Use the Chernoff Bound to derive the upper bound (5.43).... | Filo
Use the Chernoff Bound to derive the upper bound (5.43).... | Filo
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SOLVED:A very slick proof of the pointwise convergence of Fourier series, due to P. Chernoff (American Mathematical Monthly, May 1980), goes as follows. (a) Let f(x) be a C^1 function of period 2 π. First 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^i x-1). Show that g(x) is a continuous function. (c) Let Cn be the (complex) Fourier coefficients of f(x) and Dn the coefficients of g(x). Show that Dn →0. (
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