Vol. 55, No. 1, 1974

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On the regularity of the Pn-integral and its application to summable trigonometric series

S. N. Mukhopadhyay

Vol. 55 (1974), No. 1, 233–247

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 B2m2 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.

Mathematical Subject Classification 2000
Primary: 26A39
Secondary: 42A16
Received: 7 January 1974
Revised: 25 September 1974
Published: 1 November 1974
S. N. Mukhopadhyay