The Stone-Weierstrass theorem is an approximation theorem for continuous functions on closed intervals. It says that every continuous function on the interval can be approximated as accurately desired by a polynomial function. Polynomials are far easier to work with than continuous functions and allow mathematicians and computers to quickly get accurate approximations for more complex functions.
Specifically, let denote the ring of continuous functions . The Stone-Weierstrass theorem classifies certain subrings of that are dense in . In particular, it implies that any continuous function in may be arbitrarily well-approximated by polynomial functions. This corollary is known as the Weierstrass Approximation theorem.
Any infinitely differentiable (a.k.a smooth) function is arbitrarily well-approximated by its Taylor polynomials. That is, if denotes the degree Taylor polynomial, then for any , there exists such that .
Unfortunately, not all continuous functions are smooth (or even differentiable), so one cannot just always approximate using Taylor polynomials. Nonetheless, it is somewhat reasonable to assume that any continuous function might be arbitrarily well-approximated by smooth functions. If this is true, then continuous functions are well-approximated by polynomials, since the smooth functions are certainly well-approximated by polynomials.
To see why continuous functions should be well-approximated by smooth functions, suppose is an arbitrary continuous function. If is a smooth function, then is itself smooth, by differentiation under the integral sign. One can think of this as assigning infinitesimal weights to at each . The function is called the convolution of and , and is usually denoted . A particularly useful property of convolution is that, by a change of variables, one can see that .
Since is essentially a weighted version of , one could hope to find smooth functions that make very close to by spreading the weight of more and more evenly over the interval . This idea is made rigorous in the following proof:
Suppose is a smooth function that is positive on and zero elsewhere, which also satisfies This is a function with "total weight" 1, whose weight is entirely concentrated on the interval . Set each also has , but the weight becomes more evenly distributed over this interval as i.e., the graphs of become flatter
Choose . Since is a closed interval, is uniformly continuous on , so there exists such that whenever and .
Thus, we have implicitly using the fact that , and the triangle inequality Since is zero outside , this is really an integral over the interval , i.e. If we take large enough so that , then for all . We conclude for all .
It remains to construct . Define One can check is smooth, positive on , and zero elsewhere. By scaling and translating , we have constructed the desired .
The Weierstrass approximation theorem states that the polynomials are dense in . More generally, the Stone-Weierstrass theorem gives a classification of all the subrings of that are dense in .
A subring is called a subalgebra if, for any , the product is also in .
A subring is said to separate points if, for any distinct , , there is some such that .
(Stone-Weierstrass) Suppose is a subalgebra containing some nonzero constant function. Then, is dense in if and only if separates points.
The following example illustrates the power of Stone-Weierstrass, showing how to recover the Weierstrass approximation theorem from it.
Recover the Weierstrass Approximation theorem from the Stone-Weierstrass Theorem.
Let denote the subalgebra of polynomials. Certainly contains a nonzero constant function, since all constant functions are degree-zero polynomials.
To see that separates points, choose , . Then is a polynomial with Thus, we conclude is dense in .
An ideal in is a subring such that for any and , the product is in .
If is an ideal such that , then is called proper.
A maximal ideal is a proper ideal such that whenever is an ideal with , in fact . In other words, is maximal with respect to the partial order given on ideals of by set inclusion.
For , define Note that is an ideal in .
Answer the following yes-no questions:
- Is is a maximal ideal of for all ?
- Does there exist a maximal ideal of that is not of the form for some ?