**SOLUTIONS**

**Chapter 1**

**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 *e*^{2πik/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 *e ^{iωt}* and

*e*terms.

^{−iωt}**1.8** Check closure under addition.

**1.9** Consider whether *c***x** is in R^{n} for a complex constant *c*.

**1.11** Use (*p* + *q*)^{2} ≤ 2*p*^{2} + 2*q*^{2} to show that

and also that **x** + **y** is in *L*^{2}(N) if **x** and **y** are in *L*^{2}(N), and thus obtain closure under addition.

To show that *L*^{2}(N) *L*^{∞}(N) argue that Σ_{k}*x*^{2}_{k} < ∞ implies the existence of some bound *M* so that |*x _{k}*| ≤

*M*for all

*k*.

**1.13** Use equation (1.22).

**1.14** Use |e^{iωt}| = 1 if *ω* and *t* are real, and *e ^{a}*

*e*=

_{b}*e*

^{a+b}.

**1.15** Use Euler’s identity on each piece, e.g., cos(*αx*) = (*e ^{iαx}* +

*e*)/2.

^{−iαx}**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 **E**_{6,1} versus **E**_{6,5} and **E**_{6,2} versus **E**_{6,4}.

**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 ...