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2m unequal absolutely integrable Analytic Continuation apply Lemma 1.3 approximated arbitrarily arbitrary interval arbitrary sum assertion follows Beurling bounded linear functional boundedness property Carleman Chapter characteristic function compact support concave function consequently Consider an arbitrary Consider an interval convergence Corollary definition denote Domar easily seen existence exponential EXPONENTIAL POLYNOMIALS finite interval finite number following lemma follows from Lemma Fourier expansion Fourier transform function f gives OO Hahn-Banach theorem harmonic transform Hence holomorphic inequality integer integral equation 1.9 interpolation is possible Kahane 9 Laplace transforms linear vector space Mean-periodic Function mean-periodic Schwartz metric Minkowski's inequality Nordlander norms notion number of intervals obvious oo oo oo<fc<oo Paley and Wiener Parseval's formula polynomial positive numbers proof is complete pseudo-periodic real number regular everywhere satisfies Schwartz's semi-regular sequence spectrum sum sES(A Suppose Theorem 2.2 tion translations f(x Uk is regular uniformly upper density zero