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This book is intended as an introduction to harmonic analysis and generalized Gelfand pairs. Starting with the elementary theory of Fourier series and Fourier integrals, the author proceeds to abstract harmonic analysis on locally compact abelian groups and Gelfand pairs. Finally a more advanced theory of generalized Gelfand pairs is developed. This book is aimed at advanced undergraduates or beginning graduate students. The scope of the book is limited, with the aim of enabling students to reach a level suitable for starting PhD research. The main prerequisites for the book are elementary real, complex and functional analysis. In the later chapters, familiarity with some more advanced functional analysis is assumed, in particular with the spectral theory of (unbounded) self-adjoint operators on a Hilbert space. From the contents Fourier series Fourier integrals Locally compact groups Haar measures Harmonic analysis on locally compact abelian groups Theory and examples of Gelfand pairs Theory and examples of generalized Gelfand pairs

Harmonic analysis. --- Fourier analysis. --- Analysis, Fourier --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematical analysis --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Generalized Gelfand Pairs. --- Harmonic Analysis. --- Locally Compact Abelian Groups.

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Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.

Dirichlet series. --- Weyl groups. --- Weyl's groups --- Group theory --- Series, Dirichlet --- Series --- BZL pattern. --- Class I. --- Eisenstein series. --- Euler product. --- Gauss sum. --- Gelfand-Tsetlin pattern. --- Kashiwara operator. --- Kashiwara's crystal. --- Knowability Lemma. --- Kostant partition function. --- Riemann zeta function. --- Schur polynomial. --- Schützenberger involution. --- Snake Lemma. --- Statement A. --- Statement B. --- Statement C. --- Statement D. --- Statement E. --- Statement F. --- Statement G. --- Tokuyama's Theorem. --- Weyl character formula. --- Weyl denominator. --- Weyl group multiple Dirichlet series. --- Weyl vector. --- Whittaker coefficient. --- Whittaker function. --- Yang-Baxter equation. --- Yang–Baxter equation. --- accordion. --- adele group. --- affine linear transformation. --- analytic continuation. --- analytic number theory. --- archimedean place. --- basis vector. --- bijection. --- bookkeeping. --- box-circle duality. --- boxing. --- canonical indexings. --- cardinality. --- cartoon. --- circling. --- class. --- combinatorial identity. --- concurrence. --- critical resonance. --- crystal base. --- crystal graph. --- crystal. --- divisibility condition. --- double sum. --- episode. --- equivalence relation. --- f-packet. --- free abelian group. --- functional equation. --- generating function. --- global field. --- ice-type model. --- inclusion-exclusion. --- indexing. --- involution. --- isomorphism. --- knowability. --- maximality. --- nodal signature. --- nonarchimedean local field. --- noncritical resonance. --- nonzero contribution. --- p-adic group. --- p-adic integral. --- p-adic integration. --- partition function. --- polynomial. --- preaccordion. --- prototype. --- reduced root system. --- representation theory. --- residue class field. --- resonance. --- resotope. --- row sums. --- row transfer matrix. --- short pattern. --- six-vertex model. --- snakes. --- statistical mechanics. --- subsignature. --- tableaux. --- type. --- Γ-equivalence class. --- Γ-swap.

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This book presents a development of the basic facts about harmonic analysis on local fields and the n-dimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases, with respect to the form of statements, the manner of proof, and the variety of applications.The force of the analogy between the local field and Euclidean cases rests in the relationship of the field structures that underlie the respective cases. A complete classification of locally compact, non-discrete fields gives us two examples of connected fields (real and complex numbers); the rest are local fields (p-adic numbers, p-series fields, and their algebraic extensions). The local fields are studied in an effort to extend knowledge of the reals and complexes as locally compact fields.The author's central aim has been to present the basic facts of Fourier analysis on local fields in an accessible form and in the same spirit as in Zygmund's Trigonometric Series (Cambridge, 1968) and in Introduction to Fourier Analysis on Euclidean Spaces by Stein and Weiss (1971).Originally published in 1975.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Fourier analysis. --- Local fields (Algebra) --- Fields, Local (Algebra) --- Algebraic fields --- Analysis, Fourier --- Mathematical analysis --- Corps algébriques --- Fourier analysis --- 511 --- 511 Number theory --- Number theory --- Local fields (Algebra). --- Harmonic analysis. Fourier analysis --- Fourier Analysis --- Abelian group. --- Absolute continuity. --- Absolute value. --- Addition. --- Additive group. --- Algebraic extension. --- Algebraic number field. --- Bessel function. --- Beta function. --- Borel measure. --- Bounded function. --- Bounded variation. --- Boundedness. --- Calculation. --- Cauchy–Riemann equations. --- Characteristic function (probability theory). --- Complex analysis. --- Conformal map. --- Continuous function. --- Convolution. --- Coprime integers. --- Corollary. --- Coset. --- Determinant. --- Dimension (vector space). --- Dimension. --- Dirichlet kernel. --- Discrete space. --- Distribution (mathematics). --- Endomorphism. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier series. --- Fourier transform. --- Gamma function. --- Gelfand. --- Haar measure. --- Haar wavelet. --- Half-space (geometry). --- Hankel transform. --- Hardy's inequality. --- Harmonic analysis. --- Harmonic function. --- Homogeneous distribution. --- Integer. --- Lebesgue integration. --- Linear combination. --- Linear difference equation. --- Linear map. --- Linear space (geometry). --- Local field. --- Lp space. --- Maximal ideal. --- Measurable function. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Modular form. --- Multiplicative group. --- Norbert Wiener. --- P-adic number. --- Poisson kernel. --- Power series. --- Prime ideal. --- Probability. --- Product metric. --- Rational number. --- Regularization (mathematics). --- Requirement. --- Ring (mathematics). --- Ring of integers. --- Scalar multiplication. --- Scientific notation. --- Sign (mathematics). --- Smoothness. --- Special case. --- Special functions. --- Subgroup. --- Subring. --- Support (mathematics). --- Theorem. --- Topological space. --- Unitary operator. --- Vector space. --- Analyse harmonique (mathématiques) --- Analyse harmonique (mathématiques) --- Corps algébriques

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In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.

Research & information: general --- Mathematics & science --- condensing function --- approximate endpoint criterion --- quantum integro-difference BVP --- existence --- fractional Kadomtsev-Petviashvili system --- lie group analysis --- power series solutions --- convergence analysis --- conservation laws --- symmetry --- weighted fractional operators --- convex functions --- HHF type inequality --- fractional calculus --- Euler–Lagrange equation --- natural boundary conditions --- time delay --- MHD equations --- weak solution --- regularity criteria --- anisotropic Lorentz space --- Sonine kernel --- general fractional derivative of arbitrary order --- general fractional integral of arbitrary order --- first fundamental theorem of fractional calculus --- second fundamental theorem of fractional calculus --- ρ-Laplace variational iteration method --- ρ-Laplace decomposition method --- partial differential equation --- caputo operator --- fractional Fornberg–Whitham equation (FWE) --- Riemann–Liouville fractional difference operator --- boundary value problem --- discrete fractional calculus --- existence and uniqueness --- Ulam stability --- elastic beam problem --- tempered fractional derivative --- one-sided tempered fractional derivative --- bilateral tempered fractional derivative --- tempered riesz potential --- collocation method --- hermite cubic spline --- fractional burgers equation --- fractional differential equation --- fractional Dzhrbashyan–Nersesyan derivative --- degenerate evolution equation --- initial value problem --- initial boundary value problem --- partial Riemann–Liouville fractional integral --- Babenko’s approach --- Banach fixed point theorem --- Mittag–Leffler function --- gamma function --- nabla fractional difference --- separated boundary conditions --- Green’s function --- existence of solutions --- Caputo q-derivative --- singular sum fractional q-differential --- fixed point --- equations --- Riemann–Liouville q-integral --- Shehu transform --- Caputo fractional derivative --- Shehu decomposition method --- new iterative transform method --- fractional KdV equation --- approximate solutions --- Riemann–Liouville derivative --- concave operator --- fixed point theorem --- Gelfand problem --- order cone --- integral transform --- Atangana–Baleanu fractional derivative --- Aboodh transform iterative method --- φ-Hilfer fractional system with impulses --- semigroup theory --- nonlocal conditions --- optimal controls --- fractional derivatives --- fractional Prabhakar derivatives --- fractional differential equations --- fractional Sturm–Liouville problems --- eigenfunctions and eigenvalues --- Fredholm–Volterra integral Equations --- fractional derivative --- Bessel polynomials --- Caputo derivative --- collocation points --- Caputo–Fabrizio and Atangana-Baleanu operators --- time-fractional Kaup–Kupershmidt equation --- natural transform --- Adomian decomposition method --- condensing function --- approximate endpoint criterion --- quantum integro-difference BVP --- existence --- fractional Kadomtsev-Petviashvili system --- lie group analysis --- power series solutions --- convergence analysis --- conservation laws --- symmetry --- weighted fractional operators --- convex functions --- HHF type inequality --- fractional calculus --- Euler–Lagrange equation --- natural boundary conditions --- time delay --- MHD equations --- weak solution --- regularity criteria --- anisotropic Lorentz space --- Sonine kernel --- general fractional derivative of arbitrary order --- general fractional integral of arbitrary order --- first fundamental theorem of fractional calculus --- second fundamental theorem of fractional calculus --- ρ-Laplace variational iteration method --- ρ-Laplace decomposition method --- partial differential equation --- caputo operator --- fractional Fornberg–Whitham equation (FWE) --- Riemann–Liouville fractional difference operator --- boundary value problem --- discrete fractional calculus --- existence and uniqueness --- Ulam stability --- elastic beam problem --- tempered fractional derivative --- one-sided tempered fractional derivative --- bilateral tempered fractional derivative --- tempered riesz potential --- collocation method --- hermite cubic spline --- fractional burgers equation --- fractional differential equation --- fractional Dzhrbashyan–Nersesyan derivative --- degenerate evolution equation --- initial value problem --- initial boundary value problem --- partial Riemann–Liouville fractional integral --- Babenko’s approach --- Banach fixed point theorem --- Mittag–Leffler function --- gamma function --- nabla fractional difference --- separated boundary conditions --- Green’s function --- existence of solutions --- Caputo q-derivative --- singular sum fractional q-differential --- fixed point --- equations --- Riemann–Liouville q-integral --- Shehu transform --- Caputo fractional derivative --- Shehu decomposition method --- new iterative transform method --- fractional KdV equation --- approximate solutions --- Riemann–Liouville derivative --- concave operator --- fixed point theorem --- Gelfand problem --- order cone --- integral transform --- Atangana–Baleanu fractional derivative --- Aboodh transform iterative method --- φ-Hilfer fractional system with impulses --- semigroup theory --- nonlocal conditions --- optimal controls --- fractional derivatives --- fractional Prabhakar derivatives --- fractional differential equations --- fractional Sturm–Liouville problems --- eigenfunctions and eigenvalues --- Fredholm–Volterra integral Equations --- fractional derivative --- Bessel polynomials --- Caputo derivative --- collocation points --- Caputo–Fabrizio and Atangana-Baleanu operators --- time-fractional Kaup–Kupershmidt equation --- natural transform --- Adomian decomposition method

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In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia.

condensing function --- approximate endpoint criterion --- quantum integro-difference BVP --- existence --- fractional Kadomtsev-Petviashvili system --- lie group analysis --- power series solutions --- convergence analysis --- conservation laws --- symmetry --- weighted fractional operators --- convex functions --- HHF type inequality --- fractional calculus --- Euler–Lagrange equation --- natural boundary conditions --- time delay --- MHD equations --- weak solution --- regularity criteria --- anisotropic Lorentz space --- Sonine kernel --- general fractional derivative of arbitrary order --- general fractional integral of arbitrary order --- first fundamental theorem of fractional calculus --- second fundamental theorem of fractional calculus --- ρ-Laplace variational iteration method --- ρ-Laplace decomposition method --- partial differential equation --- caputo operator --- fractional Fornberg–Whitham equation (FWE) --- Riemann–Liouville fractional difference operator --- boundary value problem --- discrete fractional calculus --- existence and uniqueness --- Ulam stability --- elastic beam problem --- tempered fractional derivative --- one-sided tempered fractional derivative --- bilateral tempered fractional derivative --- tempered riesz potential --- collocation method --- hermite cubic spline --- fractional burgers equation --- fractional differential equation --- fractional Dzhrbashyan–Nersesyan derivative --- degenerate evolution equation --- initial value problem --- initial boundary value problem --- partial Riemann–Liouville fractional integral --- Babenko’s approach --- Banach fixed point theorem --- Mittag–Leffler function --- gamma function --- nabla fractional difference --- separated boundary conditions --- Green’s function --- existence of solutions --- Caputo q-derivative --- singular sum fractional q-differential --- fixed point --- equations --- Riemann–Liouville q-integral --- Shehu transform --- Caputo fractional derivative --- Shehu decomposition method --- new iterative transform method --- fractional KdV equation --- approximate solutions --- Riemann–Liouville derivative --- concave operator --- fixed point theorem --- Gelfand problem --- order cone --- integral transform --- Atangana–Baleanu fractional derivative --- Aboodh transform iterative method --- φ-Hilfer fractional system with impulses --- semigroup theory --- nonlocal conditions --- optimal controls --- fractional derivatives --- fractional Prabhakar derivatives --- fractional differential equations --- fractional Sturm–Liouville problems --- eigenfunctions and eigenvalues --- Fredholm–Volterra integral Equations --- fractional derivative --- Bessel polynomials --- Caputo derivative --- collocation points --- Caputo–Fabrizio and Atangana-Baleanu operators --- time-fractional Kaup–Kupershmidt equation --- natural transform --- Adomian decomposition method