Previous |  Up |  Next

Article

Keywords:
Stieltjes integral; Kurzweil integral; Henstock integral; ${\mathrm H}_1$-integral; Riemann-Lebesgue theorem; variational measure; adjoint classes
Summary:
Using the concept of the $ {\mathrm H}_1$-integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.
References:
[1] H. Chen: A pair of adjoint classes of Riemann-Stieltjes integrable functions. Real Anal. Exch. 23 (1998), 235–240. MR 1609806
[2] H. Chen: Adjoint classes of generalized Stieltjes integrable functions. Real Anal. Exch. 24 (1999), 139–148. MR 1691741
[3] H. Chen: Adjoint classes of Lebesgue-Stieltjes integrable functions. Real Anal. Exch. 26 (2001), 421–427. MR 1825521 | Zbl 1023.26007
[4] I. J. L. Garces, P. Y. Lee: Cauchy and Harnack extensions for the $H_1$-integral. Matimyás Mat. 21 (1998), 28–34. MR 1710941
[5] I. J. L. Garces, P. Y. Lee: Convergence theorems for the $H_1$-integral. Taiwanese J.  Math. 4 (2000), 439–445. MR 1779108
[6] I. J. L. Garces, P. Y. Lee, and D. Zhao: Moore-Smith limits and the Henstock integral. Real Anal. Exch. 24 (1999), 447–455. MR 1691764
[7] A. Maliszewski, P. Sworowski: Uniform convergence theorem for the $H_1$-integral revisited. Taiwanese J.  Math. 7 (2003), 503–505. MR 1998771
[8] A. Maliszewski, P. Sworowski: A characterization of $H_1$-integrable functions. Real Anal. Exch. 28 (2003), 93–104. MR 1973971
[9] K. A. Ross: Another approach to Riemann-Stieltjes integrals. Am. Math. Mon. 87 (1980), 660–662. DOI 10.2307/2320958 | MR 0600928 | Zbl 0446.26005
[10] S. Saks: Theory of the Integral. G. E. Stechert, New York, 1937. Zbl 0017.30004
[11] Š. Schwabik: On the relation between Young’s and Kurzweil’s concept of Stieltjes integral. Cas. Pest. Mat. 98 (1973), 237–251. MR 0322113 | Zbl 0266.26006
[12] P. Sworowski: On $H_1$-integrable functions. Real Anal. Exch. 27 (2002), 275–286. MR 1887858 | Zbl 1015.26017
[13] P. Sworowski: Some comments on the $H_1$-integral. Real Anal. Exch. 29 (2004), 789–797. MR 2083813 | Zbl 1078.26008
[14] P. Sworowski: Adjoint classes for generalized Riemann-Stieltjes integrals. 27th Summer Symposium Conference Reports, Opava  2003. Real Anal. Exch. (2003), 41–45.
[15] B. S. Thomson: Real Functions. Lecture Notes in Mathematics, Vol. 1170. Springer-Verlag, Berlin, 1985. MR 0818744
Partner of
EuDML logo