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Fourier transform of a tetration function - Ivars - 08/30/2009
Hi,
I was wondering.... If all the various forms of tetration that grow faster than exponential at as x-> infinity are Fourier transformed, they should have an extra high frequency peak(s) representing that extra speed as Fourief multiplier can not damp it at same value.
1) Would it be than possible to order all various tetrations ( finite right, left, mixed, infinite versions) by their Fourier spectra or even peak frequencies, making classification easier?
2) Since a pulse is usually depicted as Heaviside function, or delta function, with infinite growth speed of front, and, since it is known that no discontinuity can propagate faster than light, for modeling of real life pulses moving via systems could we use tetration approximations to the front? Such front can have any speed > c ( since it will represent phase or group speed) and, for a given (even linear) system, one could perhaps create a tetration ( or higher) front that propagates with any given speed?
3) Can we have a mathematical pulse with backward leaning front where top of pulse enters system before the bottom- so to say, the signal front is negative?
Ivars
RE: Fourier transform of a tetration function - Ivars - 12/17/2009
It seems to me, that
Sinusoid is only a small angle approximation of any pendullum, top , elastica ( which are all governed by the same equations) . So sinusoidal, Fourier representation of phenomena lacks most of its real properties at high power, high amplitude oscillations.
If the signal pulse front is fast increasing so that its NOT Fourier decomposable in linear superposition of infinite range monohromatic modes ( that is , increasing faster than exponential , e.g. like t^t instead of e^t) than such signal is NOT decomposable in linear superposition of Fourier modes and hence, dispersion relation of such physical phenomena (energy/impulse) may be totally different than usual linear one. And quite rich variety of them.
That is where I think the usefulness of tetration etc is, to study such ultra fast physical phenomena.
I still do not know what differential equation has a solution f(t) = t^t?
Best regards,
Ivars
RE: Fourier transform of a tetration function - bo198214 - 12/17/2009
(12/17/2009, 08:33 AM)Ivars Wrote: I still do not know what differential equation has a solution f(t) = t^t?
\( f'(x)=f(x)(\ln(x)+1) \)
RE: Fourier transform of a tetration function - tommy1729 - 12/17/2009
(08/30/2009, 10:19 PM)Ivars Wrote: Hi,
(snip)
2) Since a pulse is usually depicted as Heaviside function, or delta function, with infinite growth speed of front, and, since it is known that no discontinuity can propagate faster than light, for modeling of real life pulses moving via systems could we use tetration approximations to the front? Such front can have any speed > c ( since it will represent phase or group speed) and, for a given (even linear) system, one could perhaps create a tetration ( or higher) front that propagates with any given speed?
Ivars
forgive me for saying this , but the above is nonsense.
what does lightspeed has to do with tetration or fourier transform ?
that is a rhetoric question , dont answer it.
regards
tommy1729
RE: Fourier transform of a tetration function - Ivars - 12/18/2009
(12/17/2009, 11:45 PM)tommy1729 Wrote: what does lightspeed has to do with tetration or fourier transform ?
Shorter: What is the Fourier transform of t^t or other hyperfast operations?
Does there exist set of complex periodic functions (not necesserily e^+-Iwt) whose linear summation (integration) gives t^t and faster hyperoperations?
Mechanisms that happen faster than exponential growth could well be employed in various physical mechanisms, like phase transform.
@Bo: Thank You!
Ivars
RE: Fourier transform of a tetration function - Ivars - 01/11/2010
Really,pls explain what is wrong with what I am asking in the last post?
I may be able then to reformulate the question.
Ivars
RE: Fourier transform of a tetration function - bo198214 - 01/11/2010
I think there is no expert of fourier transforms on the forum to answer your question.
Also it is not clear how this would help with tetration.
RE: Fourier transform of a tetration function - Ivars - 01/12/2010
(01/11/2010, 01:23 PM)bo198214 Wrote: I think there is no expert of fourier transforms on the forum to answer your question.
Also it is not clear how this would help with tetration.
Truly, I did not know I have asked something complicated. I am Just still curios about potential possibilities to use tetration for physical phase transition or other processes.e.g. How such superfast signal would pass through e.g. linear system?
RE: Fourier transform of a tetration function - tommy1729 - 01/13/2010
(01/12/2010, 06:57 PM)Ivars Wrote:(01/11/2010, 01:23 PM)bo198214 Wrote: I think there is no expert of fourier transforms on the forum to answer your question.
Also it is not clear how this would help with tetration.
Truly, I did not know I have asked something complicated. I am Just still curios about potential possibilities to use tetration for physical phase transition or other processes.e.g. How such superfast signal would pass through e.g. linear system?
im not sure , but maybe you would learn from 'particle accelerator theory' or whatever that is called.
your on the wrong forum , go to a physics forum !!
just kidding , your welcome here
regards
tommy1729