Solved: This problem indicates a proof of convergence of a Fourier series under

Chapter 10, Problem 18

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This problem indicates a proof of convergence of a Fourier series under conditions more restrictive than those in Theorem 10.3.1. (a) If f and f are piecewise continuous on L x < L, and if f is periodic with period 2L, show that nan and nbn are bounded as n . Hint: Use integration by parts. (b) If f is continuous on L x L and periodic with period 2L, and if f and f are piecewise continuous on L x < L, show that n2an and n2bn are bounded as n . If f is continuous on the closed interval, then it is continuous for all x. Why is this important? Hint: Again, use integration by parts. (c) Using the result of part (b), show that n=1 |an| and n=1 |bn| converge. (d) From the result in part (c), show that the Fourier series (4) converges absolutely7 for all x.