transform pair

[AntonioLao]

Time can be controlled but not the same ways control freaks want to control space. Anyway, most don’t believe in controlling seemingly chaotic-uniform temporal motion. But controlling static or dynamic space is another story. What they really wanted to control is spatial frequency: being here, there, and everywhere but often ended up being nowhere. If someone is bored, time seems to pass away very, very slowly. If someone is busy, time seems to pass away very, very quickly. Between boredom and business the chaotic movement: slow-fast-slower-fast-faster-slowest –fastest-slow-slow is definitely unpredictable, as unpredictable as those truly natural random processes. Unfortunately, this unpredictability is subjective. But fortunately, objectively time is controlled by its association as a transform pair with spatial frequency.

Using the mathematic of Fourier analysis this branch of control theory made possible the science of communications. The truth constituting a Fourier transform pair is that as the spatial frequency domain approaches zero the time domain approaches infinity, vice versa. However, since the zero-infinity domain is an open set: 0 < t < infinity or 0 < frequency < infinity and the product of time and frequency is a constant (inversely proportional), it is equivalent to an uncertainty principle such that wherever and whenever this uncertainty principle is applied to the spectral analysis of signal processing the product of the spectral bandwidth and the time duration of a signal cannot be less than a specified minimum value. The smallest value attainable is Planck’s constant of action. However, according to classical analytical mechanics action is equivalent to the time integral of the Lagrangian energy function. Nonetheless, the square of the Lagrangian would properly serve as the integrability of space-time quantization of square of energy which is time independence.

[SteveA]

Do you happen to know of forms other than a gaussian that are transform invariant (stable) in both frequency and time domains?

[AntonioLao]

Just a wild guess, could that be the exponential distribution?