Is the Set of Cosine and Polynomial Functions Dense in C(S, ℝ)?

[Oxymoron]

Question

Let $S = \{(x,y) \in \mathbb{R}^2\,:\,x\in[0,\pi],\,y\in[0,1]\}$. Deduce whether or not,

$$\left\{\sum_{m,n=0}^M a_{m,n}\cos(mx)y^n\,:\,a_{m,n} \in \mathbb{R}\right\}$$

a subset of $C(S,\mathbb{R})$ is dense.


I was thinking no. And this is not a guess.

My reasoning is as follows. S can be thought of as some 'surface' whose projection onto the real plane is bounded by the rectangle $[0,\pi] \times [0,1]$.

So, I was led to believe this is kind of like a Fourier analysis problem. Can I construct every possible surface, hence making it dense, using my subset. Well, obviously no, because, for example, I cannot approximate $\sin$ using just linear combinations of $\cos$.

However, let's just say that our subset was

$$\left\{\sum_{m=0}^M\sum_{n=0}^N\,a_{m,n}x^my^n\,:\,a_{m,n} \in \mathbb{R}\right\}$$

Then I can approximate, by sums, every possible 'surface', or polynomial using the given subset. So I would say that this particular subset IS dense in $C(S,\mathbb{R})$.

Let's change it a little more. Consider the subset

$$\left\{\sum_{m=0}^M\sum_{n=0}^N\,a_{m,n}x^{5m}y^{2n}\,:\,a_{m,n} \in \mathbb{R}\right\}$$

In this case, this subset is NOT dense in $C(S,\mathbb{R})$ because some terms are missing, i.e. 5 and 2 are coprime, so some combinations, i.e. $x^4$ can never be made. That is, this polynomial cannot approximate, by sums, every possible surface within the bounded region. Hence not dense.


I know this is pretty complicated. But if anyone has the guts I would appreciate some feedback.

 

[AKG]

Well you have more than just linear combinations of cos. You can linear combinations of cos(mx). cos(2x) = cos²x - sin²x. It's got a 'sin' in there. Of course, it isn't sin itself but can you be sure that no linear combination of cos(mx)'s with m varying will not give you something close to sin? Also, what is M?

For me, it seems like it might be tough to come up with a specific function in C(S,R) and show that none of the functions in your set approximate it. Maybe that's what you'll have to do. To show it's not dense, you want to show that:

There exists f in C(S,R) such that there is an e > 0 such that for all g in your set, there is some x in S such that |f(x) - g(x)| > e.

If you want to prove it dense:

For each f in C(S,R) and for each e > 0, there is a g in your set such that for all x in S, |f(x) - g(x)| < e.

Well how about this: I think I remember in my QM class saying that you could approximate any single-valued continuous function with the sum of a bunch of cos curves. And you can do any single-valued function with a polynomial. So maybe we should look for functions in C(S,R) that aren't of the form h(x)j(y). Try cos(xy). To be honest, though, I don't really know how to do this question.