TschebyscheffSkalarsierung

Chebyshev, in the transliteration Tschebyscheff, is the surname of the 19th‑century Russian mathematician Pafnuty Lvovich Chebyshev (1821–1894) and of several concepts in mathematics named after him. The best known are Chebyshev polynomials, the Chebyshev distance, and Chebyshev's inequality. The name is used in many languages with slight spelling variants.

Chebyshev polynomials: There are two families commonly called Chebyshev polynomials of the first and second kind. The first kind T_n(x) is defined by T_0(x)=1, T_1(x)=x, and T_{n+1}(x)=2x T_n(x)−T_{n−1}(x). They satisfy T_n(cos θ)=cos(nθ) and are orthogonal on [-1,1] with respect to the weight (1−x^2)^{-1/2}. The second kind U_n(x) obey U_0=1, U_1=2x, and a similar recurrence; they appear in Fourier-type expansions and approximation problems. Both families minimize deviation in certain norms and underpin spectral methods and polynomial interpolation, notably through Chebyshev nodes that reduce Runge’s phenomenon.

Chebyshev distance: Also called the L∞ norm, this measures the greatest coordinatewise difference: d∞(x,y)=max_i |x_i−y_i|. It is used in robust optimization, clustering, and error analysis, and contrasts with Euclidean distance.

Chebyshev’s inequality: A foundational probability bound states that if X has finite mean μ and variance σ^2, then for any t>0, P(|X−μ|≥t) ≤ σ^2/t^2. It gives a nonparametric gauge of dispersion and underpins limit theorems and distribution-free confidence estimates.

The term Chebyshev remains a touchstone in analysis, numerical methods, and applied mathematics, reflecting Chebyshev’s broad influence.