chapter1

It is important that we work in a vector space that lends itself to applications in digital imaging and signal processing. Unlike $mathbb{R}^{N}$, where elements of the space are $N$-tuples $mathbf{v} = (v_1, ldots,v_N)^T$, elements of our space will be functions.

Definition 1.3 (The Space $L^{2}(mathbb{R})$): We define the space $L^{2}(mathbb{R})$ to be the set

$$L^{2}(mathbb{R}) = left{f:mathbb{R}to mathbb{C}|,, int_{mathbb{R}}{|f(t)|^{2}dt}<inftyright}$$

Definition 1.4 (The $L^{2}(mathbb{R})$ Norm): Let $f(t) in L^{2}(mathbb{R})$. Then the normo of $f(t)$ is

$$|f(t)| = left(int_{mathbb{R}}{|f(t)|^2dt}right)^{frac{1}{2}}$$

The norm of the function is also referred to as the energy of the function. Since it is a measure of energy of size, then it should be nonnegative. Moreover, it is natural to expect that the only function for which $|f(t)| = 0$ is $f(t) = 0$. But hold on …

Equivalent Functions: Two functions $f(t)$ and $g(t)$ are said to be equivalent if $f(t) = g(t)$ except on a set of measure 0.

Definition 1.2 (Function for Which $|f(t)| = 0$): Suppose that $f(t)in L^{2}(mathbb{R})$. Then $|f(t| = 0$ if and only if $f(t) = 0$ except on a set of measure 0.

Our first example of elements of $L^{2}(mathbb{R})$ introduces functions that are used throughout the book.

Example 1.2 (The Box $sqcap(t)$, Triangle $land(t)$, and Sinc Functions)

We define the Box function

$$sqcap(t) = left{ begin{align*} &1, quad& 0leq t < 1 \ &0, quad& text{otherwise} end{align*} right.$$

the Triangle function

$$land(t) = left{ begin{align*} &amp;t, quad&amp; 0leq t &lt; 1 \ &amp;2 - t, quad&amp; 1 leq t &lt; 2 \ &amp;0, quad&amp; text{otherwise}</p> <p>end{align*} right.$$

the Sinc function

$$sinc(t) = left{ begin{align*} &amp;1, quad &amp; t = 0 \ &amp;frac{sin{t}}{t}, quad &amp;text{otherwise} end{align*} right.$$

Figure 1.4 The functions $sqcap(t), land(t)$, and $text{sinc}(t)$

Try to show these functions are in $L^{2}(mathbb{R})$ —> Page 9

Example 1.3 (Functions in $L^{2}(mathbb{R})$) Determine whether or not the following functions are in $L^{2}(mathbb{R})$. For those functions in $L^{2}(mathbb{R})$, compute their norm.

1. $f_1(t) = t^n$, where $n = 0, 1, 2, dots$
2. $f_2(t) = t^2sqcap(t/4)$, where $sqcap(t)$ is the box function defined in Example 1.2
3. $f_3(t)$ = $left{begin{align} frac{i}{sqrt{t}},& quad tgeq 1 \ 0,& quadtext{otherwise} end{align}right.$
4. $f_4(t)$ = $left{begin{align} frac{1}{t},& quad tgeq 1 \ 0,& quadtext{otherwise} end{align}right.$

The support of a function plays an important role in the theory we develop.

Definition 1.5 (Support of a Function): Suppose that $f(t) in L^{2}(mathbb{R})$, We define the support of $f$, denoted $supp(f)$, to be the set

$$text{supp}(f) = left{t in mathbb{R},,|,,f(t) ne 0right}$$

Some examples to better illustrate Definition 1.5.

Example 1.4 (Examples of Function Support) Find the support of each of the following functions:

1. $f(t) = frac{1}{1 + |t|}$
2. $sqcap{(t)}$
3. $land(t)$
4. $g(t) = sum_{k = 0}^{infty}{c_kland(t-2k)}$, where $c_k ne 0, kin mathbb{Z}$, and $sum_{k = 0}^{infty}{c_k^2 < infty}$

Definition 1.6 (Functions of Compact Support) Let $f(t) in f(t) in L^{2}(mathbb{R})$. We say that $f$ is compactly supported if $text{supp}(f)$ is contained in a closed interval of finite length. In this case we say that the compact support of $f$ is the smallest closed interval $[a, b]$ such that $text{supp}(f) subseteq [a, b]$. This interval is denoted by $overline{text{supp}(f)}$.

From Example 1.4 .

$$overline{text{supp}(land)} = [0, 2]\ overline{text{supp}(sqcap)} = [0, 1] \$$

We stated that we want functions that tend to $0$ as $ttopminfty$ in such a way that $|f|^2 < infty$. The following proposal shows a connection between the rate of decay and the finite energy of a function in $L^{2}(mathbb{R})$.

Proposition 1.3 (Integrating the “Tails” of an $L^{2}(mathbb{R})$ Function) Suppose that $f(t) in L^{2}(mathbb{R})$ and let $epsilon > 0$. Then there exists a real number $L>0$ such that

$$int_{-infty}^{-L}{|f(t|^2dt} + int_{L}^{infty}{|f(t|^2dt} = int_{|t| &gt; L}{|f(t|^2dt} &lt; epsilon$$

Proof see Page 13 (Very basic!)

Since we are measuring everything using the norm, it is natural to view convergence in this light as well.

Definition 1.7 (Convergence in $L^{2}(mathbb{R})$) Suppose that $f_{1}(t), f_{2}(t), dots $ is a sequence of function in $L^{2}(mathbb{R})$. We say that $left{f_{n}(t)right}_{ninmathbb{N}}$ converges to $f(t) in L^{2}(mathbb{R})$ if for all $epsilon > 0$, there exists $L geq 0$ such that whenever $n > L$, we have $|f_{n}(t) - f(t| < epsilon$ .

We also say that the sequence of functions converges in norm to $f(t)$

Example 1.5 (Convergence in $L^{2}(mathbb{R})$) Let $f_{n}(t) = t^{n}sqcap(t)$. Show that $f_{n}(t)$ converges (in an $L^{2}(mathbb{R})$ sense) to $0$.

Solution see Page 16