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By V I Smirnov; Richard A Silverman
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Extra resources for Linear algebra and group theory
3 Fourier Series and the Poisson Summation Formula 33 Then Da f ∈ L2 (Rm /Zm ) implies that ∑ ∑m |( f , eb )|2 [(2π b)a]2 < ∞. |a|=k b∈Z Now there is a constant c > 0 such that ∑ [(2π b)a]2 ≥ c b 2k . |a|=k So the Cauchy–Schwarz inequality enables us to compare the series of absolute values of Fourier coefficients and the series Σ b −2k , which is an Epstein zeta function. To see that the series ∑ b −2k converges for k > n/2 you can proceed by developing a higher-dimensional version of the integral test.
5). Instead we choose to follow the discussion in Dym and McKean [147, p. 116]. 2 Fourier Integrals 29 where fn (x) = ( f ∗ · · · ∗ f )(x) if the number of f s in the convolution is n. √ Let dn be the density for the random variable (X1 + · · · + Xn)/ n. 2. The proof is finished using the following Exercise. 26. Complete the proof of the central limit theorem by approximating χ[a,b] (x) = 0, x ∈ [a, b] 1, x ∈ [a, b] by Schwartz functions k. It is possible to relax the hypotheses in the central limit theorem and one can also obtain error estimates.
Such theorems involve Laplace–Stieltjes transforms: ∞ 0 exp(−st)d α (t), with α of bounded variation and normalized to make α (0) = 0 and α (x) = (α (x+) + α (x−))/2. See Apostol  for a treatment of Riemann–Stieltjes integrals. Before thinking about Tauberian theorems, however, one should consider the following. 4 (An Abelian Theorem). Suppose that f (s) = 0∞ exp(−st)d α (t) for s > 0, where α is a normalized function of bounded variation and α (t) ∼ At c /Γ(c + 1), as t → ∞ (or t → 0+). Then f (s) ∼ As−c , as s → 0 + (or s → ∞).
Linear algebra and group theory by V I Smirnov; Richard A Silverman