Chebyshevapproksimaatioiden

Chebyshev approximation, often referred to as minimax approximation, is the problem of approximating a function on a closed interval by a polynomial in such a way that the maximum error is minimized. For a continuous function f on a compact interval I, the goal is to find a polynomial p of degree at most n that minimizes the sup norm of the difference |f(x) - p(x)| over I. The best uniform approximation exists and is unique.

A central tool in Chebyshev approximation is the Chebyshev polynomials of the first kind, T_n(x). These polynomials satisfy T_n(cos θ) = cos(nθ) and form an orthogonal basis on [-1,1] with respect to the weight (1 - x^2)^{-1/2}. They enable efficient construction and analysis of polynomial approximants and are used to develop near-optimal approximations and spectral methods.

The equioscillation or alternation theorem characterizes the best uniform approximation: a polynomial p_n is the best approximation to f on [-1,1] if and only if the error f(x) - p_n(x) attains its maximum absolute value at least n+2 times, with alternating signs. This property leads to practical criteria for determining minimax polynomials, often via the Remez algorithm.

Chebyshev nodes, x_k = cos((2k-1)π/(2n)) for k = 1,...,n, are the interpolation points that minimize Runge-type errors and yield near-minimax behavior for polynomial interpolation on [-1,1]. Chebyshev series express f as a sum of Chebyshev polynomials, providing efficient, spectrally accurate representations; coefficients can be computed via discrete cosine transforms and evaluated with stable recurrences like Clenshaw–Curtis.

Applications span numerical analysis, spectral methods for differential equations, data fitting, and signal processing. The approach is named after Pafnuty Chebyshev, reflecting its long-standing role in approximation theory.