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