Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 475Figure 7.6 Representation of triangular wave by Fourier series. -The right-hand member is simply the average or arithmetic mean of f (x) over theinterval -n 5 x 5 n (Section 4.1). One can also writeIn words: The line y = ;ao must be such that the area between the line and thecurve y = f (x) lying above the line equals the area between the line and the curvey = f (x) lying below the line. Thus the line y = ;ao is a sort of symmetry line forthe graph of y = f (x).From either of these points of view it is clear that in the two examples considered,one must have ;a0 = 0; the average of f (x) is 0, and there is as much areaabove the x-axis as below.Still another point of view yields the same formula for :ao. We define the totalsquare error of a function g(x) relative to f (x) as the integralThis error is 0 when g = f (or when g = f except for a finite number of points), isotherwise positive. We now seek a constant function y = go such that this error isas small as possible. In other words, we try to approximate y = f (x) as accurately-