Stone-Weierstrauss

[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