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Smoothness of functions --- Smoothness of functions. --- 517.518.8 --- Smooth functions --- Functions --- Approximation of functions by polynomials and their generalizations --- 517.518.8 Approximation of functions by polynomials and their generalizations

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Moduli theory --- Smoothness of functions --- Variétés topologiques à 4 dimensions --- Modules, théorie des --- 517.5 --- Smooth functions --- Functions --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Theory of functions --- Moduli theory. --- Smoothness of functions. --- 517.5 Theory of functions --- Variétés topologiques à 4 dimensions --- Modules, théorie des --- Fonctions d'une variable reelle --- Approximation

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Smoothness of functions. --- Smooth functions --- Functions --- Funcions --- Càlcul diferencial --- Càlcul --- Càlcul integral --- Funcions (Matemàtica) --- Funcions matemàtiques --- Anàlisi matemàtica --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Constants matemàtiques --- Convergència (Matemàtica) --- Convolucions (Matemàtica) --- Funcions algebraiques --- Funcions contínues --- Funcions discontínues --- Funcions d'ona --- Funcions especials --- Teoria de l'aproximació --- Superfícies de Riemann

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We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val­ ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of smoothness. In Part II, we study/examine the Global Smoothness Preservation Prop­ erty (GSPP) for almost all known linear approximation operators of ap­ proximation theory including: trigonometric operators and algebraic in­ terpolation operators of Lagrange, Hermite-Fejer and Shepard type, also operators of stochastic type, convolution type, wavelet type integral opera­ tors and singular integral operators, etc. We present also a sufficient general theory for GSPP to hold true. We provide a great variety of applications of GSPP to Approximation Theory and many other fields of mathemat­ ics such as Functional analysis, and outside of mathematics, fields such as computer-aided geometric design (CAGD). Most of the time GSPP meth­ ods are optimal. Various moduli of smoothness are intensively involved in Part II. Therefore, methods from Part I can be used to calculate exactly the error of global smoothness preservation. It is the first time in the literature that a book has studied GSPP.

Smoothness of functions --- Moduli theory --- Approximation theory --- 517.518.8 --- Smooth functions --- Functions --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Theory of approximation --- Functional analysis --- Polynomials --- Chebyshev systems --- Approximation of functions by polynomials and their generalizations --- Approximation theory. --- Moduli theory. --- Smoothness of functions. --- 517.518.8 Approximation of functions by polynomials and their generalizations --- Applied mathematics. --- Engineering mathematics. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Applications of Mathematics. --- Approximations and Expansions. --- Global Analysis and Analysis on Manifolds. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Computer mathematics --- Electronic data processing --- Mathematics --- 517.1 Mathematical analysis --- Mathematical analysis --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Engineering --- Engineering analysis

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These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors’ 2007 Springer monograph “Random Fields and Geometry.” While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.

Gaussian processes --- Random fields --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical Statistics --- Geometry --- Smoothness of functions. --- Stochastic processes. --- Functions. --- Analysis (Mathematics) --- Random processes --- Smooth functions --- Mathematics. --- Geometry. --- Statistics. --- Statistical Theory and Methods. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Euclid's Elements --- Math --- Science --- Differential equations --- Mathematical analysis --- Numbers, Complex --- Set theory --- Probabilities --- Functions --- Mathematical statistics. --- Statistical inference --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Statistics .