Mathematics 174 – Applied Mathematics 2 Problem Set 5 March 30, 2000

I. Compute the Fourier Transforms of each of the following functions using the indicated methods) A) (by direct computation)

f(x) = \{ \begin{matrix}1 & if |x| ≤ 1 \\ 2 & if 1 < |x| < 2 \\ 0 & otherwise \end{matrix}

B) (by direct computation)

f(x) = \{ \begin{matrix}sin(x) & if |x| ≤ \frac{π}{2} \\ 0 & otherwise \end{matrix}

C) Show that if xf(x) is in L^{1}(R), then

F[xf(x)](ω) = −i\frac{d}{dω} F[f](ω)

and use that to evaluate the Fourier Transform of f(x) = xe^{−x^{2}}. D) (by using the general scaling and modulation properties)

f(x) = cos(4x)e^{−3x^{2}}

(Answer: \frac{\sqrt{3}}{12\sqrt{π}} (e^{−(ω+4)^{2}/12} + e^{−(ω−4)^{2}/12}))

E) (by using the Fourier Inversion theorem and a known transform) f(x) = \frac{1}{1+x^{2}}

II. For real numbers a and c, let

g_{a,c} = \frac{1}{\sqrt{2πa^{2}}} e^{−(x−c)^{2}/(2a^{2})}

(a scaled shifted Gaussian). A) What are the meanings of the constants a, c in terms of the “bell-shaped” curve

y = g_{a,c}(x)? B) Show that if a, c, b, d are any real numbers, then

g_{a,c}(x) ∗ g_{b,d}(x) = g_{δ,c+d}(x)

where ∗ = convolution, and δ = \sqrt{a^{2} + b^{2}}. (Hint: Take the Fourier transform of the convolution, work with what you get, then take the inverse transform.)

III. Let f : R → C be an L^{1} function, and let F denote its Fourier transform. Show that if F (ω) = 0 for all |ω| > ω_{0}, then

f(x) ∗ \frac{sin(ax)}{πx} = f(x)

for all a > ω_{0}.

Due: In class, Thursday, April 6.