1.1 Color JPEG compressed images are typically 5 to 50 times smaller than they would be if stored “naively,” so the ratio of naively stored to JPEG-stored might range from a low of 0.02 to 0.2.
1.3 Work out e2πik/8 = cos(kπ/4) + i sin(kπ/4).
1.4 Use equation (1.23).
1.5 Apply Euler’s identity to cos(ωt) and sin(ωt), group the eiωt and e−iωt terms.
1.8 Check closure under addition.
1.9 Consider whether cx is in Rn for a complex constant c.
1.11 Use (p + q)2 ≤ 2p2 + 2q2 to show that
and also that x + y is in L2(N) if x and y are in L2(N), and thus obtain closure under addition.
To show that L2(N) L∞(N) argue that Σkx2k < ∞ implies the existence of some bound M so that |xk| ≤ M for all k.
1.13 Use equation (1.22).
1.14 Use |eiωt| = 1 if ω and t are real, and eaeb = ea+b.
1.15 Use Euler’s identity on each piece, e.g., cos(αx) = (eiαx + e−iαx)/2.
1.18 Recall that u · v = ||u|| ||v|| cos(θ), and that vectors are orthogonal when u · v = 0. The peak-to-peak distance in part (e) should be .
1.19 Use equation (1.22). Examine E6,1 versus E6,5 and E6,2 versus E6,4.
1.22 Use Theorem 1.8.3 for parts (b) and (d).
1.23 Use Theorem 1.8.3 for parts (a) and (c), and don’t forget ...