UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 3

Transformer för beräkningar 2001 04 16 01trp3

3.1. Let a and b be two vectors in l^{2}(Z_{3}), a given by its Fourier transform â = (1, 2, 0), and b = (1, ω, ω^{2}), where ω = e^{−2πi/3} = −\frac{1}{2} − \frac{i}{2} \sqrt{3}.

(a) Calculate b̂.

(b) Calculate a ∗ b. Tenta 2000 05 22:1

3.2. Study the convolution equation b ∗ z = w, where b is a given vector in l^{1}(Z) (the black box), z is the input signal and w is the output signal. The problem is to decide which output signals w are possible with certain black boxes b.

(a) Let b = δ_{0} +δ_{1} +δ_{2}, w = δ_{0} +δ_{1}. Prove that there is no solution z to the equation b ∗ z = w such that z(j) = 0 for all large |j|.

(b) Now let b = δ_{0} + δ_{1}, w = δ_{0} + 2δ_{1} + δ_{2}. Prove that there exists an input signal z such that b ∗ z = w and such that z(j) = 0 for all large values of |j|.

(c) Finally let b = δ_{0} + δ_{1}, w = δ_{0}. Find all solutions z ∈ l^{∞}(Z) to b ∗ z = w.

Tenta 2000 05 22:2

3.3. Suppose we know that b ∈ l^{2}(Z_{6}) satisfies b̂(0) = 0, b̂(k) ≠ 0, k = 1, ..., 5.

(a) What is the dimension of the space of input signals producing the zero output signal, in other words, what is the dimension of the space

{z ∈ l^{2}(Z_{6}); b ∗ z = 0}?

(b) Determine the dimension of all possible output signals, i.e., the dimension of the

space

{b ∗ z; z ∈ l^{2}(Z_{6})}.

Tenta 2000 05 22:3

3.4. Let u = (2, 0, 1), v = (0, 4, 3), z = (2, 0, 0, 4, 1, 3). Compute û, v̂ and then ẑ using the fast Fourier transform. Tenta 2000 05 22:4

3.5. Prove that the convolution equation a ∗ z = δ cannot be solved with z ∈ l^{2}(Z) if a(0) = a(1) = 1 and a(k) = 0 for k ∈ Z, k ≠ 0, 1. Here δ is the Dirac δ, the vector with 1 at position zero and 0 at all other slots. Are there solutions in some other space l^{p}(Z)?