If int_(RR) |f(x)|dx and int_(RR) |g(x)|dx are bounded, show that int_(RR) |(f ***g)(x)|dx <= int_(RR) |f(x)|dx * int_(RR) |g(x)|dx.

babuliaam 2022-09-23 Answered
If R | f ( x ) | d x and R | g ( x ) | d x are bounded, show that
R | ( f g ) ( x ) | d x R | f ( x ) | d x R | g ( x ) | d x
From the definition we have ( f g ) ( x ) = R f ( x τ ) g ( τ ) d τ. I guess I need to do work on this right side, but I don't see how to simplify to get the desired inequality
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Answers (1)

vyhlodatis
Answered 2022-09-24 Author has 14 answers
Applying the integral to the convolution:
| R ( f g ) ( x ) d x | = | R R f ( x τ ) g ( τ ) d τ d x | = | R R f ( x τ ) g ( τ ) d x d τ |
R R | f ( x τ ) | | g ( τ ) | d x d τ = R | f ( x ) | d x R | g ( x ) | d x
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