(adsbygoogle = window.adsbygoogle || []).push({}); Question

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

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

a subset of [itex]C(S,\mathbb{R})[/itex] 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 [itex][0,\pi] \times [0,1][/itex].

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 [itex]\sin[/itex] using just linear combinations of [itex]\cos[/itex].

However, lets just say that our subset was

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

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 [itex]C(S,\mathbb{R})[/itex].

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

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

In this case, this subset is NOT dense in [itex]C(S,\mathbb{R})[/itex] because some terms are missing, i.e. 5 and 2 are coprime, so some combinations, i.e. [itex]x^4[/itex] 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Hard question concerning denseness

**Physics Forums | Science Articles, Homework Help, Discussion**