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Review of Fourier Series

Part 1: Approximation of 2 pi-Periodic Functions

We'll begin with the approximation of 2 pi-periodic functions. The formulas are simplest in this case since the building blocks are the sine and cosine functions. So suppose f is a continuous 2 pi-periodic function. We approximate f by "trigonometric polynomials" of the form

.

The coefficients are defined by

  1. In your worksheet, graph the given 2 pi-periodic function.

  2. Look at the Fourier approximations from 1 to 10. How closely do the trigonometric polynomials approximate the function? Which points on the graph of the given function are most difficult to approximate?

  3. Let P be the trigonometric polynomial

    .

    Use the integral formulas to calculate the Fourier coefficients a0, a1, a2, b1, and b2. How are the Fourier coefficients related to the polynomial?

  4. Calculate

    .

    Where did the factor of pi come from? Explain why the formulas for ak have the appropriate factor (1/pi) in front of the integral.

  5. Give a justification for the factor of 1/pi in the formula for bk that is similar to the one given in Step 4 for ak.

  6. Now we consider the approximation of a discontinuous function. Let f be the function defined by

    for x between -pi and pi (where int is the greatest integer function) and extended to be 2 pi-periodic. Here is the graph of the extended function.

    Graph of f

    Graph of f

    Compare the graphs of the first 10 approximations with the graph of f. Pay attention to the points of discontinuity. To what do the approximating functions converge at these points? How do the approximations behave near the points of discontinuity?

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