Fall 2002 Final Homework MATH 640
DUE: November 12, 2002
1. Let Vbe the subspace of L2(R)of all functions such that fand f0are piecewise continuous and vanish at ±∞.
Define the two operators Dand Mby:
(Df)(t) = f0(t),M f (t) = t f(t),f∈V.
(a) Simplify the expression DM −MD.
(b) Simpify the inner product h(M−aI)f|(D−iαI)fiusing the identity ha+b|c+di=ha|ci+ha|di+hb|ci+
hb|diand other basic properties of the inner product.
(c) Simplify the inner product h(D−iαI)f|(M−aI)fi, using the above idea.
(d) Verify the identity that was used in proving the Heisenberg Uncertainity Principle that is found at page 92
of the notes:
−h(M−aI)f|(D−iαI)fi − h(D−iαI)f|(M−aI)fi=kfk2.
2. Let f[n] = rnfor n=0,1,...,N−1 and r∈C. Suppose that ris not an N-th root of unity. Find its discrete
Fourier transform. Supply all details.
3. Write the Discrete Fourier Transform for N=6 in terms of the DFT with N=3 as in the notes on page 99; that
is, separate out the even and odd components. Next, write out the matrix factorization on page 100, equation
(4.10) explicitly as well.
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