## Real Analysis Exchange

### An n-th order integral and its integration by parts with applications to trigonometric series.

#### Abstract

An $n$-th order symmetric Perron type integral is defined and its properties are studied. An integration by parts formula is proved and applied to solve problems related to summable trigonometric series.

#### Article information

Source
Real Anal. Exchange, Volume 30, Number 2 (2004), 451- 494 .

Dates
First available in Project Euclid: 15 October 2005

https://projecteuclid.org/euclid.rae/1129416475

Mathematical Reviews number (MathSciNet)
MR2177413

Zentralblatt MATH identifier
1110.26008

Subjects
Primary: 26A39: Denjoy and Perron integrals, other special integrals

#### Citation

Mukhopadhyay, S. N. An n -th order integral and its integration by parts with applications to trigonometric series. Real Anal. Exchange 30 (2004), no. 2, 451-- 494. https://projecteuclid.org/euclid.rae/1129416475

#### References

• J. A. Bergin, A new characterization of the Cesàro-Perron integrals using Peano derivates, Trans. Amer. Math. Soc., 228 (1977), 287–305.
• P. S. Bullen, A criterion for $n$-convexity, Pacific J. Math., 36 (1971), 81–98.
• P. S. Bullen & S. N. Mukhopadhyay, Integration by parts formulæ for some trigonometric integrals, Proc. London Math. Soc., (3), 29 (1974), 159–173.
• P. S. Bullen & S. N. Mukhopadhyay, Generalized continuity of higher order symmetric derivatives, South-East Asian Bull. Math., 13 (1989), 127–137.
• J. C. Burkill, The Cesàro-Perron scale of integration, Proc.
• P. S. Chakrabarti & S. N. Mukhopadhyay, A scale of approximate Cesàro-Perron integrals, Bull. Inst. Math. Acad. Sinica, 10 (1982), 323–346.
• G. E. Cross, The $P^n$ integral, Canad. J. Math., 18 (1975), 493–497.
• G. E. Cross, Additivity of the $P^n$ integral, Canad. J. Math., 30 (1978), 783–796.
• G. E. Cross, Additivity of the $P^n$ integral (2), Canad. J. Math., 34 (1982), 506–512.
• R. D. James, Generalized $n$-th primitives, Trans. Amer. Math. Soc., 76 (1954), 149–176.
• R. D. James, Summable trigonometric series, Pacific J. Math., 6 (1956), 99–110.
• C. M. Lee, On integrals and summable trigonometric series, Canad.
• S. N. Mukhopadhyay, On the regularity of the $P^n$-integral and its application to summable trigonometric series, Pacific J. Math., 55 (1974), 233–247.
• S. N. Mukhopadhyay, An extension of the $SCP$-integral with a relaxed integration by parts formula, Analysis Math., 25 (1999), 103–132.
• S. N. Mukhopadhyay & S. Mitra, An extension of a theorem of Marcinkiewicz and Zygmund on differentiability, Fund. Math., 151 (1996), 21–38.
• H. W. Oliver,The exact Peano derivatives, Trans. Amer. Math. Soc., 76 (1954), 444–456.
• S. Saks, Theory of the Integral, Dover, 1937.
• V. A. Skljarenko, On integration by parts in Burkill's $SCP$-integral, Math. USSR-Sb., 40 (1981), 567–582; Mat.Sb., 112 (154) (1980), 630–646, (in Russian).
• V. A. Skvorcov, Concerning the definition of the $P^2$- and $SCP$-integrals, Vestnik Moskov. Univ. Ser. I, Mat. Mekh., 21 (1966), 12–19, (in Russian).
• F. Wolf, Summable trigonometric series; an extension of uniqueness theorem, Proc. London Math. Soc., (2) 45 (1939), 328–356.
• A. Zygmund, Trigonometric Series, Cambridge, 1968.