Stone-Weierstrauss theorem

1. Jun 2, 2006

mplltt

This has something to do with all of physics and particularly equations used to solve problems with quantum physics.

I need to solve this using Stone-Weierstrauss theorem for {sin(nx)} (n=1 to infinity) over the interval (0,pi/2)

This involves the Fourier transform which would solve the series expansion starting with a0, am, & bm. This involves the use of a0=1/(2pi) INT(f(x))dx from (0,pi/2) and am=1/pi <cos(mx), f(x)> = 1/pi INT(f(x)*cos(mx))dx and bm=1/pi<sin(mx), f(x)> =1/pi INT(f(x)*sin(mx))dx. This is the dot product of the trig function and the function in L^2.

The Stone-Weierstrauss uses three main conditions:
1) All x,y are in [a,b] there exists n,s,t Phi(sub n)(x) does not equal Phi(sub n)(y)
2) Phi(sub n)(x)*Phi(sub m)(x) = sum of Gamma(sub n)*Phi(sub n)(x) =1
3) "closed under multiplication." all n,m exists {Gnu(sub j)^(n,m)} (j=0-infinity) such that Phi(sub m)(x)*Phi(sub n)(x) = Sum of Gnu(sub j)*Phi(sub j)(x)

Phi(sub j)(x) is in this case {sin(nx)}(n=1 to inifinity)

-M

2. Jun 3, 2006

HallsofIvy

Sorry but I can't make heads or tales of this.

3. Jun 3, 2006

He *might* be trying to say that he wants to prove that any function can be aproximated by a sum of the form

$$\sum a_n\sin (nx)$$

Using Stone-Weierstrass. Just a guess though.

4. Jun 3, 2006

masudr

I concur.

There appears to be no question or problem in the OP's post.

5. Jun 4, 2006

mplltt

some ways to help

The Stone-Weierstrauss uses three main conditions:
1) All x are in [a,b] there exists n,s,t Psi(sub n)(x) does not equal Psi(sub n)(y)
2) Psi(sub n)(x)*Psi(sub m)(x) = sum of Gamma(sub n)*Psi(sub n)(x) =1
3) "closed under multiplication." all n,m exists {Gnu(sub j)^(n,m)} (j=0-infinity) such that Psi(sub m)(x)*Psi(sub n)(x) = Sum of Gnu(sub j)*Psi(sub j)(x)

Psi(sub j)(x) is in this case {sin(nx)}(n=1 to inifinity)

So my question is I want to find Gamma as a coefficient to Psi=sin(nx) =1 for all n. [as per 2)]

The next question to this is that I would like to find a sufficient dot product of <Psi(nx), Psi(mx)> that will yield sines. [as per 3)]

-M

6. Jun 4, 2006

mathwonk

here is a relevant theorem that will only be intelligible to a somewhat sophisticated reader: A compactification of a (completely regular) space, corresponds to a subalgebra (condition 3)) of the algebra of continuous functions, which is constant containing, point separating (condition 1)) and closed under uniform limits.

this si not precise but neither is the question posed. this is a o convergenve theorem. poresuymably the compactification is the closure or one point closure of the given interval. the poster may not comprehend this, but it is relevant as someone may see.

7. Jun 5, 2006

mathwonk

i think you are right deadwolfe, as perhaps the stone weierstrass he is thikning of is a set of criteria (popint separating and constant containing) for an algebra of functions to be uniformly dense in the algebra of continuous functions on a given space.

this is a standard way to prove that polynomials and trig polynomials are dense in the continuous functions on a compact interval.

If the space involved is not compact, the discussion I was giving relates to constructing various compactifications from similar considerations.

Last edited: Jun 5, 2006
8. Jun 6, 2006

mathwonk

weierstrass, not strauss.