Weierstrass approximation theorem

Every continuous real-valued function on a closed bounded interval can be approximated uniformly, to any desired accuracy, by polynomials.

Bernstein polynomials: Polynomial approximants formed from sampled values of a function and binomial weights on a closed interval. Their convergence gives a constructive proof of the theorem on [0,1].
Continuous function: A function whose values vary continuously with its input, so sufficiently small input changes produce arbitrarily small output changes. Continuity is the sole regularity assumption in the theorem.
Karl Weierstrass: A German mathematician whose work helped establish rigorous analysis and the modern theory of functions. His 1885 proof established polynomial approximation for continuous functions on an interval.
Chebyshev polynomials: A sequence of orthogonal polynomials closely connected to cosine functions and optimal uniform approximation. They provide efficient polynomial approximants with small maximum error.
Bernstein's theorem: The result that Bernstein polynomials of a continuous function converge uniformly to that function on [0,1]. It supplies the theorem’s probabilistic proof through concentration of binomial distributions.
Compact space: A topological space in which every open cover has a finite subcover. A closed bounded interval is compact, enabling uniform control of continuous functions.
Marshall H. Stone: An American mathematician known for major contributions to functional analysis, topology, and logic. His generalization extended polynomial approximation to algebras of functions on compact spaces.
Polynomial interpolation: The construction of a polynomial matching prescribed function values at a finite set of points. The theorem promises uniform approximation, while interpolation alone need not converge uniformly.
Polynomial: An expression built from variables and coefficients using finitely many additions and multiplications. These are the approximating functions guaranteed by the theorem.
Heine–Cantor theorem: The theorem that every continuous function on a compact metric space is uniformly continuous. It turns interval continuity into the uniform control needed by approximation proofs.

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