Fourier Series

C will depend upon M. The formal problem is to minimiseE M C ab|fx − gx,C| M dxby an appropriate choice of C for a given value of M. This is usually done by regarding thecoefficients in C as continuous variables, so that this is now an optimisation problem inN −variable calculus. The minimum occurs at a stationary point, i.e. when∂E M∂c j 0 j 1,...,Nand further conditions involving second or higher order derivatives are necessary todetermine whether or not the stationary points provide the required minimum.If we choose M 1 thenE 1 C ab|fx − gx,C| dxbut the difficulty with this representation is implementation. The measure of the error issatisfactory but it can be difficult to determine, since differentiating the integrand is a littlelike differentiating the function fx |x|,whose derivative is discontinuous at x 0.ThemeasurewithM 2isE 2 C abfx − gx,C 2 dxand minimising E 2 C is called the least-squares norm. The integrand is always positiveandsosmallE 2 C corresponds to a good fit. Furthermore there is no difficulty with themathematical properties and there is a balance between goodness-of-fit and ease ofdetermination.Exercise Start withNbE 2 C fx − PN x,C 2 dx P N x,C a∑ c n n xn1and assume that the n x form an orthonormal set on a,b. Show that E 2 C is aminimum when the coefficient vector C is formed by the Fourier coefficients.1.9.3 Orthogonal PolynomialsThe general relationship (1.9.3) is often used with reference to sets of functions known asorthogonal polynomials, though attention is restricted here to those that satisfy wx 1.Such polynomials play an important role in applied mathematics and physics. They are51