noo 1 Problem 5. In Chapter 3 we proved the Riemann-Lebesgue lemma: for any f E L'(T), lim \f (n)] = 0. Here we will find an alternate proof using chapter 4 material. Let f e L'(T) and let gn(x) = f(x) on(f;x), where on(f; x) = (An*f)(x) and An is the Fejr kernel. (a) Explain why if N is large enough then S \gn(x)\dx is small. (b) Explain why that implies that, for the same N, In(n) is also small for any n. (c) Show that for |n| > N, (n) = f(n). (d) Conclude that lim \f(n)] = 0. } 0 n too noo 1 Problem 5. In Chapter 3 we proved the Riemann-Lebesgue lemma: for any f E L'(T), lim \f (n)] = 0. Here we will find an alternate proof using chapter 4 material. Let f e L'(T) and let gn(x) = f(x) on(f;x), where on(f; x) = (An*f)(x) and An is the Fejr kernel. (a) Explain why if N is large enough then S \gn(x)\dx is small. (b) Explain why that implies that, for the same N, In(n) is also small for any n. (c) Show that for |n| > N, (n) = f(n). (d) Conclude that lim \f(n)] = 0. } 0 n too