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topology.continuous_function.weierstrass

The Weierstrass approximation theorem for continuous functions on [a,b] #

We've already proved the Weierstrass approximation theorem in the sense that we've shown that the Bernstein approximations to a continuous function on [0,1] converge uniformly.

Here we rephrase this more abstractly as polynomial_functions_closure_eq_top' : (polynomial_functions I).topological_closure = ⊤ and then, by precomposing with suitable affine functions, polynomial_functions_closure_eq_top : (polynomial_functions (set.Icc a b)).topological_closure = ⊤

The special case of the Weierstrass approximation theorem for the interval [0,1]. This is just a matter of unravelling definitions and using the Bernstein approximations.

The Weierstrass approximation theorem: polynomials functions on [a, b] ⊆ ℝ are dense in C([a,b],ℝ)

(While we could deduce this as an application of the Stone-Weierstrass theorem, our proof of that relies on the fact that abs is in the closure of polynomials on [-M, M], so we may as well get this done first.)

An alternative statement of Weierstrass' theorem.

Every real-valued continuous function on [a,b] is a uniform limit of polynomials.

theorem exists_polynomial_near_continuous_map (a b : ) (f : C((set.Icc a b), )) (ε : ) (pos : 0 < ε) :

An alternative statement of Weierstrass' theorem, for those who like their epsilons.

Every real-valued continuous function on [a,b] is within any ε > 0 of some polynomial.

theorem exists_polynomial_near_of_continuous_on (a b : ) (f : ) (c : continuous_on f (set.Icc a b)) (ε : ) (pos : 0 < ε) :
∃ (p : polynomial ), ∀ (x : ), x set.Icc a babs (polynomial.eval x p - f x) < ε

Another alternative statement of Weierstrass's theorem, for those who like epsilons, but not bundled continuous functions.

Every real-valued function ℝ → ℝ which is continuous on [a,b] can be approximated to within any ε > 0 on [a,b] by some polynomial.