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periods connection
this relates some recent ideas of me , gottfriend and mike3 :

lets generalize , ignoring fixpoints and branches.

for starters ignore also domains and convergence.

and also ignore trivial solutions as f(x) = C. or C*f(x) is also a solution.

let w1 w2 w3 w4 be periodic functions.

w2 trivially restricted.

w4 having period h.

w1 w2 w3 having period 1.

f(x) = f(g(x))

there are 2 ways to solve and they are EQUIVENT upto (non?)uniqueness , parameters , branches and domains.

method A

T(g(x)) = T(x) + 1. notice T(x+w2(x)) is also a solution **.

f(x) = w1(T(x+w2(x)))

because of ** , f(x) = w1(T(x+w2(x))) = w3(T(x))

and w1 and w2 are not absolute !!!

method B

f(x) = sum w4(z) g^[z/h](x) ( z runs over the integers )

now the fascinating thing is

both method A and B are the complete set of solution hence

w3(T(x)) = sum w4(z) g^[z/h](x)

and the 2 big questions are

1) uniqueness

do there exist 2 w4 functions such that

sum w4_1(z) g^[z/h](x) = sum w4_2(z) g^[z/h](x)

or equivalently

sum w4_3(z) g^[z/h](x) = 0 ?? (w4_3(z) <> 0 )

and even

sum w4_4(z) g^[z/h](x) = 0 = sum w4_5(z) g^[z/h](x) ??
(w4_4(z) <> w4_5(z) <> 0 )

which resembles strongly the fourier research work by Dmitrii Evgenevich Menshov.

2) w3(T(x)) = sum w4(z) g^[z/h](x)

hence how do we go from a certain w3 to w4 and vice versa ?

what is really nice and intresting is that neither method is strongly based upon fixpoints.

and of course we relate the superfunction of g(x) with f(x) , w3 and w4.

as for convergence of method B that is strongly dependent on branches and fixpoints of x.

gottfried uses a ' matrix continuation '.

gottfried mentions a period , but in my version ( written above ) i dont find it , i probably should reread his thread.

i dont know how far mike and gottfriend agree with me and vice versa.

anyway , about that convergence.

suppose w3(T(x)) = sum w4(z) g^[z/h](x)

but sum w4(z) g^[z/h](x) seems to converge ??

we fix that as follows ( no matrix method used here )

w3(T(x))/w3_2(T(x)) =
sum w4(z) g^[z/h](x)/sum w4_0(z) g^[z/h](x+q)

now for suitable w4_0 and small integer q the right hand converges , left hand becomes :

w3(T(x))/w3_2(T(x)) = w3_3(T(x)) for q = 0

or for eg w4 = w4_0 , w3 = w3_2 and q^2 = 1

w3(T(x))/w3(T(x +/-1))

now recover w3(T(x)) by using the continuum product of the above.

and by recovering w3(T(x)) we can in principle find the superfunction of g(x) again and also note that

w3(T(x)) seem the continuation of sum w4(z) g^[z/h](x) when that sum diverges.

this idea is not new for me.

i wrote it as a teenager in my lost notebook.

tommy's lost notebook Smile ( ramanujan anyone Wink )

i got stuck on how w3 and w4 related ...

also if w3 is analytic and w4 is not and vice versa ...

but finding the inverse superfunction T(x) by finding the continuation of the method B , and then taking the inverse of w3 was fascinating too me.

i admit i forgot about it while writing ' blunder f(x) = f(x^2) ' but its coming back 2 me now.

still strugling with my teenage dreams ... tetration ... finding periods ... math beyond pythagoras Smile

i wrote that -more or less - before i even heard about pythagoras or complex numbers as a very young teenager.

i have forgotten more than some will ever know ...

i dont know how mike and gotffried tried to recover the superfunction with their continuum sum / product and alternating iterations sums , but i gave my views understandings and methods here ...

sorry if i duplicate or misunderstand others.

for clarity , all of this is still consistant imho with my previous written stuff , for instance " using sinh " or my uniqueness condition of no bending points , giving me confidence about them ...


one more thing i forgot :

i mentioned analytic with respect to w3 and w4.

but i didnt explain analytic w4 ;

sum w4(z) g^[z/h](x)

can be generalized to

sum w4(z/h) g^[z/h](x)


sum (w4(z/h) g^[z/h](x))/Y(h)

and even lim h -> 0


integral[-oo,oo,z] w4(z) g^[z](x) dz

and now w4 can be analytic and 1 periodic.

another question occurs :

sum (w4_1(z/h) g^[z/h](x))/Y(h) = integral[-oo,oo,z] w4_2(z) g^[z](x) dz

is always solvable ??


another thing worth mentioning ,

wheither or not f(x) = f(g(x)) has singularities , poles or finite radius and how those behave.

it seems that if one solution f(x) is not entire , then neither is another.

and the other way around.

( this follows from singularities of the abel functions , and if im not mistaken thus only poles can form a counterexample , BUT abel functions never have poles ... if this implies that f(x) doesnt have poles is not yet clear to me ... )

we can express g(x) such that f(x) = f(g(x)) has (an) (only?) entire solution(s) :

- its basicly a fixpoint argument of course -

f(x) = f(exp(a(x)) + x) with a(x) an entire function.

so this equation has a special place.


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