# Assignment 6

You will write a paper about using Chebyshev polynomials of the first kind to approximate functions. This will discuss the fact that there are many functions that we cannot evaluate exactly over their domains, including trigonometry functions, and that we must therefore approximate them precisely. Your paper will describe the infinity norm of a function, and will discuss the minimality property of Chebyshev polynomials with respect to that norm. It will discuss what it means to say that the set of Chebyshev polynomials is orthogonal, in particular giving the equation for the inner product of two Chebyshev polynomials. It should also give the three-term recurrence relation for Chebyshev polynomials of the first kind.

The exciting part of your paper will be a section illustrating that the polynomial \[c(x) = -0.3042T_0(x)-0.9709T_2(x)+0.3028T_4(x)\] is a pretty good approximation to \(\cos(\pi x)\) on \([-1,1]\), as shown in Figure 1. The polymials used will appear in a table that duplicates Table 1.

Figure 1: Chebyshev approximation of \(\cos(\pi x)\)

Table 1: Some Chebyshev Polynomials

Degree | Formula |
---|---|

0 | \(T_0(x)=1\) |

1 | \(T_1(x)=x\) |

2 | \(T_2(x)=2x^2-1\) |

3 | \(T_3(x)=4x^3-3x\) |

4 | \(T_4(x)=8x^4-8x^2+1\) |

The paper will be single-spaced in a 10pt font,
with 1" margins on both left and right sides.
Observe that if you use e.g
`\title{Chebyshev Approximation}\author{Chris Cougar}`
in the header of your document, and then use
`\maketitle` inside the document, then
the document class will format the title info for you.
The following things are obvious, but we note them in passing,
nonetheless.

- The paper will have a title, author information, an introduction, conclusions, and probably be divided into sections.
- The paper will be carefully written, with lucid description, proper use of grammar, spelling, and punctuation, and few typographical errors.
- The paper will include equations. References to those equations will be by number.
- The paper will include a list of references. If no references appear, the reader will assume you made everything up, which does not inspire confidence. Obviously you will not plagiarize any references.
- Papers shorter than a couple of pages or longer than five will probably not make a positive impression on the reader.
- There are some hints about mathematical writing here.
- You will read this assignment carefully to be sure that you have included everything required.

Equally obviously, the typesetting for this paper will be done using LaTeχ. The assignment is worth 60 points and is due at 9AM on Tuesday, 10 October. It is turned in when the .tex file appears as an attachment to an email message in the instructors inbox. Note that the instructor has a copy of the image, so you do not need to send that.

The "final exam" for this course will take place
at 8:00 AM on Tuesday, 12 December. This will be an ordinary
50 minute test. It will be comprehensive, but weighted toward
the latter half of the semester. As always, paper notes will
be permitted, but no electronic devices will be allowed.
A sample exam is available.

A
Solution example is available
for the quiz. The solution to
Test 1 is still available too.

The ultimate assignment is posted.