Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 527In general, we are led in this way to generalized functions of form 4f (x) + a16(x - c1) + a26(x - c2) + . -. + blS1(x - dl) + . . . + elSU(x - pl) + . . . ,(7.112)that is, of the form ordinary function (of the type described) plus a linear combinationof translated delta functions and translated derivatives up to a certain order of thedelta function. Terms with 0 coefficient are to be suppressed. We now confine ourattention to generalized function of this type.Operations on generalized functions. We add or subtract generalized functions ofthe type considered in the obvious way. For example,&q(ex + 6(x)) + (sinx + 6(x) + S'(x - 1)) = ex + sinx + 26(x) + 6'(x - 1).We can also multiply such a function by a scalar constant, by multiplying eachterm by the scalar. Multiplication of arbitrary generalized functions is not defined;in particular, 6(x) . 6(x) is not defined. However, if one of the two factors is anordinary function f (x) satisfying certain continuity conditions, then one can multiplyf by each term of the other factor. In particular, one defines f (x)8(x) to equalf (O)S(x), provided that f (x) is continuous at x = 0; one defines f (x)S1(x) to equalf (0)6'(x)- f '(O)S(x), provided that f and f' are continuous at x = 0. Theserules can be justified by the limit process described before; the latter rule can alsobe justified by the rule for differentiating products. In general, one is led to thedefinition:provided that f, . . . , f (k) are continuous at x = c. For example,f (x)~"(x - c) = f (c)8"(x- c) - 2 f '(c)6'(x- c) + f "(c)S(x- c).One can differentiate a generalized function of the type considered, by differentiatingeach term: The derivative of ~6(~)(x - c) is ~6'~+"(x - c); the derivativeof f (x) is the ordinary derivative plus contributions from the jump discontinuities.If c is such a discontinuity, then we add a term[ lim f (x)- lim f (.x)]8(x - c).x-+c+X'c-Thus if f (x) = 0 for x < 0, f (x) = cos x for 0 < x < n/2, and f (x) = - 1 for x >n/2, then f '(x) = g(x) + 6(x) - 6(x - (n/2)), where g(x) = 0 for x < 0, g(x) =-sin x for 0 < x < n/2, g(x) = 0 for x > n/2. One can verify that the usual rulesfor derivatives of sums and products hold.Generalized functions can also be integrated. & consider here only definiteintegrals from a to b (where a may be -oo and b may be oo) and exclude the case