Re: Symmetries reflect unilaterality and vice versa

On 2/22/2007 9:30 PM, Christopher J. Henrich wrote:
In article <erki6h$545$1@xxxxxxxxxxxxxxxx>, Eckard Blumschein
<blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

Fourier transform of a unilateral function f(x), e.g. f(x)= 0 for x<0,
yields a complex-valued F(y) with symmetrical real part and
antisymmetrical imaginary part. Also vice versa. For physical
implications cf.
http://iesk.et.uni-magdeburg.de/~blumsche/M283.html
Is there any word that aptly denotes a pair of two values which just
differ in sign: positive or negative? What about a bivalent value?

If f is real-valued, and F is its Fourier transform, then F(-y) is the
complex conjugate of F(y). Equivalently, the real part of F is even,
and the imaginary part is odd.

Of course.
Perhaps you did not yet realize that there is a reasonable restrictions
to many quantities of our real world: they do not have positive as well
as negative values x. I wrote 'unilateral function f(x)'.

Am I using the term "Fourier transform" in the same way as you? I
assume something like this:

F(y) = \integral_{-\infinity}^{+\infinity} f(x) e^{ixy} dx.

This is the common use, yes. However, comparing the clever evolutional
cochlear solution to the frequency analysis with its clumsy mathematical
counterpart, I got aware of the possibility to drop just the anyway
arbitrary point of reference (Christ's birth) and perform instead a
real-valued frequency analysis within IR+ instead of IR.

The loss is huge: I got rid of the need to interpret negative frequency.
I got rid of 4-fold redundancy, and a lot more, including serious
discrepancy between measured and theoretical physiological data.
No essential information is missing.

Admittedly, application of cosine transform is not new but already very
successful in MPEG coding.

My message to the physicians is as simple as brutal:
If there is a pair of two just positive quantities like e.g. radius r
and wave number k which are related to each other by complex Fourier
transform, then one of the two must be accepted like unphysical in that
it has to have negative as well as positive values in order to correctly
encode the unilaterality. Neglect of this dilemma led to wrong
interpretation of the Hermitean symmetry and related mistakes.
Schroedinger wrongly believed being correct when making his wave
functions real. He just made Weyl wonder about apparent PCT symmetry.

By the way, when I performed the usual FT within IR, I realized that
intermediate values within integral tables might be misleading. Let's
restrict to a simple example:
(I will write w for omega and choose just dt but dw/2pi)

Given a function f(t)=1 for -T/2<t<+T/2, f(t)=0 else.
FT yields, as well known, F(w)= 2 sin(wT/2)/w with ?oo<w<oo
Inverse Fourier transform correctly returns
f(t)=dw/pi [{integl_{-infty^0}+integl_{0^+infty}] sin(wT/2) cos(wt)=
= 2/pi {pi/2 for |t|<T/2 but 0 else}
if we ignore the intermediate value pi/4 for |t|=pi/2

The intermediate value could not be considered correctly returned
because it was missing within the given function.
Does it matter at all?
It originates from the tacit assumption/definition |sign(0)=0|.
At least a freshman who looks for the integral dx [0,oo] sin(x)cos(x)/x
is tempted to trust in the value pi/4 and get the wrong expression
f(t)=1/2 for ?T/2<t<+T/2, f(t)=0 else.
I recall rumoured pertaining confusion among physicians reported more
than 40 years ago and suspect a recent case.

Is there really any useful application of such intermediate values
belonging to jumping quantities in IR? Or is there some mathematics
(maybe denying LEM for reals) which follows Brouwer in that |sign(0)=1|
might be not wrong for irrationals? I am not familiar with constructive
set theory.

Eckard Blumschein,
Uni Magdeburg

.

Relevant Pages

  • Re: .999... ?= 1
    ... > the positive reals, some to denote the nonnegative reals ... > for the set of positive reals and [0,infinity) for the set ... As long as mathematics is considered to provide the very basis, ... Fourier spectrum exhibits Hermitian symmetry: ...
    (sci.math)
  • Re: The infintely small number b
    ... doesn't exhibit some perceived possibility for symmetry, ... it exists in the original system it is reflected in several ... Or for the reals? ... continuum or whatever? ...
    (sci.math)
  • Re: .999... ?= 1
    ... >> the positive reals, some to denote the nonnegative reals ... > As long as mathematics is considered to provide the very basis, ... > Fourier spectrum exhibits Hermitian symmetry: ... > Your words enunciate and claptrap are unknown to me. ...
    (sci.math)
  • Re: The infintely small number b
    ... doesn't exhibit some perceived possibility for symmetry, ... So your next qustions are - what is the original system, ... Or for the reals? ... continuum or whatever? ...
    (sci.math)
  • Re: Maths question - Multi-dimensional numbers
    ... numbers that you can convert the complex form: ... where x and y are reals ... sines.and a transform back. ... (First complex plane on I) ...
    (sci.fractals)
Loading