FUNCTION NORMS

$$\begin{eqnarray*} L_1-\mbox{norm} \qquad \vert\vert f-q\vert\vert _1&=& \int^b_... ...q\vert\vert _{\infty} &=& \max_{a\le x\le b}\vert f(x)-q(x)\vert \end{eqnarray*}$$

$w(x)$ is a ``weight'' function, which provides some flexibility in measuring closeness. In all cases consider $w(x)$ continuous and non-negative on $(a,b), \displaystyle \int^b_a w(x)dx$ exists and $$\int^b_a w(x)dx>0$$.

Remark: A sequence of functions $\{g_k(x)\}^{\infty}_{k=1}$ is said to converge to $g(x)$ with regards to a given norm $\vert\vert\cdot\vert\vert\iff$

$$\lim_{k\to\infty}\vert\vert g_k-g\vert\vert=0$$

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Example First, a definition. Suppose we want to approximate the ``zero'' function

def: $Z(x)\equiv 0$ (zero function)     $x\in [a,b]$.

Let $f(x)=Z(x)$ let $w(x)=1$ for $x\in [a,b]\quad a=0, b=3$. For any $k>0$ let $f_k(x)$ be given by

$$f_k(x)\left\{\begin{array}{ll} k(k^2x-1) & \frac{1}{k^2}\le ... ...k^2}\le x\le 3/k^2\\ 0 & \mbox {otherwise} \end{array}\right.$$

here $k \ge 1$. See Figure 27 for an illustration.

Figure 27: Approximation of $Z(x)$ with $f_k(x)$.

We obtain the following from these norms:

$$\vert\vert Z-f_k\vert\vert _1=\frac{1}{k}$$

$$\vert\vert Z-f_k\vert\vert _2=\frac{\sqrt{2}}{\sqrt{3}}$$

$$\vert\vert Z-f_k\vert\vert _{\infty}=k.$$

So, the $L_1$-norm drops as $k$ increases, the $L_2$-norm stays constant, the $L_{\infty}$-norm grows as $k$ increases. Which norm should we use?

Now, consider $\left\{\vert\vert f-f_k\vert\vert\right\}^{\infty}_{k=1}$ a sequence of real numbers and see

$$\begin{eqnarray*} &&\lim_{k\to \infty} \vert\vert Z-f_k\vert\vert _1=0\\ &&\l... ...&&\lim_{k\to \infty} \vert\vert Z-f_k\vert\vert _{\infty}=\infty \end{eqnarray*}$$

therefore the sequence $$\{f_k(x)\}^{\infty}_{k=1}$$ converges to $Z(x)$ ONLY in the $\vert\vert\cdot\vert\vert _1$, norm.

In many practical cases we would probably accept $f_k(x)$ for large $k$ as a ``good'' approximation for $f=Z(x)$ since it is only ``bad'' in a small neighborhood of a single point. However, there are practical problems in which even moderate errors in a small neighborhood are unacceptable.

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A VERY USEFUL FACT ABOUT THESE 3 NORMS

(I)

$$\begin{eqnarray*} \vert\vert f-q\vert\vert _1 =\int^b_a\vert f(x)-g(x)\vert w(x... ...\cdot\vert\vert _{\infty} \left(\int^b_a w(x)dx\right)^{\frac12} \end{eqnarray*}$$

$\therefore$ if $\vert\vert\cdot\vert\vert _{\infty}$ small then both $\vert\vert\cdot\vert\vert _1$ and $\vert\vert\cdot\vert\vert _2$ are smaller. $\therefore \vert\vert\cdot\vert\vert _{\infty}$ is stronger than the other 2 norms. Usually, we strive for this in our work.

(N.B. Advanced calculus students recognize $\vert\vert\cdot\vert\vert _\infty$-norm is equivalent to uniform convergence.) $\Box$