Re: :: generalised trigonometry and fourier analysis ::


galathaea wrote:
galathaea wrote:
galathaea wrote:
the classic fourier transform
uses the kernel

-i omega t
e

a form of the circle mapping

it is well known that any such mapping
can be decomposed into components
of the trigonometric functions

sin and cos

which select out the even and odd components

now
in terms of multisections

|0 ix
cos x = | e
|2

and

|1 ix
sin x = | e
|2

of course this is i sin x...

the transform properties that make this useful
arise from the fact that the exponential
suitably rotated
obeys

d
-- y = y
dx

and the trigonometric functions
form a basis of solutions for

2
d
-- y = y
2
dx

ie. the transform is able to provide information
on the differential deviation from idempotency
and this structural information
helps in solving differential equations

so

being a multisection freak
my very first investigations in this theory
were to expand it to general multisections

|m x
| e
|n

which
for a given n
form a basis for the equations

n
d
-- y = y
n
dx

now using simpson's multisection formula
one can define multiplication and addition formuli
for these functions

and build a generalised trigonometry

additionally
these provide higher symmetry data for fourier analysis

like my more general hypergeometric theorems
though
i have not been able to find previous discovery
in the literature

this is the most basic projective decomposition
of holonomy
though!

and the matrices formed
when building inversion relations
are well known in combinatorics
so what gives?

i expected my hypergeometric relations
to be common to graduate students
but these have to be regularly discovered by undergrads!

at least this has to be well known
to students of simpson

the simpson decomposition at roots of unity
gives these function very basic algebraic relationships
over the cyclotomic fields

but i have no references that go into this

does anyone know of prior art here?

as a side note
a recent comment by confutus on these newsgroups
has shown me that these generalised trigonometrics
form a post algebra with differentiation as negation

is this also known?

higher symmetry fourier analysis seems too obvious
to credit me for its discovery...

here is a link that describes simpson's multisection formula
for those who might be interested

http://mathworld.wolfram.com/SeriesMultisection.html

for instance

cosh x = (1/2) (e^x + e^(-x))
sinh x = (1/2) (e^x - e^(-x))

and trigonometric relations derived
using basic properties of exponents

the addition and product formulas just fallout

now doing the same with higher multisections is easy
and gives some very well known matrix relationships

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar

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