Weierstrassapproximációs

The Weierstrass approximation theorem is a fundamental result in mathematical analysis. It states that for any continuous real-valued function defined on a closed interval, there exists a polynomial that can approximate the function arbitrarily closely. More formally, if f is a continuous function on the interval [a, b], then for any arbitrarily small positive number ε, there exists a polynomial P(x) such that |f(x) - P(x)| < ε for all x in [a, b].

This theorem is significant because it implies that continuous functions can be represented by polynomials. Polynomials are often much simpler to work with than general continuous functions, as they are easy to differentiate, integrate, and evaluate. The existence of such approximating polynomials means that many analytical problems involving continuous functions can be simplified by considering their polynomial approximations instead.

The proof of the Weierstrass approximation theorem typically involves constructing a sequence of polynomials that converge uniformly to the given continuous function. A common method of construction utilizes Bernstein polynomials. These polynomials are defined as B_n(f; x) = Σ_{k=0}^n f(k/n) * C(n, k) * x^k * (1-x)^(n-k), where C(n, k) is the binomial coefficient. As n approaches infinity, the Bernstein polynomials converge uniformly to f(x) on [0, 1]. The theorem can then be extended to any closed interval [a, b] by a simple linear transformation.