Ebook: Dimensionality Reducing Expansion of Multivariate Integration

Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form.

Key features of this self-contained monograph include:

* fine exposition covering the history of the subject

* up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis

* presentation of DRE techniques using a broad array of examples

* good balance between theory and application

* coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals

* excellent and comprehensive bibliography and index

This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

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Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form.

Key features of this self-contained monograph include:

* fine exposition covering the history of the subject

* up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis

* presentation of DRE techniques using a broad array of examples

* good balance between theory and application

* coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals

* excellent and comprehensive bibliography and index

This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

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Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form.

Key features of this self-contained monograph include:

* fine exposition covering the history of the subject

* up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis

* presentation of DRE techniques using a broad array of examples

* good balance between theory and application

* coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals

* excellent and comprehensive bibliography and index

This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

Content:
Front Matter....Pages i-xi
Dimensionality Reducing Expansion of Multivariate Integration....Pages 1-23
Boundary Type Quadrature Formulas with Algebraic Precision....Pages 25-78
The Integration and DREs of Rapidly Oscillating Functions....Pages 79-108
Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions....Pages 109-149
DREs Over Complex Domains....Pages 151-179
Exact DREs Associated With Differential Equations....Pages 181-216
Back Matter....Pages 217-227

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Multivariate integration has been a fundamental subject in mathematics, with broad connections to a number of areas: numerical analysis, approximation theory, partial differential equations, integral equations, harmonic analysis, etc. In this work the exposition focuses primarily on a powerful tool that has become especially important in our computerized age, namely, dimensionality reducing expansion (DRE). The method of DRE is a technique for changing a higher dimensional integration to a lower dimensional one with or without remainder. To date, there is no comprehensive treatment of this subject in monograph or textbook form.

Key features of this self-contained monograph include:

* fine exposition covering the history of the subject

* up-to-date new results, related to many fields of current research such as boundary element methods for solving PDEs and wavelet analysis

* presentation of DRE techniques using a broad array of examples

* good balance between theory and application

* coverage of such related topics as boundary type quadratures and asymptotic expansions of oscillatory integrals

* excellent and comprehensive bibliography and index

This work will appeal to a broad audience of students and researchers in pure and applied mathematics, statistics, and physics, and can be used in a graduate/advanced undergraduate course or as a standard reference text.

Content:
Front Matter....Pages i-xi
Dimensionality Reducing Expansion of Multivariate Integration....Pages 1-23
Boundary Type Quadrature Formulas with Algebraic Precision....Pages 25-78
The Integration and DREs of Rapidly Oscillating Functions....Pages 79-108
Numerical Quadrature Formulas Associated with the Integration of Rapidly Oscillating Functions....Pages 109-149
DREs Over Complex Domains....Pages 151-179
Exact DREs Associated With Differential Equations....Pages 181-216
Back Matter....Pages 217-227
....