1. Fourier Approximation Related Matters Concerning Fourier Series

2. Approximating Functions A function f(x) that approximates a set of data {( x 1, y 1 ),( x 2, y 2 ),( x 3, y 3 ),…,( x n, y n )} does not require that the function when evaluated at each x j value agree with the corresponding y j value like interpolation does. Approximating functions try to have each f(x j ) come close to equaling each y j as possible with respect to some limit you put on the calculations for f(x). Most commonly for polynomial or trigonometric polynomials this is the degree of the polynomial. In all the interpolating polynomials we have studied (Lagrange, Newton and Fourier) we have seen that if you do not restrict the degree of the polynomial for a data set you can always get an interpolating polynomial if you take the degree to be large enough. The problems that arise from having the degree of these polynomials get large are: It increases how complicated the polynomial is. This makes calculations with it very difficult if you do them by hand. It increases how many calculations need to be done thus increasing the amount of time required for a calculation if doing them by machine.

3. Measuring How “Closely” a Function Approximates Data In order to measure how closely a function f(x) approximates a set of data {( x 1, y 1 ),( x 2, y 2 ),( x 3, y 3 ),…,( x n, y n )} we measure the error between the data and the function. Ideally (as in interpolation) we want this to be zero. There are many different ways this can be done but one of them that is widely used because it has many nice properties is what is called the sum of differences of squares. Notice the only way that E =0 is for f(x j ) = y j. This means that the function f(x) is an interpolating function. In statistics when you use the mean to approximate a set of data (i.e. f(x) = you call this value the variance. When you approximate the data with a line the line that makes E a minimum is called the regression line. We can apply this same idea to Fourier series.

4. Fourier Approximating Polynomials If we choose n equally spaced points { t 0, t 1, t 2,…, t n-1 } in the interval [0,2  ) then the value E below: will be minimal for the data set { x 0, x 1, x 2,…, x n-1 } (Fourier Form) when the coefficients a k and b k are chosen as shown below: This allows for any data set to be approximated with a trigonometric polynomial of degree m. Generally we assume 2 m +1 < n.

5. For example, lets approximate the data set {1,1,-1,2} with a trigonometric polynomial of degree 1(i.e. m =1). Notice that for exact interpolation this would require a trigonometric polynomial of degree 2.

6. The approximating polynomial is: