Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**CaptainBlack** Your

is

under my assumption about the unspecified index set for the

s

I don't understand that. The OP's is given by

My f is given by

They are orthogonal to each other.

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Opalg** I don't understand that. The OP's

is given by

My f is given by

They are orthogonal to each other.

I'm lost, is in but arn't you looking for a function not expressible in terms of these functions?

CB

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**CaptainBlack** I'm lost,

is in

but arn't you looking for a

function not expressible in terms of these functions?

Sorry to keep harping on about this. There was a typo (now corrected) in my previous comment, which made nonsense of it. What I should have said is

The difference is that has a 1 in its definition where has an x.

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Opalg** Sorry to keep harping on about this. There was a typo (now corrected) in my previous comment, which made nonsense of it. What I should have said is

The difference is that

has a 1 in its definition where

has an x.

OK that makes sense, I still prefer , which is obviously not in the space spanned by the (though not orthogonal to ) and does not require any great though to take the FT of.

CB

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Opalg** For (b), I would start by taking f to be a function in

that is orthogonal to all the functions

. For example,

Parseval's theorem then tells you that

is orthogonal to all the functions

.

Parseval's Theorem: An orthonormal sequence {xn} in H is complete iff:

llxll^2 = (sum n=1 to inf) l(x,xn)l^2.. for every x in H.

Using your premise above, how does Parseval's theorem tell you is incomplete? ie, how do you show " is orthogonal to all the functions "

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Hartlw** Parseval's Theorem: An orthonormal sequence {xn} in H is complete iff:

llxll^2 = (sum n=1 to inf) l(x,xn)l^2.. for every x in H.

Using your premise above, how does Parseval's theorem tell you

is incomplete? ie, how do you show "

is orthogonal to all the functions

"

There is more than one result that carries Parseval's name. The result that you state as Parseval's Theorem is what I call Parseval's Identity. For me, Parseval's Theorem is the result which says that if f, g are functions in with Fourier transforms then

It follows that if you have a (nonzero) function such that for all n, then for all n.

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Opalg** There is more than one result that carries Parseval's name. The result that you state as Parseval's Theorem is what I call Parseval's Identity. For me, Parseval's Theorem is the result which says that if f, g are functions in

with Fourier transforms

then

It follows that if you have a (nonzero) function

such that

for all n, then

for all n.

That wraps it up. Thanks, very nice.

I note belatedly that f is given by you in post #3, which also invokes Parseval's theorem.

The theorem that if ( , ) = 0 for all n and some not zero implies is incomplete, might help the novice.

Finally, I found many versions of Parsevals theorem, with various (formula, theorem, identity) interchanged names, including for Fourier Series and one that states the Fourier transform is bijective, but none with your version. Could you please give a source for your version, preferably internet?

Re: Fourier transform and orthogonal system

Quote:

Originally Posted by

**Hartlw** That wraps it up. Thanks, very nice.

I note belatedly that f is given by you in post #3, which also invokes Parseval's theorem.

The theorem that if (

,

) = 0 for all n and some

not zero implies

is incomplete, might help the novice.

Finally, I found many versions of Parsevals theorem, with various (formula, theorem, identity) interchanged names, including for Fourier Series and one that states the Fourier transform is bijective, but none with your version. Could you please give a source for your version, preferably internet?

Parseval's theorem - Wikipedia, the free encyclopedia (see especially the section headed *Equivalence of the norm and inner product forms*.