


In the theory of Taylor series we investigate the approximation of general functions by polynomials. Now we want to consider the approximation of general 2*piperiodic functions by functions A of the form
By analogy with the Taylor approximation theory, these functions are called trigonometric polynomials.
In this part we concentrate on the trigonometric polynomials themselves. Suppose we "know" the polynomial  in the sense that we have a list of values for the polynomial for a range of values of t. How can we find the coefficients a_{0}, a_{1}, b_{1}, ... ? Our approach will be to calculate (numerically) various integrals involving the function. In Part 3 of this module you established the following:
for any integer n,  
for n a nonzero integer,  
for all integers m and n,  
for all integers m and n with m not equal to n,  
for all nonzero integers m and n. 
So suppose we know that a trigonometric polynomial A is of the form
For simplicity, we will start by assuming that A has period 2*pi, that is w = 1. We will also assume that A can be written as a combination of just the first two harmonics, that is,
You will need the following information about C:
Once you have determined the coefficients, use them to define a trigonometric polynomial and compare its graph to that of C.
3. Write general formulas for finding the values of a_{0}, and a_{k} and b_{k} for k greater than or equal to 1.



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