à l'occasion de son retour
il y avait de bengaluru fini d'orages

i had given him up on and off many times over the past year
but though he frustrated me
taunted me
relentlessly
persistently driving me away in anger
i kept returning
a little broken
stubbornly expecting some deeper connection to form

On Tue, 08 Apr 2008 09:12:29 -0700, galathaea <galath...@veawb.coop
wrote:

à l'occasion de son retour
il y avait de bengaluru fini d'orages

i had given him up on and off many times over the past year
but though he frustrated me
taunted me
relentlessly
persistently driving me away in anger
i kept returning
a little broken
stubbornly expecting some deeper connection to form

Good to have you back, even though apparently, you're not yet actually
back in a physical sense. In any case, I see you're in fine form.

Perhaps you've awoken the spirit of Ramanujan.

On Apr 8, 1:17 pm, quasi<qu...@null.set> wrote:
On Tue, 08 Apr 2008 09:12:29 -0700, galathaea<galath...@veawb.coop
wrote:

à l'occasion de son retour
il y avait de bengaluru fini d'orages
i had given him up on and off many times over the past year
but though he frustrated me
taunted me
relentlessly
persistently driving me away in anger
i kept returning
a little broken
stubbornly expecting some deeper connection to form
Good to have you back, even though apparently, you're not yet actually
back in a physical sense. In any case, I see you're in fine form.

Perhaps you've awoken the spirit of Ramanujan.

it would have been nice to visit kumbakonam
but i was stuck on the west side of the continent when down south

i really wanted to see places of past inspiration
to see what magic they might still hold
but i couldn't get over to ramanujan's

instead i saw the buddha's places
even though i grew tired of the buddha
after reading more of the pali canon

now i'm sick of the buddha
i've generally become quite antibodhi
and wish i could have spent some days out in the southeast

the company i work for also has offices out in chennai
so maybe one of these days...

do you see how to express x^n in terms of these generalised forms?

g:

a warrior
you!

no

i had never thought it possible
that "sometimes they trick you"

i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons

that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair

but stay strong
friend

at least they removed only inessential organs

-+-+-

you had so many questions
i wonder if my other "friends" finally got to you

if so
i apologise
but yes i can try to go back some

the goal is a fourier system

in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n

oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m

which i abandoned long ago

the zeroes of these functions
are almost periodic like the besselian
in that

lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n

where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n

they all have zeroes that approach some multiple of pi in spacing

but the peaks
unlike the t_2 series
grow exponentially n > 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)

a back-of-the-envelope calculation shows
using m=0, n=3 as an example

w x
( ) ( |0 6 )
Re< t (x) > = Re< | e
( 0 3 ) ( |3 )

x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e > + e + e Re< e > |
3 \ ( ) ( ) /

x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /

which is zero when

x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /

or

-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /

as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes

the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)

and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...

so you see
a generalised fourier analysis must not use the

/
| t (r x) t (s x) dx
/ m n m' n

because roots are not integer multiples of some base periodicity

similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...

these are of course interleaved
as one would expect from rolle's theorem

these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.

n
d
--- t (x) = - t (x)
n m n m n
dx

)
is one of the things that makes tchebyshef in the hyperbolics attractive

but let me show you how
my increasingly absentminded comrade at arms
you can still lay the obvious foundations

^^.

first
though
i realise i haven't told you of nausea

you know
sartre's nausea

i reread it on my recent travels to india

i tell you with full earnestness
my easily fooled one
there is no better way to read nausea
than reading nausea in india

there is this great passage
where roquentin is talking of his travels to varanasi
(then benaras)
and how we gather these experiences to sell to others
for their admiration and respect as "one with experience"

i didn't want to see the taj mahal
i swear to you i did not plan any visit through agra

it's just
i started in delhi because that's where my driver was based
and i wanted to see khajuraho
and
well
just look at a map

so my driver suggested it
for the first night's stop

#######@@@4..7*****)

the first step is to look for the paths of integration for zeroes
and since these are entire functions
all paths with real endpoints are equivalent to the real interval

the zeroes of the integrated forms are simply
(m+1)_?_n(j)
(where "m+1" takes place in Z_n)
so any interval ending on zeroes provide obvious zero periods

of course
we can slide the endpoints to any equal values of the antiderivative
but the zeroes form a "basis" of "types" of zero-valued intervals
using the equivalence class of "continuous sliding"
because each "hump" reaches an absolute magnitude greater than each

in some very important ways
these intervals
m_I_n(r,s) = [ m_?_n(r), m_?_n(s) ]
are representative elements of homotopy equivalencies

this isn't quite true in the 0_t_1 case
(e^(-x) has no zero periods - but also no zeroes)
or the m_t_2 cases
(sin/cos have peaks all 1
so "sliding" equates I(r,s) with I(r+n,s+n)
and the equivalences are represented by
m_I_2(r) = [ m_?_2(0), m_?_2(r) ])
but the definition of I(r,s) works in these cases as well
because the main point is

/
| t (x) dx = 0
/ I (r,s) m n
m+1 n

and

/
| t (x) dx in general has interesting values
/ I (r,s) m n for all p
m' n

for one reason
zeroes of m_t_n are critical points of (m+1)_t_n

so if we label these critical values

? (j) = the jth critical value of m_t_n (greater than or equal to 0)
m n

= t ( ? (j) )
m n m-1 n

then

/
| t (x) dx = ? (s) - ? (r)
/ I (r,s) m n m+1 n m+1 n
m n

example calculations for 0_t_3 give

? (j) = 1, -2.5847397, 16.055418, -98.4739, 604.01033, -3704.8236
0 3

with the basic properties of alternating sign and exponential growth

numerically, these are growing as 6.1337
but the actual value can be calculated from the formula presented earlier

one has that the leading term of the real positive part is

x
-
2 / '\/3 \
2 e cos | ---- x |
\ 2 /

and since the half-period approaches 2 pi / '\/3
the ratio of maxima is

( x + 2 pi / '\/3 )
-------------------
2
e
--------------------
x
-
2
e

pi / '\/3
= e

as one analyses these various critical values
a map of feature values begins growing

by an easy application of hermite-lindemann-weierstrass
all m_g_n are transcendental at all algebraic points
and so it is interesting to consider when these critical values are also transcendental

but that is a different direction than i want to go here

$^..~~~~~

varanasi is the oldest continuously living town in the world
over 3000 years of continuous occupation and rebuilding

it is quite an amazing discovery to learn what 3000 years smell like

waste water thrown from the upper floors of the buildings
keeping the ground wet to moisten endless piles of cow and goat shit

out of the recesses
nag champa and sandalwood mixing with something decomposing

the burning ghats blowing their occasional peoplesmoke through the narrow alleys

the stagnant and polluted ganga
controlling the winds with her schizophrenic desires

the deep black carbonsmoke of a million 2-stroke tuktuks
and sweat
and ghee
and...

thousands of years in one experience...

nnnn-v-mmmm

now
where in the fourier case one switches to
I = [ -pi, pi ] or [ 0, 2 pi ]
and brings the sin( r x ) and cos( s x ) as the scale mapping
we can generalise this procedure to the zeroes here

turning to a period agnostic form
we can arbitrarily scale all intervals to [ 0, 1 ]

/ \
? (x) = t | ( ? (s) - ? (r)) x + ? (r) |
m n m n \ m+1 n m+1 n m+1 n /
r s

now one can work only in the range I = [0, 1]
and single integrals are simple

/
| ? (x) dx
/I m n
r s

? (x)
m+1 n
/ r s \
= | ------------------- |
\ ? (s) - ? (r) /I
m+1 n m+1 n

which by design equals 0 at 1 and 0

&&&&&..&&&&

products have a lot more structure still

the products lose a strict per-zero almost-periodicity
and store information in larger scale zero patterns

for instance
multiplying 0_t_3 by 1_t_3
gives zeroes at 0, 1.8498128, 3.0167442, 5.4412334, 6.6506245, 9.0689975, 10.278196, ..
of differences approaching long, short, long, short, long, short, ..
where long = 2.4183992 = 4 pi / ( 3 '\/3 )
short = 1.2091996 = 2 pi / (3 '\/3 )
a patterned partitioning of the 2 pi / '\/3 periods of the m_t_3 family

(0,3)x(2,3) gives short, long, short long, ..
wheras (1,3)x(2,3) gives long, short, long, short again..

the same is seen in (0,4)x(1,4)
(long, short, long, short)
and (0,4)x(3,4) (short, long, short, long, ..)
but (0,4)x(2,4) is singly almost periodic of asymptotic zero period

it is easy to prove that there are integrals of these "wildly almost periodic" products
over nontrivial intervals
with value zero
(directly from its differential structure)

and so the differential structure of the critical points
(discretely) can be tied directly to the structure of integral periods

this is the whole point of fourier analysis
where integration over the periods is used to decompose into discrete sums the function transformed

but naive use of the product formula doesn't get a lot done

/
| t (x) t (x) dx
/ m n m' n

n-1
---
/ 1 \ -m' j j
= | - / w t ( ( 1 + w ) x ) dx
/ n --- n m+m' n n
j=0

n-1 -m' j
--- w
1 \ n j
= - / -------- t ( ( 1 + w ) x )
n --- j m+m'+1 n n
j=0 1 + w

the products have an interplay of the critical points
and effectively the entire zeroed-value d^m differential structure
transfered discretely across the structure of their critical points
from this ur-form of the sum of two unit cyclotomics

this is real
so it can projected term by term to reals
and summed as strange algebraics of cosines
related to the real and imaginary projections of the cycloctomics
in the straightforward way

but it becomes difficult to prove things about these expressions

some obvious symmetries suggest that
like the fundamental connection between
cosine series and the classical first tchebyshef
(the coefficients of their series are equal)
illustrating basis freedom
the differential structure could be more clearly seen in the tchebyshef language

this is why my attention has been so lately focused on the mysterious mr t
because then we map cleanly to a multiplicative composition

we don't work with the almost-periodic and wildly almost periodic structure of the critical points
we work directly in the cyclotomic rings and fields

so perhaps the simplicity would allow better characterisation
of the basis structure of a generalised analysis?

are there some collections of intervals from the I
that form more naturally complete sets than others?

understanding the tchebyshef translation would explain much of this structure

+----%,.$~~~+

the secrets are of course in some of the identities i have revealed to you
("sometimes they trick you"??? what the hell is wrong with you!?!)
but the real understanding comes in decomposing each analysis with a differential structure
and describing the transformation better
that relates the generalised fourier analysis with it's tchebyshef theory

the secret lies in the projection property of multisections
and the logical interplay of the multisection operator and the differential operator

d |m |m-1 d
-- | = | --
dx |n |n dx

quite a bit of this structure applies to more general classes of functions still
many interesting collections of hypergeometrics and their q deformations
and if i had the strength to explain tori to you
i would certainly be relieved of some of your barking

but i think that this darkening day has stolen any time for that

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

galathaea wrote:
g:
a warrior
you!

no

i had never thought it possible
that "sometimes they trick you"

i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons

that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair

but stay strong
friend

at least they removed only inessential organs

-+-+-

you had so many questions
i wonder if my other "friends" finally got to you

if so
i apologise
but yes i can try to go back some

the goal is a fourier system

in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n

oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m

which i abandoned long ago

the zeroes of these functions
are almost periodic like the besselian
in that

lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n

where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n

they all have zeroes that approach some multiple of pi in spacing

but the peaks
unlike the t_2 series
grow exponentially n> 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)

a back-of-the-envelope calculation shows
using m=0, n=3 as an example

w x
( ) ( |0 6 )
Re< t (x)> = Re< | e
( 0 3 ) ( |3 )

x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e> + e + e Re< e> |
3 \ ( ) ( ) /

x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /

which is zero when

x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /

or

-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /

as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes

the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)

and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...

so you see
a generalised fourier analysis must not use the

/
| t (r x) t (s x) dx
/ m n m' n

because roots are not integer multiples of some base periodicity

similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...

these are of course interleaved
as one would expect from rolle's theorem

these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.

n
d
--- t (x) = - t (x)
n m n m n
dx

)
is one of the things that makes tchebyshef in the hyperbolics attractive

[...]

I had a look at the Wikipedia article on orthogonal polynomials. I
didn't know
there were so many kinds of families: Tchebycheff, Laguerre, Legendre,
Hermite
and others:

http://en.wikipedia.org/wiki/Orthogonal_polynomials

It seems to me you're looking for non-polynomial
non-trigonometric functions orthogonal for
the Hilbert space L^2(R, mu) , where mu is a measure on R
perhaps arising from some weight function. Maybe you've
heard of wavelets (Daubechies, others) .

a warrior
you!

g:
a warrior
you!

no

i had never thought it possible
that "sometimes they trick you"

i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons

that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair

but stay strong
friend

at least they removed only inessential organs

-+-+-

you had so many questions
i wonder if my other "friends" finally got to you

if so
i apologise
but yes i can try to go back some

the goal is a fourier system

in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n

oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m

which i abandoned long ago

the zeroes of these functions
are almost periodic like the besselian
in that

lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n

where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n

they all have zeroes that approach some multiple of pi in spacing

but the peaks
unlike the t_2 series
grow exponentially n> 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)

a back-of-the-envelope calculation shows
using m=0, n=3 as an example

w x
( ) ( |0 6 )
Re< t (x)> = Re< | e
( 0 3 ) ( |3 )

x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e> + e + e Re< e> |
3 \ ( ) ( ) /

x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /

which is zero when

x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /

or

-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /

as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes

the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)

and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...

so you see
a generalised fourier analysis must not use the

/
| t (r x) t (s x) dx
/ m n m' n

because roots are not integer multiples of some base periodicity

similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...

these are of course interleaved
as one would expect from rolle's theorem

these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.

n
d
--- t (x) = - t (x)
n m n m n
dx

)
is one of the things that makes tchebyshef in the hyperbolics attractive
[...]

On Apr 20, 9:25 pm, David Bernier<david...@teranews.com> wrote:
galathaea wrote:
g:
a warrior
you!
no
i had never thought it possible
that "sometimes they trick you"
i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons
that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair
but stay strong
friend
at least they removed only inessential organs
-+-+-
you had so many questions
i wonder if my other "friends" finally got to you
if so
i apologise
but yes i can try to go back some
the goal is a fourier system
in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n
oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m
which i abandoned long ago
the zeroes of these functions
are almost periodic like the besselian
in that
lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n
where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n
they all have zeroes that approach some multiple of pi in spacing
but the peaks
unlike the t_2 series
grow exponentially n> 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)
a back-of-the-envelope calculation shows
using m=0, n=3 as an example
w x
( ) ( |0 6 )
Re< t (x)> = Re< | e
( 0 3 ) ( |3 )
x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e> + e + e Re< e> |
3 \ ( ) ( ) /
x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /
which is zero when
x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /
or
-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /
as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes
the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)
and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...
so you see
a generalised fourier analysis must not use the
/
| t (r x) t (s x) dx
/ m n m' n
because roots are not integer multiples of some base periodicity
similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...
these are of course interleaved
as one would expect from rolle's theorem
these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.
n
d
--- t (x) = - t (x)
n m n m n
dx
)
is one of the things that makes tchebyshef in the hyperbolics attractive
[...]

I had a look at the Wikipedia article on orthogonal polynomials. I
didn't know
there were so many kinds of families: Tchebycheff, Laguerre, Legendre,
Hermite
and others:

http://en.wikipedia.org/wiki/Orthogonal_polynomials

It seems to me you're looking for non-polynomial
non-trigonometric functions orthogonal for
the Hilbert space L^2(R, mu) , where mu is a measure on R
perhaps arising from some weight function. Maybe you've
heard of wavelets (Daubechies, others) .

what one eventually ends up with
after all the excruciating obfuscation is laid out
is ultimately a generalisation of polynomialness itself

orthogonality is simple

the interesting connection is that
the generalised trigonometry's orthogonality
is mirror to the orthogonality of the generalised tchebyshef

many of the classic orthogonals
including the tchebyshefs
are included in the class of jacobi polynomials

the gegenbauers are simply a subclass of jacobi
distinguished by equal upper parameters
and the tchebyshef fall in the gegenbauers
but the generalisation of tchebyshefs
falls in larger class of generalised jacobi

the trick is the switch into nonpolynomialness
through the generalised polynomial

2 n-1
w w w
n n n
y = a x + a x + a x + ... + a x
0 1 2 n-1

which
surprisingly
has some very regular zero features
but a more complex feature set

this really becomes apparent in the
generalised tchebyshef representation of x^n i posted

see

n=1 is e^x
so 1_T_n(e^x) = e^(n x)
or 1_T_n(x) is simply x^n

the taylor-mclaurin orthogonals
the most fundamental orthogs of all analysis

C 2_T_(n-1)(x) is the minimax rep of 1_T_n(x)

now there is this generalisation
and whole new set of tools for feature detection
here - more elaborate patterns in differential structure

^..^

one of the secrets hidden in the shift to tchebyshef
is the origin of the weight function

it basically arises from the presence of

|n-1 x
| e
|n

in integrals of the tchebyshefs

since in the (0,2), (1,2) case

2 2
ch x - sh x = 1

this generalises quite naturally to a function

/ |0 x |n-1 x \
f | | e , | e | = 1
\ |n |n /

where the natural question arises
is it algebraic?

the answer again is revealed in cyclotomics..

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

galathaea wrote:
On Apr 20, 9:25 pm, David Bernier<david...@teranews.com> wrote:
galathaea wrote:
g:
a warrior
you!
no
i had never thought it possible
that "sometimes they trick you"
i would have thought they needed the bodies
for all those extra uniforms
flack jackets
ammunition
and primary weapons
that is certainly unfortunate
and i know your family must be severely disappointed in the whole affair
but stay strong
friend
at least they removed only inessential organs
-+-+-
you had so many questions
i wonder if my other "friends" finally got to you
if so
i apologise
but yes i can try to go back some
the goal is a fourier system
in some ways this begs i return to t (x)
m n
-m
= w g (w x)
2n m n 2n
oo
--- j
\ (-1) nj+m
= / ------- x
--- (1)
j=0 nj+m
which i abandoned long ago
the zeroes of these functions
are almost periodic like the besselian
in that
lim ( ? (r) - ? (s) ) = ?
n->oo m n+1 m n n
where m_?_n(j) is the jth zero greater than or equal to 0
of m_t_n
they all have zeroes that approach some multiple of pi in spacing
but the peaks
unlike the t_2 series
grow exponentially n> 3
and in odd-n cases
the negative domain gives exponentially dominated growth without periodicity
where the even-n of course keep the order-2 symmetry
(even/odd implies periodic in both the positive and negative domains)
a back-of-the-envelope calculation shows
using m=0, n=3 as an example
w x
( ) ( |0 6 )
Re< t (x)> = Re< | e
( 0 3 ) ( |3 )
x '\/3 x -'\/3
- ---- i x - ----- i x
1 / 2 ( 2 ) -x 2 ( 2 ) \
= - | e Re< e> + e + e Re< e> |
3 \ ( ) ( ) /
x x
- -
1 / 2 / '\/3 \ -x 2 / -'\/3 \ \
= - | e cos| ---- x | + e + e cos| ----- x | |
3 \ \ 2 / \ 2 / /
which is zero when
x
-
-x 2 / '\/3 \
e = -2 e cos | ---- x |
\ 2 /
or
-3x
---
2 / '\/3 \
e = -2 cos | ---- x |
\ 2 /
as the exponential left hand side gets closer to y=0
it will cross the cosinus closer and closer to its own zeroes
the period of this (0,3) generalisation therefore approaches 2 pi / '\/3
(= 3.6275987...)
and this can be seen from a simple plot of this function
where 0_?_3(0) occurs at 1.8498128
then 0_?_3(j) = 5.4412334, 9.0689975, 12.696596, 16.324194, ...
so you see
a generalised fourier analysis must not use the
/
| t (r x) t (s x) dx
/ m n m' n
because roots are not integer multiples of some base periodicity
similarly
the (1,3)-t has zeroes 1_?_3(j) at
0, 3.0167442, 6.6506245, 10.278196, 13.905795, 17.533394, ...
and the (2,3)-t has 2_?_3(j) at
0 (double), 4.2332072, 7.8597929, 11.487396, 15.114995, ...
these are of course interleaved
as one would expect from rolle's theorem
these "almost symmetries"
along with the unnecessary complications of using half-periods
(so derivatives switch signs after n iterations, ie.
n
d
--- t (x) = - t (x)
n m n m n
dx
)
is one of the things that makes tchebyshef in the hyperbolics attractive
[...]

I had a look at the Wikipedia article on orthogonal polynomials. I
didn't know
there were so many kinds of families: Tchebycheff, Laguerre, Legendre,
Hermite
and others:

http://en.wikipedia.org/wiki/Orthogonal_polynomials

It seems to me you're looking for non-polynomial
non-trigonometric functions orthogonal for
the Hilbert space L^2(R, mu) , where mu is a measure on R
perhaps arising from some weight function. Maybe you've
heard of wavelets (Daubechies, others) .

what one eventually ends up with
after all the excruciating obfuscation is laid out
is ultimately a generalisation of polynomialness itself

orthogonality is simple

the interesting connection is that
the generalised trigonometry's orthogonality
is mirror to the orthogonality of the generalised tchebyshef

I seem to remember reading that Riemann wanted
tchebyshef to see his work on zeta and primes.
I might look up what tchebyshef did in
number theory.



many of the classic orthogonals
including the tchebyshefs
are included in the class of jacobi polynomials

the gegenbauers are simply a subclass of jacobi
distinguished by equal upper parameters
and the tchebyshef fall in the gegenbauers
but the generalisation of tchebyshefs
falls in larger class of generalised jacobi

the trick is the switch into nonpolynomialness
through the generalised polynomial

2 n-1
w w w
n n n
y = a x + a x + a x + ... + a x
0 1 2 n-1

Not using capital letters or punctuation is not something that
causes readability problems for me. It's OK for me.
However, formulas with exponents on the line above
don't always align well; also, towers of exponents
with no parentheses are ambiguous for me ...

So

y = a_0 x ^(n^w) + a_1 * x^(n^w) + a_2 * x^((n^w)^2) + ...

I don't think I've got it. Do you get polynomials by
setting w=1 ?

which
surprisingly
has some very regular zero features
but a more complex feature set

this really becomes apparent in the
generalised tchebyshef representation of x^n i posted

see

n=1 is e^x
so 1_T_n(e^x) = e^(n x)
or 1_T_n(x) is simply x^n

the taylor-mclaurin orthogonals
the most fundamental orthogs of all analysis

C 2_T_(n-1)(x) is the minimax rep of 1_T_n(x)

now there is this generalisation
and whole new set of tools for feature detection
here - more elaborate patterns in differential structure

^..^

one of the secrets hidden in the shift to tchebyshef
is the origin of the weight function

it basically arises from the presence of

|n-1 x
| e
|n

in integrals of the tchebyshefs

since in the (0,2), (1,2) case

2 2
ch x - sh x = 1

this generalises quite naturally to a function

/ |0 x |n-1 x \
f | | e , | e | = 1
\ |n |n /

where the natural question arises
is it algebraic?

the answer again is revealed in cyclotomics..

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

generalised polynomials have many nice properties

if f, f' e W[x]

in other words
if

---
\ w
f = / a x
--- w
w e S_f c C
n
S_f finite

and

---
\ w
f' = / b x
--- w
w e S_f' c C
n
S_f' finite

where C_n is the ring of integers
of the cyclotomic field of order n

then (f + f') e W[x]
and (f f') e W[x]

also
derivatives of e W[x]
are e W[x]

the integrals can bring in logarithmic forms
but so do simple laurent polynomials
and these are well understood

the really crazy thing
is that they seem to have an interesting zero structure
where
for instance

---
\ w
/ x = c(x)
---
n
w = 1

has n zeroes in the complex numbers

the important step
then
becomes the galois analysis

so an important consequence of this generalisation
is a whole new realm of galois study

which i am sure has already been done
if i only knew the right keywords to search