Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincare metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted canonical line bundle $\omega_M(D)$ with the induced Hermitian norm $\|\cdot\|_M$ over $\overline{M} \setminus D_M$. Let $(\xi, h^{\xi})$ be a holomorphic Hermitian vector bundle over $\overline{M}$. 
In this article we define the analytic torsion $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ associated with $(M, g^{TM})$ and $(\xi \otimes \omega_M(D)^n, h^{\xi} \otimes \|\cdot\|_M^{2n})$ for $n \leq 0$. We prove that $T(g^{TM}, h^{\xi} \otimes \|\cdot\|_M^{2n})$ is related to the analytic torsion of non-cusped surfaces. Then we prove the anomaly formula for the associated Quillen norm. The results of this paper will be used in the sequel to study the regularity of the Quillen norm and its asymptotics in a degenerating family of Riemann surfaces with cusps and to prove the curvature theorem. We also prove that our definition of the analytic torsion for hyperbolic surfaces is compatible with the one obtained through Selberg trace formula by Takhtajan-Zograf.

Analytic torsion for surfaces with cusps I. Compact perturbation theorem and anomaly formula

This is the second in a series of two articles where we study various aspects of the spectral theory associated to families of hyperbolic Riemann surfaces obtained through elliptic degeneration. In the first article, we investigate the asymptotics of the trace of the heat kernel both near zero and infinity and we show the convergence of small eigenvalues and corresponding eigenfunctions. Having obtained necessary bounds for the trace, this second article presents the behavior of several spectral invariants. Some of these invariants, such as the Selberg zeta function and the spectral counting functions associated to small eigenvalues below 1/4, converge to their respective counterparts on the limiting surface. Other spectral invariants, such as the spectral zeta function and the logarithm of the determinant of the Laplacian diverge. In these latter cases, we identify diverging terms and remove their contributions, thus regularizing convergence of these spectral invariants. Our study is motivated by a result from \cite{He 83}, which D. Hejhal attributes to A. Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we obtain a quantitative result to Selberg's remark.

Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

. We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence U = { U n ( P, Q ) } n ≥ 0 of the ﬁrst kind determined by the real numbers P, Q under certain natural assumptions on P and Q . We deduce an asymptotic expansion of the theta function θ U ( t ) as t ↓ 0 and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function ζ U ( s, z ) = ∞ U ) − s to the whole complex s − plane. Moreover, we identify the residues of ζ U ( s, z ) at all poles in the half-plane ℜ ( s ) ≤ 0.

On the Hurwitz-type zeta function associated to the Lucas sequence

This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In Garbin et al. (2018) [8], we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (Huntley et al. (1995) [14] and (1997) [15], Jorgenson et al. (1997) [20] and (1997) [17]) and on degenerating families of finite volume hyperbolic three manifolds (Dodziuk et al. (1998) [4].) Although the types of results developed here and in Garbin et al. (2018) [8], are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.

Daniel Garbin

Papers

Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

Spectral asymptotics on sequences of elliptically degenerating Riemann surfaces

Heat kernel asymptotics on sequences of elliptically degenerating Riemann surfaces

Effective bounds for Fourier coefficients of certain weakly holomorphic modular forms