What is the local analog of the Fourier Transform ?

[UndertheOcean]

Battling a hangover right now from yesterdays 6 beers with colleagues This question comes screaming into my mind :

We know that all periodic functions can be projected into the infinite dimensional but countable space with a basis of exp(i*w*t)

This is called the Fourier Series.

All L2 functions (non-periodic but finite energy so to speak) on the other hand can only be projected into an infinitely DENSE and infinite dimensional basis ==> this is the Fourier Transform.

Now let us switch to a local perspective :

In my mind I see the Taylor Series as an Analog to the Fourier series , only local.


What I can't figure out is what is the Analog of the Fourier Transform Locally.

BTW , I am no mathematician , and all this is very non rigorous , just thinking out loud , so geeks in the house , what are your 2 cents ?

[CBGLuthier]

1. Is this all part of the Random Thoughts II audition process?

:-)

[Fumesucker]

2. Bwaaah..

:rofl:

Although this actually sound more like a Boojatta thread..

[SpiralHawk]

3. Okay, okay, okay - your beer and travel money are on the way

but as for the 'various experiences,' we'll just have to check the synchronastic infundibillium.

[muriel_volestrangler]

4. I think you need to explain your usage of 'local', first (nt)

[UndertheOcean]

6. My usage is kind of a non rigorous geometric imagining of the thing

The Taylor Series converges locally first , meaning the farther we are from the point where we are deriving the series the more terms are needed.

All differential equations are solved locally.

I guess it has something to do with infinite differentiation and smoothness.


While to my mind all spectrum seems to converge more uniformly , for example the Fourier series and the sequence of reimann sums of the fourier transform of a function with ever increasing terms.

[muriel_volestrangler]

12. Beyond me, then

You would seem to need 'fractional differentiation' - not the first differential, or the second, but the 'one-and-a-halfth', and many (all?) points in between. Which doesn't seem like a valid concept to me, but neither do various bits of higher mathematics that are beyond my comprehension. You've lost me with 'reimann sums', too.

[bemildred]

11. And "analog". nt

[PCIntern]

5. Use of tensor calculus may solve your di-lemma.

Triple integrate the transform, apply the mean-value theorem to the result.

Take two more sips and call it a day...

[UndertheOcean]

7. Problem here , I have to learn Tensor Calculus , hey , hope you are not sending me on a wild goose

chase !!!!!

[PCIntern]

8. go right ahead and get back to me today...

:hi:

[Tyrs WolfDaemon]

9. It is obvious from your post that you want to know why...

Republicans believe that 1+1=3 if they were to see that in the bible.
I feel your pain in not understanding this fact.
If you do find a solution to this, then please let me know.










* I wish I could actually help, but it has been ages since I took higher level mathematics.

[bemildred]

10. 2c

So, what would be the "non-local" "analog" of the Taylor Series, i.e. the "Taylor Transform"? That would then seem to be related to Taylor Series in a way analogous to how the Fourier Series is related to the Fourier Transform.

It's been a very long time, but I do remember generalization of the Taylor Series into function spaces, and that is the direction I would start looking in.

YMMV.