Henstock-Kurzweil integration on Euclidean spaces /
The 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 Perron integral; in particular, it includes the...
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| Format: | eBook |
| Language: | English |
| Published: |
New Jersey :
World Scientific,
©2011.
|
| Series: | Series in real analysis ;
v. 12. |
| Subjects: | |
| Online Access: | CONNECT CONNECT |
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| 100 | 1 | |a Lee, Tuo Yeong, |d 1967- | |
| 245 | 1 | 0 | |a Henstock-Kurzweil integration on Euclidean spaces / |c Lee Tuo Yeong. |
| 260 | |a New Jersey : |b World Scientific, |c ©2011. | ||
| 300 | |a 1 online resource (ix, 314 pages). | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Series in real analysis ; |v v. 12 | |
| 504 | |a Includes bibliographical references (pages 295-303) and index. | ||
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a 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. [symbol]-measurable sets. 3.4. A characterization of [symbol]-measurable sets. 3.5. [symbol]-measurable functions. 3.6. Vitali covering theorem. 3.7. Further properties of Lebesgue integrable functions. 3.8. The L[symbol] 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 Jarnik. 4.3. Some necessary and sufficient conditions for Henstock-Kurzweil integrability. 4.4. Harnack extension for one-dimensional Henstock-Kurzweil integrals. 4.5. 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 Jarnik revisited. 5.5. A measure-theoretic characterization of the Henstock-Kurzweil integral. 5.6. Product variational measures. 5.7. Notes and remarks. | |
| 505 | 8 | |a 6. Multipliers for the Henstock-Kurzweil integral. 6.1. One-dimensional integration by parts. 6.2. 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. 7.1. A generalized Dirichlet test. 7.2. 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. 8.6. A convergence theorem for Henstock-Kurzweil integrals. 8.7. Applications to double Fourier series. 8.8. Another convergence theorem for Henstock-Kurzweil integrals. 8.9. A two-dimensional analogue of Boas' theorem. 8.10. A convergence theorem for double sine series. 8.11. Some open problems. 8.12. Notes and remarks. | |
| 520 | |a The 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 Perron integral; in particular, it includes the powerful Lebesgue integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood with a prerequisite of advanced calculus. | ||
| 590 | |a EBSCO eBook Academic Comprehensive Collection North America | ||
| 650 | 0 | |a Henstock-Kurzweil integral. | |
| 650 | 0 | |a Lebesgue integral. | |
| 650 | 0 | |a Calculus, Integral. | |
| 730 | 0 | |a WORLDSHARE SUB RECORDS | |
| 776 | 0 | 8 | |i Print version: |a Lee, Tuo Yeong, 1967- |t Henstock-Kurzweil integration on Euclidean spaces. |d Singapore ; Hackensack, N.J. : World Scientific, ©2011 |z 9789814324588 |w (OCoLC)724966681 |
| 830 | 0 | |a Series in real analysis ; |v v. 12. | |
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