Tschebyschow

Tschebyschow, more commonly spelled Chebyshev in English, refers to Pafnuty Lvovich Chebyshev, a Russian mathematician who lived from 1821 to 1894. He was a leading figure in 19th‑century mathematics, making foundational contributions to probability theory, numerical analysis, and approximation theory. Chebyshev helped establish rigorous methods in analysis and influenced the development of the Russian mathematical school.

Chebyshev polynomials, denoted T_n(x), were introduced by his work. They are defined by T_0(x)=1, T_1(x)=x and T_{n+1}(x)=2xT_n(x)−T_{n−1}(x). Satisfying T_n(cos θ)=cos(nθ), they form a family of polynomials that are especially suited for minimizing the maximum error in approximations on the interval [-1,1]. They are central to minimax approximation, interpolation, and spectral methods.

Chebyshev's inequality is a fundamental result in probability theory. It states that for any random variable X with finite mean μ and finite variance σ^2, the probability that X deviates from μ by at least a is at most σ^2/a^2: P(|X−μ|≥a) ≤ σ^2/a^2. This bound holds regardless of the underlying distribution and is widely used in statistics and analysis.

Other concepts bearing his name include the Chebyshev distance, defined as the maximum coordinate difference between two vectors, used in geometry and optimization, and Chebyshev systems (also called Tchebyshev systems) in approximation theory, which characterize families of functions with optimal interpolation properties.

Today, Chebyshev's work remains influential in numerical analysis, approximation theory, and probability. The German-language form Tschebyschow appears in some sources, while English references typically use Chebyshev. Numerous theorems, polynomials, and distance notions carry his name.