i=1,2,\ldots,N. [/maths]
Typically, $p$ is taken from the set of polynomials of degree $\leq d$.

[[Approximation front page|Approximation]] is the task, given a function $f:A\to X$, where $X$ is a suitable space, to find a function $p:A\to X$ taken from a certain class that is in some sense close to $f$.  Often we consider, for example, polynomial interpolation where $A$ is a bounded subset of $\R^d$ and $X=\R$.

Different measures of closeness can be used for determining how close functions are.  The most common are:
[maths]\begin{align}
\|f-p\|_\infty&=\sup_{a\in A}\|f(a)-p(a)\|,\\
\|f-p\|_2     &=\left[\int_A \|f(a)-p(a)\|^2\,da\right]^{1/2}.
\end{align}[/maths]
In [[Polynomial approximation front page|polynomial approximation]] $p$ is required to be a polynomial of degree $\leq d$ for some fixed $d$.
Often approximations can be found by interpolation: the data used is simply $(x_i,f(x_i))$, $i=1,2,\ldots,N$ for suitably chosen points $x_i$.

[PREREQUISITES]
Calculus, real analysis.

[EXAMPLE]
Polynomial interpolation in an interval with data $(x_i,y_i)$, $i=1,2,\ldots,N$ can be done with polynomials of degree $\leq d$ provided that $N\geq d+1$ and all $x_i$'s are distinct.  The interpolant is unique (amongst all polynomials of degree $\leq d$) provided in addition, $N=d+1$.  There are a number of different ways of computing the polynomial interpolant for $N=d+1$, including solving linear systems with Van der Monde matrices, Lagrange interpolation polynomials, and Newton divided differences.

If the data for polynomial interpolation comes from a smooth function $f:[a,b]\to\R$, then there is a commonly used formula for the error in the interpolant:
[maths] f(x)-p(x) = \frac{f^{(d+1)}(c)}{(d+1)!}\prod_{i=1}^N(x-x_i), [/maths]
for some $c$ between $\min(x,x_1,\ldots,x_N)$ and $\max(x,x_1,\ldots,x_N)$.

[GENERAL DISCUSSION]

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