Fourier, part 4

Fourier Approximations and Music

Part 4: Fourier Coefficients for Trigonometric Polynomials of Period 2*pi

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*pi-periodic 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 a₀, a₁, b₁, ... ?

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

. We want to discover integral formulas for the coefficients a_k and b_k.

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,

Your task will be to find the values of the a's and b's.

1. Here is the information you will need about A:

(a) Using the integral formulas developed in Part 3 along with the values of the integrals given above, calculate by hand the the integral of the function

to determine the value of a₀.

(b) Find the value of a₁. This is a two step process which first involves multiplying A(t) by cos(t) and then integrating the product from -pi to pi. Again you will have to use the formulas from Part 3 together with the given facts about A.

(d) Let B represent the function

where the coefficients are the ones you determined in (a), (b) and (c), and compare your graph to the graph of A given in your worksheet. If the two graphs do not agree, recalculate the coefficients and compare the graphs again.

2. In this part you will again calculate the values of the a's and b's for a trigonometric polynomial C. Assume that C has period 2*pi and can be written as a combination of the first three harmonics, i.e,

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 a₀, and a_k and b_k for k greater than or equal to 1.

modules at math.duke.edu

	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.