From Approximate Variation to Pointwise Selection Principles - Vyacheslav V. Chistyakov

Blick ins Buch

From Approximate Variation to Pointwise Selection Principles (eBook)

Vyacheslav V. Chistyakov (Autor)

eBook Download: PDF

2021 | 1st ed. 2021
XIII, 86 Seiten
Springer International Publishing (Verlag)
978-3-030-87399-8 (ISBN)

Lese- und Medienproben

Ebook-Leseprobe (PDF)

The book addresses the minimization of special lower semicontinuous functionals over closed balls in metric spaces, called the approximate variation. The new notion of approximate variation contains more information about the bounded variation functional and has the following features: the infimum in the definition of approximate variation is not attained in general and the total Jordan variation of a function is obtained by a limiting procedure as a parameter tends to zero. By means of the approximate variation, we are able to characterize regulated functions in a generalized sense and provide powerful compactness tools in the topology of pointwise convergence, conventionally called pointwise selection principles.

The book presents a thorough, self-contained study of the approximate variation and results which were not published previously in book form. The approximate variation is illustrated by a large number of examples designed specifically for this study. The discussion elaborates on the state-of-the-art pointwise selection principles applied to functions with values in metric spaces, normed spaces, reflexive Banach spaces, and Hilbert spaces. The highlighted feature includes a deep study of special type of lower semicontinuous functionals though the applied methods are of a general nature. The content is accessible to students with some background in real analysis, general topology, and measure theory.

Among the new results presented are properties of the approximate variation: semi-additivity, change of variable formula, subtle behavior with respect to uniformly and pointwise convergent sequences of functions, and the behavior on improper metric spaces. These properties are crucial for pointwise selection principles in which the key role is played by the limit superior of the approximate variation. Interestingly, pointwise selection principles may be regular, treating regulated limit functions, and irregular, treating highly irregular functions (e.g., Dirichlet-type functions), in which a significant role is played by Ramsey's Theorem from formal logic.

Vyacheslav V. Chistyakov is Professor of Mathematics at the National Research University Higher School of Economics in Nizhny Novgorod, Russia. He received a PhD in Mathematics in 1987 from the Moscow State (Lomonosov) University, Moscow, Russia, and a D.Sc. in Mathematics in 2002 from the Sobolev Institute of Mathematics (Siberian Branch of Russian Academy of Sciences), Novosibirsk, Russia. His areas of research are real and functional analysis, set-valued analysis, optimization, and the decision making theory. He authored more than 100 research articles and several books including Metric Modular Spaces: Theory and Applications, Springer, 2015.

Erscheint lt. Verlag	22.10.2021
Reihe/Serie	SpringerBriefs in Optimization
Reihe/Serie	SpringerBriefs in Optimization
Zusatzinfo	XIII, 86 p. 19 illus.
Sprache	englisch
Themenwelt	Mathematik / Informatik ► Mathematik
Schlagworte	approximate variation • convergent sequences • Dirichlet function • Metric Spaces • Normed linear space • pointwise selection principles • semicontinuous functionals
ISBN-10	3-030-87399-4 / 3030873994
ISBN-13	978-3-030-87399-8 / 9783030873998

Haben Sie eine Frage zum Produkt?

PDF (Wasserzeichen)
Größe: 1,6 MB

DRM: Digitales Wasserzeichen
Dieses eBook enthält ein digitales Wasserzeichen und ist damit für Sie personalisiert. Bei einer missbräuchlichen Weitergabe des eBooks an Dritte ist eine Rückverfolgung an die Quelle möglich.

Dateiformat: PDF (Portable Document Format)
Mit einem festen Seitenlayout eignet sich die PDF besonders für Fachbücher mit Spalten, Tabellen und Abbildungen. Eine PDF kann auf fast allen Geräten angezeigt werden, ist aber für kleine Displays (Smartphone, eReader) nur eingeschränkt geeignet.

Systemvoraussetzungen:
PC/Mac: Mit einem PC oder Mac können Sie dieses eBook lesen. Sie benötigen dafür einen PDF-Viewer - z.B. den Adobe Reader oder Adobe Digital Editions.
eReader: Dieses eBook kann mit (fast) allen eBook-Readern gelesen werden. Mit dem amazon-Kindle ist es aber nicht kompatibel.
Smartphone/Tablet: Egal ob Apple oder Android, dieses eBook können Sie lesen. Sie benötigen dafür einen PDF-Viewer - z.B. die kostenlose Adobe Digital Editions-App.

Buying eBooks from abroad
For tax law reasons we can sell eBooks just within Germany and Switzerland. Regrettably we cannot fulfill eBook-orders from other countries.

Print-Ausgabe

Buch | Softcover

53,49 €