Homework Set 7
Math/ECE 430
due Friday, March 15, 2013 or Monday, March 25, 2013
Show that the following properties of the Fourier transform hold. Check for errors
involving factors of 2π.
1. fb(ω) = [fb(−ω)]. Here the left side is the Fourier transform of the complex conjugate
of f .
2. If f is real-valued, then fb for negative frequencies is determined by fb for positive
frequencies. (What is the formula?)
\
3. [f (t
− t0 )](ξ) = e−i2πξt0 fb(ξ)
R
4. |fb(ω)| ≤ |f (t)|dt
\
5. λf
(λt) = fb(ω/λ)
R
6. If g(t) = h(t − t0 )f (t0 )dt0 , then gb = b
hfb. (Are the factors of 2π correct?)
\ (ω)
7. (d/dω)fb(ω) = −2πitf
Rt
8. If g(t) = −∞ f (t0 )dt0 , then gb(ξ) = (2πiξ)−1 fb(ξ) for ξ 6= 0. It might be easier to show
that gb0 (ξ) = 2πiξ fb(ξ), which is equivalent (why?).
9. The Parseval relation:
2π?)
R
f (t)g(t)dt =
R
fb(ω)b
g (ω)dω. (Should there be any factors of
10. If (d/dω)fb is integrable, then for some constant c, |f (t)| ≤ c/t for large t. Similarly,
if (d/dω)n fb is integrable, then |f (t)| ≤ c/tn for large t.
1