The approximate symmetric integral
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
By a symmetric integral is understood an integral obtained from some kind of symmetric derivation process. Such integrals arise most naturally in the study of trigonometric series and in particular to handle the following problem. Suppose that a trigonometric series (1) AQ/2 + ^ cik cos kx + bk sin kx k=\ converges everywhere to a function/. It is known that this may occur without / being integrable in any of the more familiar senses so that the series may not be considered as a Fourier series
... s a Fourier series of/; indeed Denjoy  has shown that if b n is a sequence of real numbers decreasing to zero but with ^b n jn~ +00 then the function/(JC) = ^b n sinnx is not Denjoy-integrable. It is natural to ask then for an integration procedure that can be applied to / in order that the series be the Fourier series of/ with respect to this integral. A solution, based on symmetric integrals, follows from the well known observation of Riemann: the series +00 (2) aox 2 /4 -/_\tf£ cos kx + bk sin kx)/k 2 obtained by two formal integrations of series (1) converges uniformly and absolutely to a continuous function G from which the function/ may be obtained by a second order symmetric derivation, This suggests the development of an integral that can recover a function G from its second symmetric derivative. This program has been followed by Denjoy  and James  both of whom produce second order integrals from this derivation process, James by a Perron-type approach and Denjoy by a transflnite totalization process. Taylor  introduced his AP-integral to solve this same question when the series is given to be everywhere Abel-summable and he too uses a second order derivation process, but to produce a certain kind of first order integral. The articles of Cross  and Skvorcov  and  should be consulted for the relation between Taylor's integral and the James P 2 -integral.