¿Cómo elevar la función de densidad de probabilidad transformada de Fourier a una potencia fraccionaria?

By hypothesis my measured probability density functions (PDF) result from n convolutions of an elementary distribution (E).

I have two distributions the first (F) of which is supposed to have undergone more convolutions (m_1) than the second (G) (m_2 convolutions).

In fourier space:

F' = E'^m_1
G' = E'^m_2

As the two PDFs are constituted from the same elementary distribution, I should be able to be able to calculate the PDF of G from F

G' = F'^{m_1/m_2}

Taking the IFFT i should have a distribution that overlaps well with G.

A naive way of doing this would to be simply to calculate the FT of F and raise it to the power 1/integer and testing a range of integers.

Mi pregunta are there any tricks for raising the Fourier transformed PDF to a fractional power. I have done so but the IFFT gives a distribution far from that which is expected. And strange aliasing errors.

I've included a padded vector as one might do if they were to do a convolution of two PDFS.

My normalization is based on the fact that the k=0 [ProbF(1,1)] wave vector gives the integral of the PDF which should be equal to one.

Of course, the hypothesis could be wrong, but it has all the reasons in the world to be valid.

Mi código

Inc  = INC1 ;                           % BINS 
null = zeros(1,length(Inc));            % PADDED PROB
Inc  = [ Inc.*-1 (Inc)  ];              % PADDED INC VECTOR
Prob = [ null heightProb1  ]       ;    % PADDED PROB VECTOR

ProbF     = (fft(Prob))         ;
ProbFnorm = ProbF./ProbF(1,1) ;         % NORMALIZED BY K=0 COMPONENT (integral of PDF =1)

m=.79                                   % POWER TO RAISE

ProbFtrans = ((ProbFnorm).^(m));        % 'DECONVOLUTION' IN FOURIER SPACE
ProbIF     = (ifft((ProbFtrans)).*(ProbF(1,1)));    % RETURN TO PROBABILITY SPACE

figure(2);
plot(Inc,ProbIF,'rs')    

Gracias de antemano por su ayuda