Hermite Polynomials
#4
(07/06/2014, 04:14 AM)mike3 Wrote: Yes, we need to find a condition for when the continuum sum will converge given a convergent \( f(z) \) Hermite series. And I suppose the bit at the beginning about the integral might need to be improved a little -- we could, e.g. show that it is always smaller than a known convergent integral to make the proof complete.

Hmm. Can we say that \( |f(x+yi)| < C_x e^{\alpha |y|} \) for \( 0 < \alpha < \pi/2 \)? for \( x \) belonging to the area we want to continuum sum. This will guarantee a converging continuum sum with a triple integral transfrom from FC. I have all this rigorously laid out. If Faulbaher's continuum sum takes \( s(s+1)(s+2)\cdots(s+n-1)\to \frac{1}{n}s(s+1)(s+2)\cdots(s+n) \) then My continuum sum may be the same as Faulbaher's. Now as for saying if the representation as continuum summed Hermite polynomials is convergent, I know some techniques from FC again that might work here. But they rely on the above and I'd have to take a closer look.

This is really quite interesting, I like this representation.

Are you going to show:

\( e^{\sum_{n=0}^{z-1} f(n)} = \frac{d}{dz} f(z) \)

Or do you have a different more convenient pattern for \( a_n \)



Messages In This Thread
Hermite Polynomials - by mike3 - 07/05/2014, 06:07 AM
RE: Hermite Polynomials - by JmsNxn - 07/05/2014, 02:11 PM
RE: Hermite Polynomials - by mike3 - 07/06/2014, 04:14 AM
RE: Hermite Polynomials - by JmsNxn - 07/06/2014, 11:03 AM
RE: Hermite Polynomials - by mike3 - 07/06/2014, 11:04 PM
RE: Hermite Polynomials - by JmsNxn - 07/07/2014, 12:39 PM
RE: Hermite Polynomials - by mike3 - 07/08/2014, 06:31 AM
RE: Hermite Polynomials - by fivexthethird - 07/06/2014, 02:11 PM
RE: Hermite Polynomials - by tommy1729 - 07/08/2014, 12:24 PM

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