The symmetric P2m-integral
(and P2m+1-integral) as defined by R. D. James in “Generalized n-th primitives”,
Trans. Amer. Math. Soc., 76 (1954), is useful to solve problems relating to
trigonometric series (see R. D. James: Summable trigonometric series, Pacific J.
Math., 6 (1956)). But the definition of the integral is not valid, since Lemma 5.1 of
the former paper of James, which is the basis of the whole theory, is incomplete due
to the fact that the difference of two functions having property B2m−2 may not have
this property. Therefore, all the subsequent results of James also remain incomplete
and a complete systematic definition of the integral is needed. In the present
paper a definition of the P2m-integral (and P2m+1-integral) is given and it is
shown that all the results of the later paper of James remain valid with this
integral.
