UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 1

Transformer för beräkningar 2000 04 07 00trp1

1.1. Suppose we know that a ∈ l^{2}(Z_{N}) satisfies â(0) = 0, â(k) ≠ 0, k = 1, ..., N − 1. Can you determine the dimension of

{z ∈ l^{2}(Z_{N}); a ∗ z = 0}?

Can you determine the dimension of

{a ∗ z; z ∈ l^{2}(Z_{N})}?

Generalize to an arbitrary convolution equation a ∗ z = b, where a, b are given vectors, z an unknown vector.

1.2. Calculate ẑ when z = (1, 2, 3, 4)^{T} ∈ l^{2}(Z_{4}). Calculate ŵ when

w = (1, 2, 3, 4, 5, 6)^{T} ∈ l^{2}(Z_{6}).

1.3. Define a mapping T: l^{2}(Z_{4}) → l^{2}(Z_{4}) by

T (z) = (2z(0) − z(1), iz(1) + 2z(2), z(1), 0).

Is T linear? Is T translation invariant? Calculate T (R_{1}z) and R_{1}(T z) for z = (1, 0, −2, i)^{T}.

1.4. Let S, T be two translation-invariant linear mappings of l^{2}(Z_{N}) into itself. Prove that they commute, i.e., that T ∘ S = S ∘ T. Deduce that circulant matrices always commute—try also to prove this fact by a direct calculation using the definition of circulant matrices.

1.5. Prove that T (z ∗ w) = T (z) ∗ w = z ∗ T (w) if T is a translation-invariant linear mapping of l^{2}(Z_{N}) into itself.

1.6. Prove that a linear mapping T of l^{2}(Z_{N}) into itself is translation invariant if and only if it is a polynomial in the translation operator R_{1}.

1.7. Let u = (1, 3)^{T}, v = (0, 4)^{T}, z = (1, 0, 3, 4)^{T}. Compute û, ŵ and then ẑ using the fast Fourier transform. Compute the Fourier transform of w = (0, 1, 4, 3)^{T}.

1.8. Compute the Fourier transform of w = (1, 0, 1, 0, 1, 0, 1, 0)^{T}—maybe using z = (1, 1, 1, 1)^{T}.