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For 0 ≤ r < 1, define Pr (θ) as follows. Put 2 z = reiθ and put Pr (θ) = 1−|z| |1−z|2 . 35, we have r|n| einθ . Pr (θ) = (41) n∈Z It follows from (41) that the first property of an approximate identity holds. ) The second property is immediate, as Pr (θ) ≥ 0. The third property is also easy to check. Fix > 0. If 2 |θ| ≥ and z = reiθ , then |1 − z|2 ≥ c > 0. Hence, Pr (θ) ≤ 1−r c . Thus, the limit as r increases to 1 of Pr (θ) is 0. Hence, the Poisson kernel defines an approximate identity on the circle.

2 we prove a related but more difficult result involving H¨older continuous functions. 8. 12. If all the Fourier coefficients of a Riemann integrable function f vanish, then f (x) = 0 at all points x at which f is continuous. In the theory of integration, one establishes also that the set of points at which a Riemann integrable function fails to be continuous has measure zero. Thus, we can conclude, when fˆ(n) = 0 for all n, that f is zero almost everywhere. 9. There exist continuous functions whose Fourier series do not converge at all points.

1) |wk |2 = ||w||22 . 2) j=0 N 1 2π ∞ |wk | ≤ k=0 k=0 We can therefore continue estimating (60) to get ⎞ 12 ⎛ ⎛ N cjk zj wk | ≤ M ⎝ | N |zj |2 ⎠ ⎝ j=0 j,k=0 N ⎞ 12 |wj |2 ⎠ . 2) follows by letting N tend to infinity. The computation in the proof of this theorem differs when the coefficients of the matrix C are instead given by gˆ(j − k). Suppose the sequences z and w are equal. Then we obtain ∞ j,k=0 1 cjk zj z k = 2π 2π 0 ∞ zj z k e j,k=0 −i(j−k)t 1 g(t)dt = 2π ∞ 2π 0 | j=0 zj e−ijt |2 g(t)dt. 42 1.

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