Re: Fractional Transforms


"amy666" <tommy1729@xxxxxxxxxxx> wrote in message
news:25975231.1229169752229.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
Being intrested in both fractional iterations and many transforms like
fourier and laplace transforms it naturally leads to the following idea :

" Fractional Transforms "

let T be an integral transform

let L be the Laplace integral transform


T(T(f(x)) = L(f(x)) for all f(x).

[cut]

Let T be defined by

T(f) = g(x) = int(0,oo) f(y)k(xy) dy

Let

T^-1(g)= f(x) = int(0,oo) g(y)h(xy) dy

then

H(s)*K(1-s)=1

where H and K are the mellin transforms of h and k. This is a nessecary and
sufficient condition that T(T^1)f =f once sutable restrictions are place on
the space for f.

Let f = L(v) for some function v. you want

T^1(L(v)) = T(v)

Take the mellin tranform of both sides

Gamma(1-s)V(s)H(s) = K(s)V(1-s) where V is the mellin tranform of v

This becomes

K(s) / (K(1-s)* Gamma(1-s)) = V(s) / V(1-s)

i think the above must show that a transform having a kernal of the form
k(xy) can not satisfy what you want.

of course this doesnt clear other integral transforms where the kernel takes
a different form say k(x-y) for example.




.

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