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14. Prove that the periodization of the Fejer kernel FN on the real line is equal to the Fej´er kernel for periodic functions of period 1. In other words, X∞ n=−∞ FN (x + n) = FN (x), when N ≥ 1 is an integer, and where FN (x) = X N n=−N µ 1 − |n| N e 2πinx = 1 N sin2 (Nπx) sin2 (πx) . 15. This exercise provides another example of periodization. (a) Apply the Poisson summation formula to the function g in Exercise 2 to obtain X∞ n=−∞ 1 (n + α) 2 = π 2 (sin πα) 2 whenever α is real, but not equal to an integer. (b) Prove as a consequence that (15) X∞ n=−∞ 1 (n + α) = π tan πα whenever α is real but not equal to an integer. [Hint: First prove it when 0 < α < 1. To do so, integrate the formula in (b). What is the precise meaning of the series on the left-hand side of (15)? Evaluate at α = 1/2.

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Periodization Exercises
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