108Chapter 7The Fourier Transform and its ApplicationsMultiplying both sides by±π2and using the linearity of the Fourier transform, itfollows thatF(²π2f(x))(w)=g(w).SoFg=FF³²π2f(x)´=²π2f(x),by the reciprocity relation. Using the symbolwas a variable, we getF³sinaxx´=²π2f(wµ ±π2if|w|<a,0otherwise.17.(a) Consider ±rst the casea>0. Using the de±nition of the Fourier transformand a change of variablesF(f(ax))(w1√2π¶∞-∞f(ax)e-iωxdx=1a1√2π¶∞-∞f(x)e-iωaxdx(ax=X, dx=1adX)=1aF(f)(wa).Ifa<0, thenF(f(ax))(w1√2π¶∞-∞f(ax)e-iωxdx=1a1√2π¶-∞∞f(x)e-iωaxdx=-1aF(f)(wa).Hence for alla±= 0, we can write=1|a|F(f)·ωa¸.(b) We haveF(e-|x|)(w²2π11+w2.By (a),F(e-2|x|)(w12²2π11+(w/2)2=²2π24+w2.(c) Let
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.