Some topics in complex and harmonic analysis , 2 Stephen
نویسنده
چکیده
If the limit of this as t → 0 exists, then we say that h(x) is Abel summable with the limit as the sum. If h(x) is integrable on R, then h is Abel summable with the usual integral of h being the Abel sum. One can show that the function At(x) can be expressed as an average of Gaussians. This is a quite useful fact, and basically it is a way of saying that Gaussians are more concentrated. For instance, one can use this to show that if h is Gauss summable, then h is Abel summable, and with the same sum. One can also use this to compute the Fourier transform of At, which is the Poisson kernel Pt(y) = cn t (|y|2 + t2)(n+1)/2 , (2)
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