quick math question...

[eLiu]

Is it possible to orthogonalize a "basis" of a function space? Like sin/cos (edit: in the fourier sense) form an orthogonal basis for the space of L^2 functions (i think)

i.e. with vector spaces, we can apply the gram-schmidt process (or whatever other orthogonalization algorithm you like) to turn n linearly independent vectors into n orthonormal vectors.

Is there a Gram-Schmidt analogue for functional bases?

sorry if this sounds stupid...i havent had any formal exposure to what I'm starting to call "continuous linear algebra"

 

[kogase]

Originally posted by: eLiu
sorry if this sounds stupid...

Understatement of the year! This is the most idiotic thing I've ever read. If I only could understand half the words I'm sure it would be that much stupider.

 

[eLiu]

Originally posted by: kogase

Originally posted by: eLiu
sorry if this sounds stupid...

Understatement of the year! This is the most idiotic thing I've ever read. If I only could understand half the words I'm sure it would be that much stupider.

 

[eLiu]

Well I finally figured this one out.

There is a generalized "Gram-Schmidt" process for orthogonalizing linearly independent functions. The process is basically the same in concept (but more complex in execution as now we need the L^2 inner product instead of the "standard" inner product) to the GS orthogonalization process for linearly independent vectors. Very cool.

 

[RaynorWolfcastle]

Originally posted by: eLiu
Is it possible to orthogonalize a "basis" of a function space? Like sin/cos (edit: in the fourier sense) form an orthogonal basis for the space of L^2 functions (i think)

i.e. with vector spaces, we can apply the gram-schmidt process (or whatever other orthogonalization algorithm you like) to turn n linearly independent vectors into n orthonormal vectors.

Is there a Gram-Schmidt analogue for functional bases?

sorry if this sounds stupid...i havent had any formal exposure to what I'm starting to call "continuous linear algebra"

I vaguely remember doing something like that to generate the Legendre polynomials, fortunately I haven't touched that in a couple years and will never have to do anything of the kind again

 

[eLiu]

Originally posted by: RaynorWolfcastle

Originally posted by: eLiu
Is it possible to orthogonalize a "basis" of a function space? Like sin/cos (edit: in the fourier sense) form an orthogonal basis for the space of L^2 functions (i think)

i.e. with vector spaces, we can apply the gram-schmidt process (or whatever other orthogonalization algorithm you like) to turn n linearly independent vectors into n orthonormal vectors.

Is there a Gram-Schmidt analogue for functional bases?

sorry if this sounds stupid...i havent had any formal exposure to what I'm starting to call "continuous linear algebra"

I vaguely remember doing something like that to generate the Legendre polynomials, fortunately I haven't touched that in a couple years and will never have to do anything of the kind again

Yeah there are loads of different ways to generate orthogonal polynomials...I think this process makes the 3rd or 4th one I'm aware of. woohoo for my dorkiness :/

But I guess legendre is the easiest since the associated weight function is just 1...instead of something nuts like sqrt(1-x^2).

 

[eigen]

God, I am happy I study discrete math and not analysis.

 

[Howard]

I did this before and I have no idea what you're talking about.

 

[eLiu]

Originally posted by: eigen
God, I am happy I study discrete math and not analysis.

oh it's not that bad...besides this is only like pseudo-analysis...im looking into this for some PDE stuff, not really for the sake of proving anything. I took a real analysis class...it was hard...I'm staying away.

 

[eigen]

Originally posted by: eLiu

Originally posted by: eigen
God, I am happy I study discrete math and not analysis.

oh it's not that bad...besides this is only like pseudo-analysis...im looking into this for some PDE stuff, not really for the sake of proving anything. I took a real analysis class...it was hard...I'm staying away.

Look dude If its not graphs and counting arguements I am lost.