Discrete orthogonal polynomials asymptotics and applications (Annals of mathematics studies, number 164)

• Main Authors: Baik, J, Kriecherbauer, T, McLaughlin, K. T.-R, Miller, P. D
• Format: eBook Book
• Language: English
• Published: Princeton Princeton University Press 2007
• Edition: 1
• Series: Annals of Mathematics Studies
• Subjects: Airy function; Analytic continuation; Analytic function; Ansatz; Approximation error; Approximation theory; Asymptote; Asymptotic analysis; Asymptotic expansion; Asymptotic formula; Asymptotic theory; Beta function; Boundary value problem; Calculation; Cauchy's integral formula; Cauchy-Riemann equations; Change of variables; Complex number; Complex plane; Correlation function; Degeneracy (mathematics); Determinant; Diagram (category theory); Discrete measure; Distribution function; Eigenvalues and eigenvectors; Equation; Estimation; Existential quantification; Explicit formulae (L-function); Factorization; Fredholm determinant; Functional derivative; Gamma function; Gradient descent; Harmonic analysis; Hermitian matrix; Homotopy; Hypergeometric function; Identity matrix; Inequality (mathematics); Integrable system; Invariant measure; Inverse scattering transform; Invertible matrix; Jacobi matrix; Joint probability distribution; Lagrange multiplier; Lax equivalence theorem; Limit (mathematics); Linear programming; Lipschitz continuity; MATHEMATICS; MATHEMATICS / Discrete Mathematics; Matrix function; Maxima and minima; Monic polynomial; Monotonic function; Morera's theorem; Neumann series; Number line; Orthogonal polynomials; Orthogonal polynomials > Asymptotic theory; Orthogonality; Orthogonalization; Parameter; Parametrix; Pauli matrices; Pointwise; Pointwise convergence; Polynomial; Polynômes orthogonaux; Potential theory; Probability; Probability distribution; Probability measure; Probability theory; Proportionality (mathematics); Quantity; Random matrix; Random variable; Rate of convergence; Rectangle; Rhombus; Riemann surface; Special case; Spectral theory; Statistic; Subset; Theorem; Théorie asymptotique; Toda lattice; Trace (linear algebra); Trace class; Transition point; Triangular matrix; Trigonometric functions; Uniform continuity; Unit vector; Upper and lower bounds; Upper half-plane; Variational inequality; Weak solution; Weight function; Wishart distribution; Subset; I0; Inequality (mathematics); Integrable system; Rectangle; Inverse scattering transform; Ansatz; Gradient descent; Invariant measure; Degeneracy (mathematics); Invertible matrix; Hypergeometric function; Correlation function; Calculation; Neumann series; Explicit formulae (L-function); Analytic continuation; Determinant; Orthogonality; Riemann surface; Existential quantification; Fredholm determinant; Homotopy; Weight function; Transition point; Unit vector; Diagram (category theory); Harmonic analysis; Jacobi matrix; Pauli matrices; Asymptotic formula; Lipschitz continuity; Orthogonalization; Statistic; Orthogonal polynomials; Approximation theory; Theorem; Asymptotic analysis; Trigonometric functions; Probability theory; Probability; Linear programming; Trace class; Weak solution; Spectral theory; Parameter; Uniform continuity; Polynomial; Triangular matrix; Identity matrix; Random matrix; Functional derivative; Hermitian matrix; Gamma function; Lagrange multiplier; Variational inequality; Eigenvalues and eigenvectors; Distribution function; Toda lattice; Random variable; Trace (linear algebra); Complex plane; Monotonic function; Number line; Estimation; Cauchy's integral formula; Matrix function; Boundary value problem; Proportionality (mathematics); Asymptotic expansion; Approximation error; Special case; Joint probability distribution; Probability measure; Airy function; Change of variables; Maxima and minima; Potential theory; Probability distribution; Rhombus; Asymptote; Upper and lower bounds; Pointwise; Monic polynomial; Beta function; Rate of convergence; Upper half-plane; Cauchy-Riemann equations; Lax equivalence theorem; Wishart distribution; Equation; Quantity; Analytic function; Discrete measure; Parametrix; Factorization; Pointwise convergence; Complex number; Morera's theorem; Limit (mathematics)
• ISBN: 0691127344; 9780691127330; 9780691127347; 0691127336; 9781400837137; 1400837138
• DOI: 10.1515/9781400837137

This book describes the theory and applications of discrete orthogonal polynomials--polynomials that are orthogonal on a finite set. Unlike other books, Discrete Orthogonal Polynomials addresses completely general weight functions and presents a new methodology for handling the discrete weights case. J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D. Miller focus on asymptotic aspects of general, nonclassical discrete orthogonal polynomials and set out applications of current interest. Topics covered include the probability theory of discrete orthogonal polynomial ensembles and the continuum limit of the Toda lattice. The primary concern throughout is the asymptotic behavior of discrete orthogonal polynomials for general, nonclassical measures, in the joint limit where the degree increases as some fraction of the total number of points of collocation. The book formulates the orthogonality conditions defining these polynomials as a kind of Riemann-Hilbert problem and then generalizes the steepest descent method for such a problem to carry out the necessary asymptotic analysis.