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Advances in inequalities of the Schwarz, Grüss and Bessel type in inner product spaces. (English) Zbl 1115.26019

Hauppauge, NY: Nova Science Publishers (ISBN 1-59454-202-3/hbk). viii, 249 p. (2005).
Publisher’s description (no review copy received): The theory of Hilbert spaces plays a central role in contemporary mathematics with numerous applications for Linear Operators, Partial Differential Equations, in Nonlinear Analysis, Approximation Theory, Optimization Theory, Numerical Analysis, Probability Theory, Statistics and other fields. The Schwarz, triangle, Bessel, Gram and most recently, Grüss type inequalities have been frequently used as powerful tools in obtaining bounds or estimating the errors for various approximation formulae occurring in the domains mentioned above. Therefore, any new advancement related to these fundamental facts will have a flow of important consequences in the mathematical fields where these inequalities have been used before.
The main aim of this book is to survey some recent results related to reverses of the Schwarz, triangle and Bessel inequalities. Some Grüss type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecarić inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski inequalities for sequences of real numbers and the generalization of Wagner’s inequality in inner product spaces are pointed out. Finally, some Grüss type inequalities for \(n\)-tuples of vectors in inner product spaces and their natural applications for the approximation of the discrete Fourier and Mellin transforms are given as well.
This book is of significance to researchers in different branches of Mathematical and Functional Analysis where the theory of Hilbert spaces is of relevance. Since it is self-contained and all the results are completely proved, the work would be very useful to graduate students interested in Theory of Inequalities and its Applications.
Table of Contents: Preface;
PART 1. REVERSE INEQUALITIES. Chapter 1. Reverses for the Schwarz Inequality; Chapter 2. Inequalities of the Grüss Type; Chapter 3. Reverses of Bessel’s Inequality;
PART 2. OTHER INEQUALITIES IN INNER PRODUCT SPACES. Chapter 4. Generalisations of Bessel’s Inequality; Chapter 5. Some Grüss-Type Inequalities for \(n\)-Tuples of Vectors; Chapter 6. Other Inequalities in Inner Product Spaces; Bibliography; Index.

MSC:

26D15 Inequalities for sums, series and integrals
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
46C99 Inner product spaces and their generalizations, Hilbert spaces
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