**Sample problem.**

Let us start with a simple non-periodic function
like `f(t) = 2sin(t) - 3sin(2 ^{1/2}t)`.

The graph of

Suppose that we know that the function is of the form
`f(t) = B _{1}sin(t) + B_{2}sin(2^{1/2}t)`,
and are given the curve. How to calculate the coefficients

**Tools from trigonometry.**

- any product of sines can be rewritten as a difference of cosines.
Specifically,
`sin(v`_{1}t)sin(v_{2}t) = .5cos([v_{1}-v_{2}]t) - .5cos([v_{1}+v_{2}]t). - the long-term average value of
`cos(vt)`, for any non-zero`v`, is zero.

The graphs of`cos(t)`(blue) and of the average value of the cosine function from`0`to`t`(yellow). - It follows that the long-term average of
`sin(v`is zero unless_{1}t)sin(v_{2}t)`v`or_{1}= v_{2}`v`. We can mimic the language of ordinary Fourier series and say that under these conditions the two functions_{1}= -v_{2}`sin(v`and_{1}t)`sin(v`are_{2}t)*almost orthogonal.* - the long-term average value of
`sin`is^{2}(vt)`1/2`, for any non-zero speed`v`.

The graphs of`sin`(blue) and of the average value of the sine-squared function from^{2}(t)`0`to`t`(yellow).**The solution.****To calculate**multiply`B`:_{1}`f(t)`by`sin(t)`and compute twice the long-term average value of the product. This number must be`B`. Why? The product is_{1}`f(t)sin(t) = B`._{1}`sin`+ B^{2}(t)_{2}sin(t)sin(2^{1/2}t)

The average value is the sum of the average values of the two terms. The first has average value`B`while the second has long-term avarage value zero._{1}/2**To calculate**multiply`B`:_{2}`f(t)`by`sin(2`and compute twice the long-term average value of the product. This number must be^{1/2}t)`B`._{2}The following graph shows how the multiplications and long-term averaging tease out the coefficients

`B`and_{1}= 2`B`._{2}= -3

The graph of`f(t)`is plotted in blue, the running average of`2f(t)sin(t)`is plotted in red, and the running average of`2f(t)sin(2`is plotted in green.^{1/2}t)*erratum "sin(vt)" corrected 7/14/05.*

- Setting up the problem
- Almost orthogonality
- The complete calculation
- The disk-sphere-cylinder integrator

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© Copyright 2001, American Mathematical Society - Setting up the problem