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A repost from alt.uu.future,alt.education.distance:

From: esz001@cck.coventry.ac.uk (Will Overington)
Message-ID: <dv1mbwhk@cck.coventry.ac.uk>
Date: 20 Jul 92 11:44:46 GMT
Organization: Coventry University, Coventry, UK


LEARNING MATERIAL UNDER DEVELOPMENT

CHEBYSHEV POLYNOMIALS

UNIT 1

20th July 1992

This course is about a set of polynomials called Chebyshev
Polynomials.  Chebyshev was Russian and thus the name Chebyshev
as used here is the particular transliteration into Roman
characters that is normally used in the West.  The gentleman's
original name, which was written using Cyrillic characters,
(which is the name given to the alphabet in which Russian and
some other languages are written), can also be transliterated so
as to begin with a letter T, and this letter T is, in fact, the
letter used to denote the Chebyshev Polynomials.

There are an infinite number of Chebyshev Polynomials, though one
always uses a finite number of them, maybe two, maybe seven,
maybe some other number as the application requires.

In this first unit, the first five Chebyshev Polynomials are
introduced.  It could have been four, it could have been six.
There is no great significance in the choice of five.  Five was
chosen so as to give scope to start exploring Chebyshev
Polynomials without being overburdened to start off.

In the description of these polynomials, due to using ordinary 7
bit ASCII representation, the number immediately following the T,
which should really be a subscript, is shown full size.  For
example T2 should really be T followed by a small, subscripted 2.

Here are the first five Chebyshev Polynomials.

T0(x) = 1

T1(x) = x

T2(x) = 2 * x^2  -  1

T3(x) = 4 * x^3  -  3 * x

T4(x) = 8 * x^4  -  8 * x^2  +  1

You might like to try the following exercises.

Draw graphs of y=T0(x), y=T1(x), y=T2(x), y=T3(x) and y=T4(x) in
the range x=-1.2 to x=+1.2 in steps 0.1 apart in x.  That is,
x=- 1.2, x=-1.1, x=-1.0, x=-0.9, and so on.

In use, Chebyshev Polynomials are only used in the range x=-1.0
to x=+1.0, though one can use scaling so that, for example, if
you wish to use them over a range of -10.0 to +10.0, then you may
use y=T2(x/10.0) and so on.

For each of these graphs, using only the range x=-1.0 to x=+1.0,
write down the maximum and minimum values of y.

For each of these graphs, write down the value of y for x=1.0,
x=-1.0 and x=0.0, displaying the results in a table.

A set of tables of x, x^2, x^3 and x^4 are included to help you
work out the values.

       x        x^2       x^3       x^4

   -1.2000    1.4400   -1.7280    2.0736
   -1.1000    1.2100   -1.3310    1.4641
   -1.0000    1.0000   -1.0000    1.0000
   -0.9000    0.8100   -0.7290    0.6561
   -0.8000    0.6400   -0.5120    0.4096
   -0.7000    0.4900   -0.3430    0.2401
   -0.6000    0.3600   -0.2160    0.1296
   -0.5000    0.2500   -0.1250    0.0625
   -0.4000    0.1600   -0.0640    0.0256
   -0.3000    0.0900   -0.0270    0.0081
   -0.2000    0.0400   -0.0080    0.0016
   -0.1000    0.0100   -0.0010    0.0001
    0.0000    0.0000    0.0000    0.0000
    0.1000    0.0100    0.0010    0.0001
    0.2000    0.0400    0.0080    0.0016
    0.3000    0.0900    0.0270    0.0081
    0.4000    0.1600    0.0640    0.0256
    0.5000    0.2500    0.1250    0.0625
    0.6000    0.3600    0.2160    0.1296
    0.7000    0.4900    0.3430    0.2401
    0.8000    0.6400    0.5120    0.4096
    0.9000    0.8100    0.7290    0.6561
    1.0000    1.0000    1.0000    1.0000
    1.1000    1.2100    1.3310    1.4641
    1.2000    1.4400    1.7280    2.0736