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the Poisson summation formula
the Titchmarsh convolution theorem
Appendix Integration by parts
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absolutely continuous algebra analytic extension analytic function approximate identity assertion Banach Bessel's inequality Beurling boundary function bounded function circle closed subspace completes the proof conjugate function constant contained continuous function convolution corollary defined dense subspace dual equal everywhere exponential factor Fejer Fejer kernel finite Fourier coefficients Fourier series Fourier transform Fourier-Stieltjes coefficients Fourier-Stieltjes sequence function F function in H'(T function-theoretic harmonic function Hence hypothesis inner function integrand interval isometry last section Lebesgue lemma limit linear functional LP(R LP(T mapping mean value Mercer's theorem modulus nonnegative norm null orthogonal outer function Parseval relation pointwise Poisson integral Poisson kernel positive integer positive measure positive number Problem proves the theorem Rademacher functions real axis result Riesz right side satisfies Show singular space spectral set square-summable subset summable function Suppose Tauberian theorem tends to oo topology translation trigonometric polynomials uniformly unit disk upper half-plane vanishes zeros