A Modern View of the Riemann Integral

1. Auflage 2022
ISBN: 978-3-031-11798-5
Verlag: Springer International Publishing

Buch, Englisch, Band 2309, 176 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 295 g

Reihe: Lecture Notes in Mathematics

This monograph uncovers the full capabilities of the Riemann integral. Setting aside all notions from Lebesgue’s theory, the author embarks on an exploration rooted in Riemann’s original viewpoint. On this journey, we encounter new results, numerous historical vignettes, and discover a particular handiness for computations and applications.

This approach rests on three basic observations. First, a Riemann integrability criterion in terms of oscillations, which is a quantitative formulation of the fact that Riemann integrable functions are continuous a.e. with respect to the Lebesgue measure. Second, the introduction of the concepts of admissible families of partitions and modified Riemann sums. Finally, the fact that most numerical quadrature rules make use of carefully chosen Riemann sums, which makes the Riemann integral, be it proper or improper, most appropriate for this endeavor.

A Modern View of the Riemann Integral is intended for enthusiasts keen to explore the potential of Riemann's original notion of integral. The only formal prerequisite is a proof-based familiarity with the Riemann integral, though readers will also need to draw upon mathematical maturity and a scholarly outlook.

Zielgruppe

Research

Autoren/Hrsg.

Torchinsky, Alberto

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Reelle Analysis

Inhaltsverzeichnis

Preface.- Chapter 1. Introduction.- Chapter 2. The ???–Riemann Integral.- Chapter 3. A Convergence Theorem.- Chapter 4. The Modified ????–Riemann Sums.- Chapter 5. The Pattern and Uniform Integrals.- Chapter 6. The Improper and Dominated Integrals.- Chapter 7. Coda.- Appendix I.- Appendix II.- References.- Index.

Über Autor(innen)

Alberto Torchinsky is Emeritus Professor of Mathematics at Indiana University Bloomington. His research interests are centered on harmonic and real analysis. He has authored several other books, including the widely cited LNM 1381, Weighted Hardy Spaces, with Jan-Olov Strömberg. Prior to Indiana University, he held positions at the University of Illinois and Cornell University, having received his PhD at the University of Chicago under A. P. Calderón.