Approximation theory

Approximation theory studies how functions and other objects can be represented by simpler ones, and how the resulting error can be measured and controlled.

Normed vector space: A vector space equipped with a norm that assigns lengths to vectors. Approximation often takes place in spaces where a norm measures the distance between target and approximant.
Polynomial approximation: The approximation of functions by polynomials, often with error measured over a specified domain. Polynomials are a central simple family whose approximation error can be bounded and optimized.
Polynomial interpolation: The construction of a polynomial that matches a function at specified points. Interpolation turns sampled function values into a continuous approximant, though node choice controls its error.
Runge's phenomenon: The large oscillation near interval endpoints that can occur in polynomial interpolation at equally spaced nodes. It shows that adding interpolation points need not improve an approximant when the method is poorly conditioned.
Uniform convergence: Convergence in which the maximum error across an entire domain tends to zero. It captures approximation guarantees that hold simultaneously at every point of a domain.
Fourier series: A representation of a periodic function as a sum of sine and cosine waves. Fourier expansions approximate periodic functions using orthogonal frequency components.
Numerical quadrature: The approximation of definite integrals by weighted combinations of function values. Quadrature error is often derived by approximating the integrand with a tractable function.
Gibbs phenomenon: Persistent overshoot near a jump discontinuity in truncated Fourier representations. It demonstrates that convergence of an expansion need not eliminate local error near discontinuities.
Lp space: A function space whose elements have finite p-th-power integrals, with distance measured by an Lp norm. Lp norms provide common ways to measure average approximation error.
Spline: A piecewise polynomial function whose pieces meet under prescribed smoothness conditions. Splines approximate local behavior while avoiding the oscillations common in high-degree global polynomials.

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