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Sec. 6–10 Appendix: Proof of Schwarz’s Inequality 477
6–10 APPENDIX: PROOF OF SCHWARZ’S INEQUALITY
Schwarz’s inequality is
q
2 q
q
f(t)g(t) dt ` … ƒ f(t) ƒ 2 dt ƒ g(t) ƒ 2 dt
L-q
L -q L -q
and becomes an equality if and only if
f(t) = Kg * (t)
(6–182)
(6–183)
where K is an arbitrary real constant. f(t) and g(t) may be complex valued. It is assumed that
both f(t) and g(t) have finite energy. That is,
Proof.
L
q
-q
ƒ f(t) ƒ 2 dt 6q
and
Schwarz’s inequality is equivalent to the inequality
L
q
-q
ƒ g(t) ƒ 2 dt
6q
(6–184)
Furthermore,
L
L
q
-q
and equality holds if Eq. (6–183) is satisfied. Thus, if we can prove that
L
q
-q
q
q
f(t)g(t) dt ` … ƒ f(t)g(t) ƒ dt = ƒ f(t) ƒƒg(t)ƒ dt
L L
q
-q
q
q
2 2
f(t)g(t) dt ` … ƒ f(t) ƒ dt ƒ g(t) ƒ dt
CL C L
-q
q
q
2 2
ƒ f(t) ƒƒg(t) ƒ dt … ƒ f(t) ƒ dt ƒ g(t) ƒ dt
CL CL
then we have proved Schwarz’s inequality. To simplify the notation, we replace
ƒ g(t) ƒ by the real-valued functions a(t) and b(t) where
a(t) = ƒ f(t) ƒ
and
b(t) = ƒ g(t)ƒ
Then we need to show that
-q
-q
-q
-q
-q
(6–185)
(6–186)
(6–187)
ƒ f(t) ƒ and
(6–188a)
(6–188b)
b
q
q
a(t)b(t) dt 6 a 2 (t) dt b 2 (t) dt
(6–189)
La
C L -q CL -q
This can easily be shown by using an orthonormal functional series to represent a(t)
and b(t). Let
a(t) = a 1 w 1 (t) + a 2 w 2 (t)
(6–190a)