The symmetric P 2m -integral (and P 2w+1 -integral) as defined by R. D. James in "Generalized nth 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

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... ty B 2m -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 P 2m -integral (and P 2m+1 -integral) is given and it is shown that all the results of the later paper of James remain valid with this integral. 1* Definitions and Notations* Most of the definitions and notations of [8] will be used with essential modifications. The generalized symmetric derivative [8] (also called symmetric de La Vallee Poussin derivative [18]) of even and odd orders and the generalized unsymmetric derivative [8] (also called Peano derivative [13] or unsymmetric de La Vallee Poussin derivative [11]) of a function/at x Q will be denoted by D r f(x Q ) and /< r )(cc 0 ) respectively, where r denotes the order of the respective derivatives. If D 2k f(x 0 ) exists, 0 <^ k <£ m -1, define Θ 2m (f; x 0 , h) by -J^M/; *o, h) = ±{f(x 0 + h)+ f(x Q -h) -g T he upper generalized symmetric derivate of / at x 0 of order 2m is defined as D 2m f(x 0 ) = lim sup θ zm (f; x Q , h) . A->0 Replacing Ίimsup' by Ίiminf one gets the definition of D 2m f(x Q ). The function / is said to satisfy the property <9i m at x Q , written as/e^m(£ 0 ), if lim sup hθ 2m (f; x 0 , h) ^> 0 , and fe£$ m (x 0 ) if -fe^ζ m (x 0 ). The function / is said to be smooth 233 234 S. N. MUKHOPADHYAY at x Q of order 2m if limhθ 2m (f;x 0 , h) = 0. h-*0 Clearly smoothness of order 2m implies smoothness of order 2m -2 and if / is smooth at x 0 of order 2m then fe<9ζ m (x 0 ) Π S^m(Xo)-For symmetric derivatives of odd order, 0 2w+1 (/; x Q , h), D 2m+1 f(x 0 ), D 2m+1 f(x 0 ), £% m +i(x 0 ), ^m+i(%o) are defined analogously.