Answer to Question #17536 in Real Analysis for mary

Question

a) Let f(x) be a C^1 function of period 2pi. Show that we may as well assume that f(0) = 0 and we need only show that the fourier series converges to zero at x = 0
b) Let g(x) = f(x)/(e^ix-1). Show that g(x) is a continuous function
c) Let Cn be the complex fourier coefficient of f(x) and Dn the coefficients of g(x). Show that Dn ---> 0
d) Show that Cn = Dn-1 - Dn so that the series sigma (Cn) is telescoping
e) deduce that the fourier series of f(x) at x = 0 converges to zero

Answer 1

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