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The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results.
Differential operators --- Functions of bounded variation --- Differentiaaloperatoren --- Fonctions à variation bornée --- Functies met begrensde variatie --- Integralen --- Integrals --- Intégrales --- Operateurs differentiels --- Operator theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of real variables. --- Operator Theory. --- Global Analysis and Analysis on Manifolds. --- Real Functions. --- Real variables --- Functions of complex variables --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic --- Functional analysis
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Functionals involving both volume and surface energies have a number of applications ranging from Computer Vision to Fracture Mechanics. In order to tackle numerical and dynamical problems linked to such functionals many approximations by functionals defined on smooth functions have been proposed (using high-order singular perturbations, finite-difference or non-local energies, etc.) The purpose of this book is to present a global approach to these approximations using the theory of gamma-convergence and of special functions of bounded variation. The book is directed to PhD students and researchers in calculus of variations, interested in approximation problems with possible applications.
Calculus of variations --- Functions of bounded variation --- Convergence --- Perturbation (Mathematics) --- Calcul des variations --- Convergentie --- Fonctions à variation bornée --- Functies met begrensde variatie --- Perturbatie (Wiskunde) --- Perturbation (Mathématiques) --- Variatieberekening --- Partial differential equations. --- Numerical analysis. --- Mathematical physics. --- Partial Differential Equations. --- Numerical Analysis. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Mathematical analysis --- Partial differential equations --- Mathematics --- Functions of bounded variation. --- Convergence. --- Calculus of variations. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Functions --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables
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