Chebyshev

Pafnuty Lvovich Chebyshev (also Tschebyschow) was a Russian mathematician born in 1821 in Okatovo and who died in 1894 in Saint Petersburg. He made foundational contributions across number theory, probability, and approximation theory, and several mathematical tools bear his name.

Chebyshev polynomials form a family of orthogonal polynomials named after him. The first kind, T_n(x), is defined by T_n(cos θ) = cos(nθ) with T_0 = 1, T_1 = x, and the recurrence T_{n+1}(x) = 2x T_n(x) − T_{n−1}(x). The second kind, U_n(x), satisfies U_n(cos θ) = sin((n+1)θ)/sin θ with U_0 = 1, U_1 = 2x and the recurrence U_{n+1}(x) = 2x U_n(x) − U_{n−1}(x). These polynomials are important in approximation theory, spectral methods, and numerical analysis, and they are defined on the interval [-1, 1].

Chebyshev distance, also called the L-infinity distance, between two vectors is defined as the maximum absolute difference among their coordinates: d∞(x, y) = max_i |x_i − y_i|. It is a standard metric in geometry, clustering, and computer science, influencing nearest-neighbor searches and robust geometric computations.

In probability, Chebyshev’s inequality states that for any random variable X with finite mean μ and finite variance σ^2, and any a > 0, 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 provides distribution-free bounds on fluctuations.

In number theory, Chebyshev established bounds for π(n), the prime-counting function, demonstrating that π(n) is of growth order n/log n and providing explicit constants. These results preceded the prime number theorem and influenced its eventual proof.

Chebyshev’s work thus spans polynomials, geometry, probability, and prime distribution, and his name remains attached to essential concepts across mathematics.