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Measure theory. Mathematical integration --- Measure theory --- Functions of real variables --- Integrals, Generalized --- Integrals, Generalized. --- Measure theory. --- Functions of real variables. --- Fonctions d'une variable réelle --- Calcul intégral --- Fonctions d'une variable reelle --- Calcul integral

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Functional analysis --- Differential equations

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A paean to twentieth century analysis, this modern text has several important themes and key features which set it apart from others on the subject. A major thread throughout is the unifying influence of the concept of absolute continuity on differentiation and integration. This leads to fundamental results such as the Dieudonné–Grothendieck theorem and other intricate developments dealing with weak convergence of measures. Key Features: * Fascinating historical commentary interwoven into the exposition; * Hundreds of problems from routine to challenging; * Broad mathematical perspectives and material, e.g., in harmonic analysis and probability theory, for independent study projects; * Two significant appendices on functional analysis and Fourier analysis. Key Topics: * In-depth development of measure theory and Lebesgue integration; * Comprehensive treatment of connection between differentiation and integration, as well as complete proofs of state-of-the-art results; * Classical real variables and introduction to the role of Cantor sets, later placed in the modern setting of self-similarity and fractals; * Evolution of the Riesz representation theorem to Radon measures and distribution theory; * Deep results in modern differentiation theory; * Systematic development of weak sequential convergence inspired by theorems of Vitali, Nikodym, and Hahn–Saks; * Thorough treatment of rearrangements and maximal functions; * The relation between surface measure and Hausforff measure; * Complete presentation of Besicovich coverings and differentiation of measures. Integration and Modern Analysis will serve advanced undergraduates and graduate students, as well as professional mathematicians. It may be used in the classroom or self-study.

Mathematics. --- Analysis. --- Functions of a Complex Variable. --- Measure and Integration. --- Global analysis (Mathematics). --- Functions of complex variables. --- Mathématiques --- Analyse globale (Mathématiques) --- Fonctions d'une variable complexe --- Mathematical analysis --- Functions of real variables --- Integration, Functional --- Integrals, Generalized --- Measure theory --- Electronic books. -- local. --- Functions of real variables. --- Integrals, Generalized. --- Integration, Functional. --- Mathematical analysis. --- Measure theory. --- Engineering & Applied Sciences --- Applied Mathematics --- Lebesgue measure --- Measurable sets --- Measure of a set --- 517.1 Mathematical analysis --- Functional integration --- Real variables --- Analysis (Mathematics). --- Algebraic topology --- Measure algebras --- Rings (Algebra) --- Functional analysis --- Calculus, Integral --- Functions of complex variables --- Analysis, Global (Mathematics) --- Differential topology --- Geometry, Algebraic

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Fourier analysis --- Harmonic analysis --- Sampling (Statistics) --- Tomography --- Wavelets (Mathematics) --- 519.65 --- Analysis, Fourier --- Mathematical analysis --- 519.65 Approximation. Interpolation --- Approximation. Interpolation --- Wavelet analysis --- Body section radiography --- Computed tomography --- Computer tomography --- Computerized tomography --- CT (Computed tomography) --- Laminagraphy --- Laminography --- Radiological stratigraphy --- Stratigraphy, Radiological --- Tomographic imaging --- Zonography --- Cross-sectional imaging --- Radiography, Medical --- Geometric tomography --- Random sampling --- Statistics of sampling --- Statistics --- Mathematical statistics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis

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This self-contained text provides an introduction to modern harmonic analysis in the context in which it is actually applied, in particular, through complex function theory and partial differential equations. It takes the novice mathematical reader from the rudiments of harmonic analysis (Fourier series) to the Fourier transform, pseudodifferential operators, and finally to Heisenberg analysis. Within the textbook, the new ideas on the Heisenberg group are applied to the study of estimates for both the Szegö and Poisson-Szegö integrals on the unit ball in complex space. Thus the main theme of the book is also tied into complex analysis of several variables. With a rigorous but well-paced exposition, this text provides all the necessary background in singular and fractional integrals, as well as Hardy spaces and the function theory of several complex variables, needed to understand Heisenberg analysis. Explorations in Harmonic Analysis is ideal for graduate students in mathematics, physics, and engineering. Prerequisites include a fundamental background in real and complex analysis and some exposure to functional analysis.

analyse (wiskunde) --- informatica --- Harmonic analysis. Fourier analysis --- differentiaalvergelijkingen --- Fourierreeksen --- Fourieranalyse --- Analytical spaces --- mathematische modellen --- wiskunde --- Group theory --- Mathematical analysis --- Computer science --- Partial differential equations --- Harmonic analysis.