Question

In this exercise, we compare the inner products and norms introduced in Problems 32 and 34. Let’s denote the two norms by ${||\mathbf{f}||}_{\mathbf{32}}$and ${||\mathbf{f}||}_{\mathbf{34}}$, respectively.

1. Compute ${||\mathbf{t}||}_{\mathbf{32}}$and ${||\mathbf{t}||}_{\mathbf{34}}$.Which is larger? Explain the answer conceptually. Graph the weight functions${\mathbf{w}}_{\mathbf{32}}\left(\mathbf{t}\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}$and ${\mathbf{w}}_{\mathbf{34}}\left(\mathbf{t}\right)\mathbf{=}\frac{\mathbf{2}}{\mathbf{\pi }}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}$on the same axes. Then graph the functions ${\mathbf{w}}_{\mathbf{32}}\left(\mathbf{t}\right){\mathbf{t}}^{\mathbf{2}}$and ${\mathbf{w}}_{\mathbf{34}}\left(\mathbf{t}\right){\mathbf{t}}^{\mathbf{2}}$on the same axes.

Short answer

The value of ${||t||}_{32}$is$0.577$and the value of${||t||}_{34}$is $0.5$so, ${||t||}_{32}>{||t||}_{34}$.

Step-by-step solution

The given information.

The inner product introduced in problem 32 is

And the inner product introduced in problem 34 is

To find which norm is larger t32andt34.

Consider,

${||t||}_{32}$

Substitute the value as follows:

$\begin{array}{c}{||t||}_{32}=\sqrt{\frac{1}{2}\underset{-1}{\overset{1}{\int }}t\times tdt}\\ =\sqrt{\frac{1}{2}\underset{-1}{\overset{1}{\int }}{t}^{2}dt}\\ =\sqrt{\frac{1}{2}{\left[\frac{t^3}{3}\right]}_{-1}^1}\\ =\sqrt{\frac{1}{2}\times \frac{1}{3}\left[{\left(1\right)}^3-{\left(-1\right)}^3\right]}\end{array}$

Further simplify as follows:

$\begin{array}{c}{||t||}_{32}=\sqrt{\frac{1}{6}\left[1+1\right]}\\ =\sqrt{\frac{1}{6}\times 2}\\ =\sqrt{\frac{1}{3}}\\ =0.577\end{array}$

Now, consider

${||t||}_{34}$

Substitute the value as follows:

$\begin{array}{c}{||t||}_{34}=\sqrt{\frac{2}{\pi }\underset{-1}{\overset{1}{\int }}\sqrt{1-t^2}\times t\times tdt}\\ =\sqrt{\frac{2}{\pi }\underset{-1}{\overset{1}{\int }}{t}^2\sqrt{1-t^2}dt}\end{array}$

On further simplification we get,

$\begin{array}{c}{||t||}_{34}=\sqrt{\frac{1}{4}}\\ =\sqrt{\frac{1}{2\times 2}}\\ =\frac{1}{2}\\ =0.5\end{array}$

Thus,${||t||}_{32}$ is larger.

To draw the graph of the weight functions w32t=12 andw34t=2π1-t2 on the same axes.