Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Stone-Weierstrauss theorem

  1. Jun 2, 2006 #1
    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. jcsd
  3. Jun 3, 2006 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Sorry but I can't make heads or tales of this.
     
  4. Jun 3, 2006 #3
    He *might* be trying to say that he wants to prove that any function can be aproximated by a sum of the form

    [tex]\sum a_n\sin (nx) [/tex]

    Using Stone-Weierstrass. Just a guess though.
     
  5. Jun 3, 2006 #4
    I concur.

    There appears to be no question or problem in the OP's post.
     
  6. Jun 4, 2006 #5
    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
     
  7. Jun 4, 2006 #6

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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.
     
  8. Jun 5, 2006 #7

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    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
  9. Jun 6, 2006 #8

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper
    2015 Award

    weierstrass, not strauss.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Stone-Weierstrauss theorem
  1. Is this theorem true? (Replies: 4)

  2. Torricelli's theorem (Replies: 1)

  3. Louiville theorem (Replies: 1)

  4. Divergence Theorem (Replies: 15)

  5. Uniqueness Theorem (Replies: 1)

Loading...