School of Mathematics and Statistics
Te Kura Ma¯tai Tatauranga
MATH 301 Assignment 4
Due Date: Sunday 15 October before midnight.
1. Given a function f(x) defined for 0 = x l, prove that there is a unique periodic function of period l that agrees with f on the interval [0,l).
2. If the Fourier transform of f(x) is fˆ(k):
(a) Show that the Fourier transform of is given by
.
Hint: try using the transform of an integral, plus the symmetry principle — Proposition
7.10 on p.276 of Olver.
(b) Use this general result to find the Fourier transform of 1/x.
(c) Use this general result to find the Fourier transform of e-x2/x.
3. Find the Fourier transform of:
(a) e-x2/2 (b) xe-2|x|
(c) erf
4. Use the Fourier transform to find an integral formula for a solution to the Airy differentialequation .
5. Find the adjoint of the matrix
using the Euclidean dot product on R2 for the inner product on the domain; and the weighted inner product hv,wi = 2v1w1+3v2w2 on R2, for the inner product on the target space. This is from section 9.1 of Olver.
6. Prove that the adjoint of the adjoint is the original operator, that is, that L = (L*)*. This is from section 9.1 of Olver.
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