Approximation Theory
Approximation theory studies how well functions can be represented by simpler ones — polynomials, splines, trigonometric series, or rational functions — and constructs the approximants that achieve the best or near-best accuracy.
Definition
Approximation theory is the branch of numerical analysis concerned with representing functions by simpler classes of functions and with quantifying the error of such representations under various measures of best fit.
Scope
This area covers interpolation and best approximation, the convergence and error of polynomial and spline approximants, least-squares and minimax (Chebyshev) criteria, and the theoretical results — existence, uniqueness, and rates of convergence — that quantify how approximation error decreases as more degrees of freedom are added.
Sub-topics
Core questions
- How accurately can a given function be approximated by polynomials, splines, or rational functions of a given size?
- What approximant is optimal under a chosen error measure, such as least-squares or maximum (minimax) error?
- How does the smoothness of a function control the rate at which approximation error decreases?
- When does interpolation converge to the underlying function, and when does it fail?
Key theories
- Weierstrass approximation theorem
- Every continuous function on a closed bounded interval can be uniformly approximated as closely as desired by polynomials, establishing that polynomials are dense in the space of continuous functions and motivating constructive approximation methods.
- Best approximation and equioscillation
- The best minimax polynomial approximation of a continuous function exists, is unique, and is characterized by the Chebyshev equioscillation theorem, which states that the error attains its maximum magnitude with alternating sign at enough points.
- Smoothness and convergence rates
- The decay rate of approximation error is governed by the smoothness of the target function: analytic functions admit geometric convergence of polynomial approximants, while functions with limited derivatives converge only algebraically.
Clinical relevance
Approximation theory underpins the construction of accurate numerical methods throughout scientific computing: quadrature rules, spectral and finite-element bases, data fitting and smoothing, computer-aided geometric design, and the special-function and elementary-function routines built into numerical software all rest on results about how well and how cheaply functions can be approximated.
History
The subject grew from Chebyshev's nineteenth-century work on best uniform approximation and Weierstrass's density theorem, was advanced by the study of orthogonal polynomials and Fourier series, and was reshaped in the computer era by spline theory and by the practical Chebyshev-based methods popularized in modern numerical computing.
Key figures
- Pafnuty Chebyshev
- Karl Weierstrass
- Carl Runge
- Lloyd N. Trefethen
Related topics
Seminal works
- trefethen2013
- powell1981
- cheney1966
Frequently asked questions
- What is the difference between interpolation and best approximation?
- Interpolation forces the approximant to match the function exactly at chosen points, whereas best approximation minimizes an overall error measure (such as maximum or least-squares error) without necessarily matching at any point. A best approximant is usually more accurate overall but harder to construct.
- Why does using more interpolation points sometimes make things worse?
- High-degree polynomial interpolation at equally spaced points can oscillate wildly near the ends of the interval — the Runge phenomenon — so error can grow rather than shrink. Choosing Chebyshev-distributed points or using splines avoids this.