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,
FN (x + n) = FN (x),
when N ≥ 1 is an integer, and where
FN (x) = X
15. This exercise provides another example of periodization.
(a) Apply the Poisson summation formula to the function g in Exercise 2 to
(n + α)
whenever α is real, but not equal to an integer.
(b) Prove as a consequence that
(n + α)
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.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.