## A Nonlinear Tauberian Theorem in Function Theory and Some Results on Tauberian Oscillations |

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abelian theorem arg K(a assume bounded Chapter choose completely regular growth completes the proof contour integration convolution corollary cp(t cp(x deduce definition Edrei and Fuchs Edrei-Fuchs lemma entire function exists f e Q f is slowly F(re F(reia finite fixed follows form 19 form 35 form 40 Fourier transform function f harmonic measure hence holds implies regular variation integral integral transform Jensen's theorem Karamata Lebesgue Lebesgue integral Lebesgue measurable Let f lim f lim f(x lim f*K-a lim inf f(x lim sup f(x log f log f(r Mellin transform meromorphic function negative zeros Nevanlinna non-decreasing obtain Osc f Osc f*K Parseval's theorem Pitt's theorem positive proof of theorem proposition proved rAL(r result rL(r satisfies sequence slowly decreasing slowly varying function statement tauberian theorem thesis Valiron's formula Valiron's theorem varies regularly varies slowly vn(z Wiener's theorem z e Dn