By I. M. Gel'fand and G. E. Shilov (Auth.)
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Equation (6) may be written te, - 9 > ' ) = (/,9>) for every φ in K. But this then means that the functional g is defined for every φ in Κ which is the derivative of some other function φ in K; in other words, g is defined on the set 0Q defined earlier. W e now wish to extend g to all of K. This can be done, for instance, by considering first some φι{χ) in Κ such that J_ φι{χ) dx = I and writing any φ in Κ in the form 9 = 9i ί 9dx +φο. where φο is in Φ^. In this way with every φ we associate uniquely its **projection" φο onto Φ^, W e now define (g,,9)-{g, P) = - ( / . ' ' ) is again a continuous linear functional in S, But it is clear that when restricted to Κ this functional coincides with the derivative of / . In other words, the derivative / ' of / restricted to Κ can be extended to S together with / , which is what was asserted. The generalization of this result to higher derivatives and to several variables presents no difliculties. 1 Functions with Algebraic Singularities 45 W e have seen before how all regular functionals corresponding to functions increasing no faster than some power of the variable (which we shall call slowly increasing functions) can be extended from Κ to S, W e now see that this is true also of the derivatives of such functionals. We shall find that the one corresponding to -i[sin2(xj2)]-1 will be the derivative of that corresponding to t cot (x/2), etc. In other words, Eqs. (4) will be given meaning in terms of generalized functions. i and of numbers. 7 fJi I-'i n' Q' + I-'i-l n' Q '-1 + ... + Q 1-'0 fl-l Sn + - n + 2" n ' are constants, and {Yn} and {Sn} are bounded sequences e Fikhtengofts, "Calculus," Vol. III, p. 550. See also K. Knopp, "Theory and Application of Infinite Series," p. 378. Blackie & Son, London, 1928.