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1. Fourier analysis on finite abelian groups 2. Appendix: spectral theorem for unitary operators 3. Appendix: cancellation lemma There are Fourier expansions on finite abelian groups essentially identical in form to Fourier expansions of periodic functions on the real line. This follows from the spectral theory of unitary operators on finite-dimensional(More)
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Local authorities, in England, implemented the Framework for the Assessment of Children in Need and their Families in April 2001. The Framework is the first 'official' standard assessment model intended for use in the initial assessment of all 'children in need' under Part 111, section 17 of the 1989 Children Act. This new 'conceptual map' needs to be(More)
1. Zonal spherical harmonics on SL 2 (C) 2. Formula for triple integrals of eigenfunctions 3. Asymptotics of triple integrals 4. Asymptotics of integrals of n-fold products We determine precise asymptotics in spectral parameters for integrals of n-fold products of zonal spherical harmonics on SL 2 (C). In a variety of situations, integrals of products of(More)
We obtain second integral moments of automorphic L–functions on adele groups GL 2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL 1 and GL 2. This requires complete reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL 2 (Q) or(More)
1. Cautionary example 2. Criterion for essential self-adjointness 3. Examples of essentially self-adjoint operators 4. Appendix: Friedrichs' canonical self-adjoint extensions 5. The following has been well understood for 70-120 years, or longer, naturally not in contemporary terminology. The differential operator T = d 2 dx 2 on L 2 [a, b] or L 2 (R) is a(More)