By B. A. Fuks
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T If F (t) := 0 f (s) ds converges to F∞ as t → ∞, then limλ→0 fˆ(λ) = F∞ . This Abelian theorem is easy to see. The converse is false in general: If limλ→∞ fˆ(λ) = t F∞ exists, then 0 f (s) ds need not converge as t → ∞. But if a theorem says that it does under some additional hypothesis then we call it a Tauberian theorem and the additional hypothesis a Tauberian condition. An interesting Tauberian theorem is the following. 1 (Newman–Korevaar–Zagier). Assume that fˆ has a holomorphic extension to an open set containing C+ .
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4), cf. 2]. We finish this section by going back to the origins of Tauberian theory. Given ∞ a bounded sequence (an )n∈N0 consider the power series p(z) = n=0 an z n which ∞ is defined for |z| < 1. If n=0 an =: b∞ exists then Abel showed in 1826 that lim p(x) = b∞ . 8) 1 The converse is not true in general. Additional assumptions are needed. 9) lim n an = 0, n→∞ thus proving the first “Tauberian theorem”. 9) can be relaxed to supn∈N n an < ∞. Another Tauberian theorem is due to Riesz. 6). 5 (Riesz).