Weierstrass Approximation Theorem

The Weierstrass approximation theorem assures us that polynomial approximation can get arbitrarily close to any continuous function as the polynomial order is increased.

Let be continuous on a real interval . Then for any , there exists an th-order polynomial , where depends on , such that

For a proof, see, *e.g.*, [62, pp. 146-148].

Thus, any continuous function can be approximated arbitrarily well by
means of a polynomial. This does not necessarily mean that a Taylor
series expansion can be used to find such a polynomial since, in
particular, the function must be differentiable of all orders
throughout . Furthermore, there can be points, even in infinitely
differentiable functions, about which a Taylor expansion will not
yield a good approximation, as illustrated in the next section. The
main point here is that, thanks to the Weierstrass approximation
theorem, we know that good polynomial approximations *exist* for
any continuous function.

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