 # 1. Let f be a function on the circle. For each N ≥ 1 the discret...

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1. Let f be a function on the circle. For each N ≥ 1 the discrete Fourier coefficients of f are defined by aN (n) = 1 N X N k=1 f(e 2πik/N )e −2πikn/N , for n ∈ Z. We also let a(n) = Z 1 0 f(e 2πix)e −2πinx dx denote the ordinary Fourier coefficients of f. (a) Show that aN (n) = aN (n + N). (b) Prove that if f is continuous, then aN (n) → a(n) as N → ∞. G 4. Let e be a character on G = Z(N), the additive group of integers modulo N. Show that there exists a unique 0 ≤ ` ≤ N − 1 so that e(k) = e`(k) = e 2πi`k/N for all k ∈ Z(N). Conversely, every function of this type is a character on Z(N). Deduce that e` 7→ ` defines an isomorphism from ˆ to G. [Hint: Show that e(1) is an Nth root of unity.] 5. Show that all characters on S 1 are given by en(x) = e 2πinx with n ∈ Z, and check that en 7→ n defines an isomorphism from Sc1 to Z. [Hint: If F is continuous and F(x + y) = F(x)F(y), then F R is differentiable. To see this, note that if F(0) 6= 0, then for appropriate δ, c = δ 0 F(y) dy 6= 0, and cF(x) = R δ+x x F(y) dy. Differentiate to conclude that F(x) = e Ax for some A.

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