Page 383 - Schaum's Outline of Theory and Problems of Advanced Calculus

374 FOURIER INTEGRALS [CHAP. 14

2
@U @ U
(b)Prove directly that the function in (a)satisfies ¼ and the conditions of Problem 14.13.
@t @x 2

1 jxj < 1
0 jxj > 1
14.27. Verify the convolution theorem for the functions f ðxÞ¼ gðxÞ¼ .
14.28. Establish equation (4), Page 364, from equation (3), Page 364.
14.29. Prove the result (12), Page 365.
ð ð
1 1 i u 1 1 i v
Hint: If Fð Þ¼ p ffiffiffiffiffiffi f ðuÞ e du and Gð Þ¼ p ffiffiffiffiffiffi gðvÞ e dv, then
2 1
2 1
1 ð 1 ð 1
e i ðuþvÞ f ðuÞ gðvÞ du dv
Fð Þ Gð Þ¼
1
2 1

Now make the transformation u þ v ¼ x:
1 ð 1 ð 1 i x
e f ðuÞ gðx uÞ du dx
ffiffiffi
Fð Þ Gð Þ¼ p
1
1
Define
1 ð 1
f ðuÞ gðx uÞ du
f g ¼ p ffiffiffi ð f g is a function of xÞ
1
then
1 ð 1 ð 1 i x
e f gdx
Fð Þ Gð Þ¼ p ffiffiffi
1
1
Thus, Fð Þ Gð Þ is the Fourier transform of the convolution f g and conversely as indicated in (13)
f g is the Fourier transform of Fð Þ Gð Þ.
14.30. If Fð Þ and Gð Þ are the Fourier transforms of f ðxÞ and gðxÞ respectively, prove (by repeating the pattern of
Problem 14.29) that
ð ð
1 1
f ðxÞ gðxÞ dx
Fð Þ Gð Þ d ¼
1 1
where the bar signifies the complex conjugate. Observe that if G is expressed as in Problem 14.29 then
1 ð 1
e i x f ðuÞ gðvÞ dv
g
G Gð Þ¼ 1
14.31. Show that the Fourier transform of gðu xÞ is e i x , i.e.,
1 ð 1
e i x Gð Þ¼ p ffiffiffi 1 e i u f ðuÞ gðu xÞ du
Hint: See Problem 14.29. Let v ¼ u x.

14.32. Prove Riemann’s theorem (see Problem 14.10).