UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 2

Transformer för beräkningar 2000 05 01 00trp2

2.1. Let U, V and W be subspaces of an inner product space X, and suppose that U ⊥ V, U ⊥ W, and that U + V = U + W. Prove that V = W.

2.2. Prove that for any z ∈ l^{2}(Z_{N}), we have { \atop U(z)}(n) = ẑ(n) for all n, and that, if N is even, { \atop D(z)}(n) = \frac{1}{2}(ẑ(n) + ẑ(n + \frac{1}{2}N)) for all n. Here U and D denote upsampling and downsampling, respectively.

2.3. Let u ∈ l^{2}(Z_{N}) and define v(n) = u(n) + u(n + \frac{1}{2}N), assuming N even. Prove that v̂(n) = û(2n) for all n.

2.4. Define z = (1, 2, 3, 4, 5, 6, 7, 8) ∈ l^{2}(Z_{8}). Calculate P_{−j}(z) and Q_{−j}(z), j = 1, 2, 3, in the Haar wavelet system. Notation as in Frazier.

2.5. For any z ∈ l^{∞}(Z) let us denote by K(z) the smallest interval I ⊂ Z such that z(j) = 0 for all j ∉ I. (K(0) = Ø.) Prove that if K(z) and K(w) are of finite length, then

K(z ∗ w) = K(z) + K(w).

(An easy form of Titchmarsh’ theorem on supports.) Show by an example that this may not be true if K(z) or K(w) is infinitely long. But it is true if K(z) and K(w) are contained in the set of nonnegative integers.

2.6. 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)?

2.7. Prove that ẑ is a continuous function, ẑ ∈ C(R), if z ∈ l^{1}(Z). Here ẑ(t) = ∑ z(k)e^{ikt}, t ∈ R.

2.8. Prove that if f ∈ C^{2}(R) is periodic with period 2π, then f̌ ∈ l^{1}(Z). Here f̌(n) = (2π)^{−1} ∫_{−π}^{π} f(t)e^{−int}dt, t ∈ R.

2.9. Prove that the convolution equation a ∗ z = b can be solved with z ∈ l^{1}(Z) if â, b̂ ∈ C^{2}(R) and have period 2π, and if in addition we assume that there exists an ε > 0 such that |â(t)| ⩾ ε for all t such that b̂(t) ≠ 0.

2.10. Prove that the convolution equation z∗z = δ has exactly two solutions z ∈ l^{1}(Z), but infinitely many solutions z ∈ l^{2}(Z).