LEADER 05253nam 2200709 a 450 001 9910788961403321 005 20230725052651.0 010 $a1-283-23477-7 010 $a9786613234773 010 $a981-4324-59-0 035 $a(CKB)3400000000016253 035 $a(EBL)840570 035 $a(OCoLC)748215459 035 $a(SSID)ssj0000537511 035 $a(PQKBManifestationID)12251896 035 $a(PQKBTitleCode)TC0000537511 035 $a(PQKBWorkID)10553370 035 $a(PQKB)10682822 035 $a(MiAaPQ)EBC840570 035 $a(WSP)00007933 035 $a(Au-PeEL)EBL840570 035 $a(CaPaEBR)ebr10493518 035 $a(CaONFJC)MIL323477 035 $a(EXLCZ)993400000000016253 100 $a20110608d2011 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aHenstock-Kurzweil integration on euclidean spaces$b[electronic resource] /$fLee Tuo Yeong 210 $aSingapore ;$aHackensack, N.J. $cWorld Scientific$dc2011 215 $a1 online resource (325 p.) 225 1 $aSeries in real analysis ;$vv. 12 300 $aDescription based upon print version of record. 311 $a981-4324-58-2 320 $aIncludes bibliographical references and indexes. 327 $aPreface; Contents; 1. The one-dimensional Henstock-Kurzweil integral; 1.1 Introduction and Cousin's Lemma; 1.2 Definition of the Henstock-Kurzweil integral; 1.3 Simple properties; 1.4 Saks-Henstock Lemma; 1.5 Notes and Remarks; 2. The multiple Henstock-Kurzweil integral; 2.1 Preliminaries; 2.2 The Henstock-Kurzweil integral; 2.3 Simple properties; 2.4 Saks-Henstock Lemma; 2.5 Fubini's Theorem; 2.6 Notes and Remarks; 3. Lebesgue integrable functions; 3.1 Introduction; 3.2 Some convergence theorems for Lebesgue integrals; 3.3 ?m-measurable sets; 3.4 A characterization of ?m-measurable sets 327 $a3.5 ?m-measurable functions3.6 Vitali Covering Theorem; 3.7 Further properties of Lebesgue integrable functions; 3.8 The Lp spaces; 3.9 Lebesgue's criterion for Riemann integrability; 3.10 Some characterizations of Lebesgue integrable functions; 3.11 Some results concerning one-dimensional Lebesgue integral; 3.12 Notes and Remarks; 4. Further properties of Henstock-Kurzweil integrable functions; 4.1 A necessary condition for Henstock-Kurzweil integrability; 4.2 A result of Kurzweil and Jarn ??k; 4.3 Some necessary and su cient conditions for Henstock- Kurzweil integrability 327 $a4.4 Harnack extension for one-dimensional Henstock-Kurzweil integrals4.5 Other results concerning one-dimensional Henstock- Kurzweil integral; 4.6 Notes and Remarks; 5. The Henstock variational measure; 5.1 Lebesgue outer measure; 5.2 Basic properties of the Henstock variational measure; 5.3 Another characterization of Lebesgue integrable functions; 5.4 A result of Kurzweil and Jarn ??k revisited; 5.5 A measure-theoretic characterization of the Henstock- Kurzweil integral; 5.6 Product variational measures; 5.7 Notes and Remarks; 6. Multipliers for the Henstock-Kurzweil integral 327 $a6.1 One-dimensional integration by parts6.2 On functions of bounded variation in the sense of Vitali; 6.3 The m-dimensional Riemann-Stieltjes integral; 6.4 A multiple integration by parts for the Henstock-Kurzweil integral; 6.5 Kurzweil's multiple integration by parts formula for the Henstock-Kurzweil integral; 6.6 Riesz Representation Theorems; 6.7 Characterization of multipliers for the Henstock-Kurzweil integral; 6.8 A Banach-Steinhaus Theorem for the space of Henstock- Kurzweil integrable functions; 6.9 Notes and Remarks; 7. Some selected topics in trigonometric series 327 $a7.1 A generalized Dirichlet test7.2 Fourier series; 7.3 Some examples of Fourier series; 7.4 Some Lebesgue integrability theorems for trigonometric series; 7.5 Boas' results; 7.6 On a result of Hardy and Littlewood concerning Fourier series; 7.7 Notes and Remarks; 8. Some applications of the Henstock-Kurzweil integral to double trigonometric series; 8.1 Regularly convergent double series; 8.2 Double Fourier series; 8.3 Some examples of double Fourier series; 8.4 A Lebesgue integrability theorem for double cosine series; 8.5 A Lebesgue integrability theorem for double sine series 327 $a8.6 A convergence theorem for Henstock-Kurzweil integrals 330 $aThe Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Per 410 0$aSeries in real analysis ;$vv. 12. 606 $aHenstock-Kurzweil integral 606 $aLebesgue integral 606 $aCalculus, Integral 615 0$aHenstock-Kurzweil integral. 615 0$aLebesgue integral. 615 0$aCalculus, Integral. 676 $a515.43 686 $aSK 430$2rvk 686 $aSK 620$2rvk 700 $aLee$b Tuo Yeong$f1967-$01574159 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910788961403321 996 $aHenstock-Kurzweil integration on euclidean spaces$93850264 997 $aUNINA