## An integral Tauberian theorem and other topics |

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### Contents

Preliminary definitions and outline of proof | 4 |

Preliminary lemmas | 7 |

The main lemma | 13 |

10 other sections not shown

### Common terms and phrases

analytic function assume Bers bilateral Laplace transform choose chosen closed graph theorem compact subset computation continuous inclusion converges absolutely Cornell University corollary cusp forms defined denote discontinuous group Divide both sides dm(z dual Dual Space Earle epimorphism exist f Az)A f is holomorphic f(rm f(um finite number follows fundamental domain fundamental region g(sk g(xk g(xn Hahn-Banach theorem identically zero implies inequality inner product 5.1 INTEGRAL TAUBERIAN THEOREM left-hand derivatives lemma Let f lim inf lim sup log f(z lower a-exponential peaks m+l)a meromorphic function nonnegative function peaks for f Petersson Petersson inner product Pfcf proof of lemma Proof of theorem prove r-rQ relative Poincare series reproducing formulas result rQ+a sequence solutions of 2.9 t,rQ t(rQ Tm+1 upper a-exponential peaks upper half plane valid xm-ma y)dy