


In our discussion in Parts 25, we have focused on periodic functions with period 2*pi. This was because the sine and cosine functions represent the most basic periodic motions, and those functions have period 2*pi. However, most periodic functions have periods other than 2*pi. Think about the period of the pressure function corresponding to a note played on a guitar that we examined in Part 2.5.
This function has a fundamental period approximately 1/440 seconds.
In this part of the module, we will investigate how to extend our approximation strategy to include periodic functions with arbitrary periods.
As we noted previously, we used 2*piperiodic trigonometric polynomials to approximate 2*piperiodic functions. Now suppose we have a function that is 2Lperiodic. (Even for an arbitrary period, we will find it convenient to express the period as a multiple of 2.) In particular, suppose the function we want to approximate has period 10, i.e., L = 5. So, if we look at the graph of this periodic function, it will complete one whole period on the interval from 5 to 5.
We'll begin by deciding what basic trigonometric functions should we use for our approximations in this setting.
The question now is how to define and calculate the coefficients a_{0}, a_{k }, and b_{k} in this general setting.
Give a geometrical interpretation of the value of a_{0}, i.e., what are you finding when you integrate a function over an interval and divide the result by the length of the interval?
The 1/pi factor in the front of the integrals resulted from the calculations
by using a substitution to transform the integrals to ones previously evaluated. Use your symbolic algebra system to verify your calculations.
or
using the formulas
Now we examine the approximations of periodic functions of arbitrary period obtained using these trigonometric polynomials.
and extended to be periodic of period 2. Construct the approximating polynomials, and compare the graphs of the polynomials to the graph of the original function.
and extended to be periodic of period 1/110. Construct the approximating polynomials, and compare the graphs of the polynomials to the graph of the original function.
Sometimes the "natural" domain of a periodic function is not an interval that is symmetric about the origin. For example, in the next part of this module, we will consider the periodic variations in the Canadian lynx population during the 19^{th} century. Here, a natural periodic interval of the population as a function of time is the interval between the years 1821 to 1861. We now consider how to go about defining approximating trigonometric polynomials in this case.
To be more precise, suppose that we can tell that a function f is periodic and that it completes one fundamental variation over the interval [c, d]. Thus f is periodic with period d  c and not periodic with any shorter period.
and extended to be periodic of period 3. Construct the approximating polynomials, and compare the graphs of the polynomials to the graph of the original function. Show at least three such comparisons.



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