Functional Algebra and Hypercalculus in Infinite Dimensions: Hyperintegrals, Hyperfunctionals and Hyperderivatives

Mark Burgin
UCLA, California, USA

Series: Theoretical and Applied Mathematics
BISAC: MAT003000, MAT002000




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The theory of hypernumbers and extrafunctions is further development in distribution theory inspired by contemporary physics and influenced by problems in mathematical physics. It makes more functions differentiable and provides new kinds of derivatives and hyperderivatives aimed at solving more differential and operator equations than ever before possible.

In the book, extrafunctions are extended to hyperfunctionals and hyperoperators in infinite-dimensional vector spaces. Due to its development, many problems in contemporary physics, as well as in modern linear and nonlinear analysis have an infinite-dimensional nature, and the infinite-dimensional theory of extrafunctions, hyperfunctionals and hyperoperators provides new tools for solving many of these problems.

The book describes new mathematical structures such as hyperderivatives and hyperintegrals of real and complex functions, hyperprobability and hyperexpectation of random processes and some others, essentially increasing power of functional analysis and probability applications. It presents the key parts of calculus – number systems, function spaces, the differential calculus and the integral calculus – in the setting of hypernumbers, extrafunctions, hyperfunctionals and hyperoperators in finite-dimensional and infinite-dimensional vector spaces. In addition, functional algebra, which employs algebraic operations with extrafunctions, hyperfunctionals and hyperoperators is developed. New relations between hyperdifferentiation and continuity of functions and operators are explicated. As differentiation and integration are special cases of hyperdifferentiation and hyperintegration, respectively, hypercalculus includes calculus as its part or subtheory.

It is possible to use this book for enhancing traditional courses of calculus for undergraduates, as well as for teaching separate courses for graduate and undergraduate students at colleges and universities. To achieve these goals, exposition in the book goes from simple topics to more and more advanced topics, while proof of some statements are left as exercises for the students. (Imprint: Nova)


Chapter 1. Introduction: Challenges of Infinity

Chapter 2. Number Hyperspaces over Normed Fields

Chapter 3. Hyperfunctionals and Hyperoperators as Extrafunctions

Chapter 4. Hyperdifferentiation as a Hyperoperator

Chapter 5. Hyperintegration as a Hyperfunctional

Chapter 6. Hyperprobability as a Comprehensive Characteristic of Random Phenomena

Chapter 7. Conclusion: New Opportunities

Appendix: Notation and Rudimentary Constructions



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