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Topic: Discrete convolution and the integral
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Tomi J Laakso

Posts: 1
Registered: 12/12/04
Discrete convolution and the integral
Posted: Jun 18, 1996 7:16 AM
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Do you know any literature or articles considering the following or
related topics?

Let f be Riemann-integrable in [0,oo[. Let f_m be the sequence
(f(0),f(1/m),f(2/m),...). Take ANY sequence g and form covolution
h_m=g*f_m. Define function k_m(x)=m^(-r) h_m(x/m rounded to integer),
where r is positive real number and h_m(i) denotes ith element of h_m.
Now it is relatively easy to prove that the operator A_g that send f
to lim m->oo k_m is either always divergent or converges to a
constant multiple of fractional integral I^(r-1).
Standard texts on Fractional Calculus do not consider this issue.

Thanks for your help.

Tomi J Laakso
University of Helsinki


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