-- retour de tchebyshef --

08/04/2008 - 18:12 von galathaea | Report spam
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

tchebyshef was the natural next step for me
but as i wrote to you so desperately in the past
he was not polynomial
and so i troubled over his generalised abilities

tchebyshef was the next step in the generalised fourier
he had to be

mais il y avait d'autres beautés
que je ne pourrais pas apprécier
jusqu'à  ce jour fatidique de marche

..

as i have mentioned prior
he is easy to describe in the generalised trigonometry

the first kind has an obvious definition

/ \
T | g ( theta ) | = g ( n theta )
m n \ m 0 / m 0

where g (x) is the generalised (hyperbolic) trigonometric
m 0
oo j
x
|0 x x \ -
| e = (0, m) multisection of e = / (1)
|m j
j=0(mod m)

-+-+-

it had occurred to me early on
that there might be a secret benefit to the nonpolynomialness

a theorem in the classical tchebyshef
gave the (0, 2)-form the benefit of uniqueness
for polynomial minimax approximations
so nonpolynomialness could possibly be a path towards preserving minimax qualities

aber ich benötigte eine frische annà¤herung...

-+-+-

there are obvious theorems one can immediately prove from the definitions

the composition theorem

/ \
T | T (x) | = T (x)
m n \ m n' / m n n'

is simple and in many ways uninteresting
but i had toyed with it from very early on

it reveals the game
the substitution of x for g ( theta )
0 m
which is so useful to the tchebyshef generalisation

und doch nach diesem stà¼rmischen bengaluru tag
ich sah die graue rà¤tselfalte weg...

-+-+-

so many things to relate
i choose now only one to state

a single example to inflate
until the next
and its unending
disappointing
contribution to conflate

a simple one to start with

recall the product rule for generalised trigonometrics
comme j'ai écrit environ tellement il y a bien longtemps

watch the trick

m-1
j
/ |0 (n-1)theta \ / |0 theta \ 1 / |0 n theta \ |0 (n-1+w )theta \
| | e | | | e | = - | | e + / | e m |
\ |m / \ |m / m \ |m |m /
j=1

through the secret substitution

|0 theta
| e --> x
|m

|0 n theta
| e > T (x)
|m m n

es wurde aufgedeckt

m-1

\
T (x) = m x T (x) - / T (x)
m n m n-1 j
j=1 m n-1+w
m

which reduces to the classical recurrence

T (x) = 2 x T (x) - T (x)
n n-1 n-2

for T = T (x)
n 2 n

mais la généralisation n'était pas une récurrence!

and though i have displayed here this
in the format of it's malformed tradition
the symmetry inside can be revealed

-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-
| |
+ m-1 +
| |
+ \ +
| m x T (x) = / T (x) |
+ m n j +
| j=0 m n+w |
+ m +
| |
-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-

interlocking wheels of dhamma binding the whole cyclotomic ring of places tight!

^..^

i must rest now friend
and let my newfound parasites retire

there is a much greater tale here
though
ignited by the fever of that bengalurian night

i will reveal these over the coming weeks
as my strength returns and the toxins of bharat subside

but imagine
if you will
the great power that comes from controlling the magic of this generalisation

eine zauberei von symbolen

finally
the generalised fourier theory is maturing

this new cyclic type of relation is similar
but much more general and natural
to the relations you are familiar with in modular forms and other special hypergeometrics

can you see the generalisation of jacobi polynomials?

perhaps a simpler task until i write again
can fill your exercises and prepare you for the methods

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

a basic task for you
dear friend
to prepare the theory of representations and approximations

kya?

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

Lesen sie die antworten

#1 quasi
08/04/2008 - 22:17 | Warnen spam
On Tue, 08 Apr 2008 09:12:29 -0700, galathaea
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.

quasi

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