Muntz-Szasz Theorem

The Muntz-Szasz theorem, a basic result of approximation theory proved by Herman Muntz in 1914 and Otto Szasz in 1916, shows the extent to which the Weierstrass theorem on polynomial approximation can be extended by restricting certain coefficients of the polynomials to be zero. In a special case, the theorem states that the monomials with exponents drawn from a set S of natural numbers span a dense subset of the continuous functions on a closed interval [a, b] with a greater than zero if and only if the sum of the reciprocals of the elements of S diverges, a condition that also applies, with an adjustment for the constant term, on intervals starting at zero, and the result extends to exponents drawn from any strictly increasing sequence of positive real numbers. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/