Applications of q-Calculus in Operator Theory
The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of d
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Applications of q-Calculus in Operator Theory
Applications of q-Calculus in Operator Theory
Ali Aral • Vijay Gupta • Ravi P. Agarwal
Applications of q-Calculus in Operator Theory
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Ali Aral Department of Mathematics Kırıkkale University Yahs¸ihan, Kirikkale, Turkey
Vijay Gupta School of Applied Sciences Netaji Subhas Institute of Technology New Delhi, India
Ravi P. Agarwal Department of Mathematics Texas A&M University-Kingsville Kingsville, Texas, USA
ISBN 978-1-4614-6945-2 ISBN 978-1-4614-6946-9 (eBook) DOI 10.1007/978-1-4614-6946-9 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013934278 Mathematics Subject Classification (2010): 41A36-41A25-41A17-30E10 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Simply, quantum calculus is ordinary classical calculus without the notion of limits. It defines q-calculus and h-calculus. Here h ostensibly stands for Planck’s constant, while q stands for quantum. A pioneer of q-calculus in approximation theory is the former Professor Alexandru Lupas [117], who first introduced the qanalogue of Bernstein polynomials. Ten years later Phillips [133] introduced another generalization on Bernstein polynomials [113] based on q-int
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