Fourier Series

hx 12 fx − t gt dt , (1.7.7)−then what is its FS?If the FS for hx is written in the form ∑ n−c n e inx , then the aim is to determine thecoefficients c n . Direct substitution giveshx ~ 12 hx 12 12 12 −−∑n−∑n− ∑ ∑n− m−∑n− n e inx−t∑m− m e imt∑ n m e inx−t e imtm− n m e inx−∑ n m e inx 2 mn m−eim−ntdtdtdtfrom 1.8.4~ ∑ n n e inx . (1.7.8)n−Thus c n n n and this result has an important corollary. Suppose that a FS is obtainedwith the coefficients written in a product form, as in (1.7.8) and with each component formrecognisable. Then the function that the FS represents is given by the integral (1.7.7).Another corollary stemming from (1.7.8) and (1.7.7) is that fx − t gt dt −−ft gx − t dt . (1.7.9)This must be true, since the resultant (1.7.8) does not distinguish between the two possibleinput forms. The structure (1.7.7) is known as a Convolution.39