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Calculus. --- Càlcul --- Càlcul infinitesimal --- Límits (Matemàtica) --- Aritmètica --- Anàlisi harmònica --- Anàlisi p-àdica --- Càlcul diferencial --- Càlcul fraccional --- Càlcul mental --- Corbes --- Curvatura --- Equacions diferencials --- Sèries de Fourier --- Superfícies (Matemàtica) --- Teories no lineals --- Anàlisi matemàtica --- Concepte de nombre --- Funcions --- Geometria infinitesimal --- Mètode ABN --- Numeració --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Mathematical analysis --- Functions --- Geometry, Infinitesimal

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Fourier series. --- Hardy spaces. --- Martingales (Mathematics) --- Sèries de Fourier --- Espais de Hardy --- Martingales (Matemàtica) --- Stochastic processes --- Spaces, Hardy --- Functional analysis --- Functions of complex variables --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Fourier analysis --- Harmonic analysis --- Harmonic functions --- Processos estocàstics --- Hardy, Espacios de --- Anàlisi funcional --- Integrals de Fourier --- Sèries trigonomètriques --- Anàlisi de Fourier --- Càlcul --- Integrals de Dirichlet

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This monograph presents the summability of higher dimensional Fourier series, and generalizes the concept of Lebesgue points. Focusing on Fejér and Cesàro summability, as well as theta-summation, readers will become more familiar with a wide variety of summability methods. Within the theory of higher dimensional summability of Fourier series, the book also provides a much-needed simple proof of Lebesgue’s theorem, filling a gap in the literature. Recent results and real-world applications are highlighted as well, making this a timely resource. The book is structured into four chapters, prioritizing clarity throughout. Chapter One covers basic results from the one-dimensional Fourier series, and offers a clear proof of the Lebesgue theorem. In Chapter Two, convergence and boundedness results for the lq-summability are presented. The restricted and unrestricted rectangular summability are provided in Chapter Three, as well as the sufficient and necessary condition for the norm convergence of the rectangular theta-means. Chapter Four then introduces six types of Lebesgue points for higher dimensional functions. Lebesgue Points and Summability of Higher Dimensional Fourier Series will appeal to researchers working in mathematical analysis, particularly those interested in Fourier and harmonic analysis. Researchers in applied fields will also find this useful.

Fourier analysis. --- Sequences (Mathematics). --- Measure theory. --- Fourier Analysis. --- Sequences, Series, Summability. --- Measure and Integration. --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Mathematical sequences --- Numerical sequences --- Algebra --- Mathematics --- Analysis, Fourier --- Mathematical analysis --- Summability theory. --- Sequences (Mathematics) --- Series --- Sèries de Fourier --- Sumabilitat --- Teoria de sumabilitat --- Sèries (Matemàtica) --- Successions (Matemàtica) --- Integrals de Fourier --- Sèries trigonomètriques --- Anàlisi de Fourier --- Càlcul --- Integrals de Dirichlet

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Engineering mathematics. --- Geometry, Differential. --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Mathematical analysis --- Functions --- Geometry, Infinitesimal --- Differential geometry --- Engineering --- Engineering analysis --- Mathematics --- Matemàtica per a enginyers --- Geometria diferencial --- Càlcul --- Càlcul infinitesimal --- Límits (Matemàtica) --- Aritmètica --- Anàlisi harmònica --- Anàlisi p-àdica --- Càlcul diferencial --- Càlcul fraccional --- Càlcul mental --- Corbes --- Curvatura --- Equacions diferencials --- Sèries de Fourier --- Superfícies (Matemàtica) --- Teories no lineals --- Anàlisi matemàtica --- Concepte de nombre --- Funcions --- Geometria infinitesimal --- Mètode ABN --- Numeració --- Geometria --- Càlcul de tensors --- Connexions (Matemàtica) --- Coordenades --- Cossos convexos --- Dominis convexos --- Espais de curvatura constant --- Espais simètrics --- Estructures hermitianes --- Formes diferencials --- G-estructures --- Geodèsiques (Matemàtica) --- Geometria de Riemann --- Geometria diferencial global --- Geometria integral --- Geometria simplèctica --- Hiperespai --- Subvarietats (Matemàtica) --- Topologia diferencial --- Varietats (Matemàtica) --- Varietats de Kähler --- Matemàtica en l'electrònica --- Mecànica aplicada --- Anàlisi dimensional

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This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science. The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function. Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Mathematics. --- Predicate calculus. --- Logic, Symbolic and mathematical. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Calculus, Predicate --- Quantification theory --- Logic, Symbolic and mathematical --- Math --- Science --- Càlcul --- Lògica matemàtica --- Lògica formal --- Lògica simbòlica --- Lògica simbòlica i matemàtica --- Lògica universal --- Logística (Filosofia) --- Matemàtica --- Atzar --- Càlcul lambda --- Categories (Matemàtica) --- Decidibilitat (Lògica matemàtica) --- Funcions recursives --- Independència (Matemàtica) --- Lògica algebraica --- Lògica borrosa --- Lògica combinatòria --- Lògica de primer ordre --- Lògica informàtica --- Matemàtica constructiva --- Metodologia de la ciència --- Nombres cardinals --- Pragmàtica (Lingüística) --- Probabilitats --- Semàntica (Filosofia) --- Teoria axiomàtica de conjunts --- Teorema de Gödel --- Teoria de la commutació --- Teoria de la prova --- Teoria de la recursió --- Teoria de màquines --- Teoria de models --- Teoria de tipus --- Àlgebra abstracta --- Metamatemàtica --- Programació lògica --- Sil·logisme --- Teoria de conjunts --- Càlcul infinitesimal --- Límits (Matemàtica) --- Aritmètica --- Anàlisi harmònica --- Anàlisi p-àdica --- Càlcul diferencial --- Càlcul fraccional --- Càlcul mental --- Corbes --- Curvatura --- Equacions diferencials --- Sèries de Fourier --- Superfícies (Matemàtica) --- Teories no lineals --- Anàlisi matemàtica --- Concepte de nombre --- Funcions --- Geometria infinitesimal --- Mètode ABN --- Numeració --- Lògica difusa --- Mathematical logic. --- Computer science --- Mathematical Logic and Foundations. --- Mathematics of Computing. --- Computer mathematics --- Electronic data processing --- Lògica matemàtica.