IMA Journal of Numerical Analysis, ISSN 0272-4979, 07/2018, Volume 38, Issue 3, pp. 1085 - 1118

Abstract Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form...

asymptotic analysis | numerical software | Hermite polynomials | Laguerre polynomials | Riemann-Hilbert problems | MATHEMATICS, APPLIED | STEEPEST DESCENT METHOD | ALGORITHM | GAUSS-LEGENDRE | RESPECT | QUADRATURE NODES | WEIGHTS | ROOTS | COMPUTATION

asymptotic analysis | numerical software | Hermite polynomials | Laguerre polynomials | Riemann-Hilbert problems | MATHEMATICS, APPLIED | STEEPEST DESCENT METHOD | ALGORITHM | GAUSS-LEGENDRE | RESPECT | QUADRATURE NODES | WEIGHTS | ROOTS | COMPUTATION

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2010, Volume 47, Issue 6, pp. 4326 - 4355

We obtain exponentially accurate Fourier series for nonperiodic functions on the interval [−1, 1] by extending these functions to periodic functions on a...

Numerical quadratures | Approximation | Numerical methods | Least squares | Eigenvalues | Polynomials | Mathematical functions | Fourier series | Gaussian quadratures | Fourier coefficients | Frames | Orthogonal polynomials | MATHEMATICS, APPLIED | frames | GAUSS QUADRATURE | CLENSHAW-CURTIS QUADRATURE | C-INFINITY | HIGH-ORDER | ALGORITHMS | least squares | ASYMPTOTICS | RULES | EQUATION | CONTINUATION | orthogonal polynomials | TRANSFORMS | Intervals | Numerical analysis | Periodic functions | Mathematical analysis | Fourier analysis | Mathematical models | Convergence

Numerical quadratures | Approximation | Numerical methods | Least squares | Eigenvalues | Polynomials | Mathematical functions | Fourier series | Gaussian quadratures | Fourier coefficients | Frames | Orthogonal polynomials | MATHEMATICS, APPLIED | frames | GAUSS QUADRATURE | CLENSHAW-CURTIS QUADRATURE | C-INFINITY | HIGH-ORDER | ALGORITHMS | least squares | ASYMPTOTICS | RULES | EQUATION | CONTINUATION | orthogonal polynomials | TRANSFORMS | Intervals | Numerical analysis | Periodic functions | Mathematical analysis | Fourier analysis | Mathematical models | Convergence

Journal Article

Advances in Computational Mathematics, ISSN 1019-7168, 8/2016, Volume 42, Issue 4, pp. 791 - 822

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞ $\infty $ . These are defined...

Jacobi polynomials | Visualization | Computational Mathematics and Numerical Analysis | 30E10 | 33C45 | Mathematical and Computational Biology | Numerical software | Mathematics | Computational Science and Engineering | Riemann–Hilbert problems | 35Q15 | 26C04 | Mathematical Modeling and Industrial Mathematics | 65D15 | Asymptotic analysis | MATHEMATICS, APPLIED | ALGORITHM | PARAMETERS | GAUSS-LEGENDRE | NUMERICAL-SOLUTION | Riemann-Hilbert problems | QUADRATURE NODES | WEIGHTS | STEEPEST DESCENT | COMPUTATION | Asymptotic expansions | Analysis | Polynomials

Jacobi polynomials | Visualization | Computational Mathematics and Numerical Analysis | 30E10 | 33C45 | Mathematical and Computational Biology | Numerical software | Mathematics | Computational Science and Engineering | Riemann–Hilbert problems | 35Q15 | 26C04 | Mathematical Modeling and Industrial Mathematics | 65D15 | Asymptotic analysis | MATHEMATICS, APPLIED | ALGORITHM | PARAMETERS | GAUSS-LEGENDRE | NUMERICAL-SOLUTION | Riemann-Hilbert problems | QUADRATURE NODES | WEIGHTS | STEEPEST DESCENT | COMPUTATION | Asymptotic expansions | Analysis | Polynomials

Journal Article

SIAM REVIEW, ISSN 0036-1445, 09/2019, Volume 61, Issue 3, pp. 443 - 473

Functions of one or more variables are usually approximated with a basis: a complete, linearly independent system of functions that spans a suitable function...

RIESZ BASIS | MATHEMATICS, APPLIED | SMOOTH FUNCTIONS | RECOVERING EXPONENTIAL ACCURACY | frames | ill-conditioning | FOURIER EXTENSIONS | FINITE SECTION METHOD | ALGORITHMS | OPERATOR | function approximation | SPHEROIDAL WAVE-FUNCTIONS | TIGHT FRAMES | UNCERTAINTY | singular value decomposition

RIESZ BASIS | MATHEMATICS, APPLIED | SMOOTH FUNCTIONS | RECOVERING EXPONENTIAL ACCURACY | frames | ill-conditioning | FOURIER EXTENSIONS | FINITE SECTION METHOD | ALGORITHMS | OPERATOR | function approximation | SPHEROIDAL WAVE-FUNCTIONS | TIGHT FRAMES | UNCERTAINTY | singular value decomposition

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2018, Volume 56, Issue 3, pp. 1360 - 1385

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding...

Arbitrary domain | Fourier extension | Fourier series | Plunge region | IRREGULAR REGION | MATHEMATICS, APPLIED | plunge region | HIGH-ORDER | C-INFINITY | TIME | ALGORITHMS | EIGENVALUE DISTRIBUTION | SPHEROIDAL WAVE-FUNCTIONS | UNCERTAINTY | SMOOTH DOMAINS | arbitrary domain | CONTINUATION

Arbitrary domain | Fourier extension | Fourier series | Plunge region | IRREGULAR REGION | MATHEMATICS, APPLIED | plunge region | HIGH-ORDER | C-INFINITY | TIME | ALGORITHMS | EIGENVALUE DISTRIBUTION | SPHEROIDAL WAVE-FUNCTIONS | UNCERTAINTY | SMOOTH DOMAINS | arbitrary domain | CONTINUATION

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 2009, Volume 231, Issue 2, pp. 933 - 947

Newton–Cotes quadrature rules are based on polynomial interpolation in a set of equidistant points. They are very useful in applications where sampled function...

Least squares approximation | Discrete orthogonal polynomials | Numerical integration | MATHEMATICS, APPLIED | ORTHOGONAL POLYNOMIALS

Least squares approximation | Discrete orthogonal polynomials | Numerical integration | MATHEMATICS, APPLIED | ORTHOGONAL POLYNOMIALS

Journal Article

International Journal of Mechanical Sciences, ISSN 0020-7403, 01/2019, Volume 150, pp. 691 - 704

•Description of an equilibrium state of sound wave reflections between multiple scattering obstacles.•Computation of the asymptotic behaviour of the...

High-frequency scattering | Periodic orbits | Phase extraction | Oscillatory integration | Multiple obstacles | Boundary element method | APPROXIMATION | ENGINEERING, MECHANICAL | MECHANICS | BOUNDARY-ELEMENT METHODS | HIGH-FREQUENCY | QUANTIZATION | EQUATION

High-frequency scattering | Periodic orbits | Phase extraction | Oscillatory integration | Multiple obstacles | Boundary element method | APPROXIMATION | ENGINEERING, MECHANICAL | MECHANICS | BOUNDARY-ELEMENT METHODS | HIGH-FREQUENCY | QUANTIZATION | EQUATION

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2016, Volume 38, Issue 2, pp. A899 - A922

Fourier series of smooth, nonperiodic functions on [-1, 1] are known to exhibit the Gibbs phenomenon, and they exhibit overall slow convergence. One way of...

Fourier continuation | Fourier extension | Fourier series | Prolate Spheroidal Wave functions | MATHEMATICS, APPLIED | SEQUENCES | RESOLUTION | SIGNALS | HIGH-ORDER | EXTRAPOLATION | SPHEROIDAL WAVE-FUNCTIONS | UNCERTAINTY | ERROR | CONTINUATION

Fourier continuation | Fourier extension | Fourier series | Prolate Spheroidal Wave functions | MATHEMATICS, APPLIED | SEQUENCES | RESOLUTION | SIGNALS | HIGH-ORDER | EXTRAPOLATION | SPHEROIDAL WAVE-FUNCTIONS | UNCERTAINTY | ERROR | CONTINUATION

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 1/2006, Volume 44, Issue 3, pp. 1026 - 1048

We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived...

Numerical quadratures | Interpolation | Approximation | Integrands | Mathematical integrals | Polynomials | Region of integration | Mathematical functions | Gaussian quadratures | Degrees of polynomials | Numerical quadrature | Steepest descent method | Oscillatory functions | MATHEMATICS, APPLIED | oscillatory functions | PLANE | steepest descent method | numerical quadrature | DERIVATIVES | TRANSFORMS

Numerical quadratures | Interpolation | Approximation | Integrands | Mathematical integrals | Polynomials | Region of integration | Mathematical functions | Gaussian quadratures | Degrees of polynomials | Numerical quadrature | Steepest descent method | Oscillatory functions | MATHEMATICS, APPLIED | oscillatory functions | PLANE | steepest descent method | numerical quadrature | DERIVATIVES | TRANSFORMS

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 2/2019, Volume 78, Issue 2, pp. 710 - 745

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically...

Computational Mathematics and Numerical Analysis | Compression | 65R20 | Theoretical, Mathematical and Computational Physics | 65N38 | 45M05 | High-frequency scattering | Mathematics | Oscillatory integration | 65F50 | 45A05 | Algorithms | Mathematical and Computational Engineering | Condition number | Boundary element method | Smooth window functions | MATHEMATICS, APPLIED | HELMHOLTZ-EQUATION | DECOMPOSITION | QUADRATURE | INTEGRAL-EQUATIONS | SCATTERING | Wave propagation

Computational Mathematics and Numerical Analysis | Compression | 65R20 | Theoretical, Mathematical and Computational Physics | 65N38 | 45M05 | High-frequency scattering | Mathematics | Oscillatory integration | 65F50 | 45A05 | Algorithms | Mathematical and Computational Engineering | Condition number | Boundary element method | Smooth window functions | MATHEMATICS, APPLIED | HELMHOLTZ-EQUATION | DECOMPOSITION | QUADRATURE | INTEGRAL-EQUATIONS | SCATTERING | Wave propagation

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 04/2014, Volume 260, pp. 312 - 336

Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon....

Oscillatory functions | Fourier extension | Fourier series | Function approximation | IRREGULAR REGION | MATHEMATICS, APPLIED | APPROXIMATION | NONPERIODIC FUNCTIONS | BOUNDARY-CONDITIONS | STABILITY | EQUATIONS | EMBEDDED DOMAIN METHODS | C-INFINITY | SPHEROIDAL WAVE-FUNCTIONS | QUADRATURE | Intervals | Algebra | Periodic functions | Mathematical analysis | Oscillations | Fourier analysis | Mathematical models

Oscillatory functions | Fourier extension | Fourier series | Function approximation | IRREGULAR REGION | MATHEMATICS, APPLIED | APPROXIMATION | NONPERIODIC FUNCTIONS | BOUNDARY-CONDITIONS | STABILITY | EQUATIONS | EMBEDDED DOMAIN METHODS | C-INFINITY | SPHEROIDAL WAVE-FUNCTIONS | QUADRATURE | Intervals | Algebra | Periodic functions | Mathematical analysis | Oscillations | Fourier analysis | Mathematical models

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 11/2014, Volume 83, Issue 290, pp. 2893 - 2914

computational scheme for barycentric weights.]]>

Hypergeometric functions | Interpolation | Roots of functions | Approximation | Polynomials | Hermite polynomials | Legendre polynomials | Gaussian quadratures | Weighting functions | Degrees of polynomials | JACOBI | NODES | MATHEMATICS, APPLIED | GAUSS-LEGENDRE | APPROXIMATION | STABILITY | DIFFERENTIAL-EQUATIONS | COMPUTATION | FORMULAS | LAGRANGE INTERPOLATION

Hypergeometric functions | Interpolation | Roots of functions | Approximation | Polynomials | Hermite polynomials | Legendre polynomials | Gaussian quadratures | Weighting functions | Degrees of polynomials | JACOBI | NODES | MATHEMATICS, APPLIED | GAUSS-LEGENDRE | APPROXIMATION | STABILITY | DIFFERENTIAL-EQUATIONS | COMPUTATION | FORMULAS | LAGRANGE INTERPOLATION

Journal Article

Wave Motion, ISSN 0165-2125, 04/2019, Volume 87, pp. 92 - 105

The Wave Based Method (WBM) is a Trefftz method for the simulation of wave problems in vibroacoustics. Like other Trefftz methods, it employs a non-standard...

Wave based method | Frames | Helmholtz | Collocation | APPROXIMATION | PHYSICS, MULTIDISCIPLINARY | FOURIER EXTENSIONS | ACOUSTICS | MECHANICS | CONVERGENCE | DOMAINS | FUNDAMENTAL-SOLUTIONS | SCATTERING | Methods | Differential equations | Helmholtz equations | Trefftz method | Vibration | Approximation | Partial differential equations | Computer simulation | Redundancy | Waveform analysis | Acoustics | Ill-conditioning (mathematics) | Stability analysis | Discretization | Collocation methods | Conditioning | Vibroacoustics | Numerical stability

Wave based method | Frames | Helmholtz | Collocation | APPROXIMATION | PHYSICS, MULTIDISCIPLINARY | FOURIER EXTENSIONS | ACOUSTICS | MECHANICS | CONVERGENCE | DOMAINS | FUNDAMENTAL-SOLUTIONS | SCATTERING | Methods | Differential equations | Helmholtz equations | Trefftz method | Vibration | Approximation | Partial differential equations | Computer simulation | Redundancy | Waveform analysis | Acoustics | Ill-conditioning (mathematics) | Stability analysis | Discretization | Collocation methods | Conditioning | Vibroacoustics | Numerical stability

Journal Article

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2007, Volume 29, Issue 6, pp. 2305 - 2328

We consider two-dimensional scattering problems, formulated as an integral equation defined on the boundary of the scattering obstacle. The oscillatory nature...

Oscillatory integrals | High-frequency scattering | Steepest descent | Integral equations | steepest descent | integral equations | MATHEMATICS, APPLIED | DIFFRACTION | high-frequency scattering | oscillatory integrals

Oscillatory integrals | High-frequency scattering | Steepest descent | Integral equations | steepest descent | integral equations | MATHEMATICS, APPLIED | DIFFRACTION | high-frequency scattering | oscillatory integrals

Journal Article

Journal of Sound and Vibration, ISSN 0022-460X, 03/2014, Volume 333, Issue 6, pp. 1796 - 1817

In a recent work, a strategy was proposed which exploits the residue theorem as an efficient tool for evaluating the frequency averaged input power into...

ACOUSTICS | MECHANICS | MODELS | ENGINEERING, MECHANICAL | CAUSALITY | Residues | Theorems | Dynamic tests | Vibration | Dynamics | Band theory | Strategy | Dynamical systems | Planes | Tools | Quadratures

ACOUSTICS | MECHANICS | MODELS | ENGINEERING, MECHANICAL | CAUSALITY | Residues | Theorems | Dynamic tests | Vibration | Dynamics | Band theory | Strategy | Dynamical systems | Planes | Tools | Quadratures

Journal Article

SIAM Journal on Applied Mathematics, ISSN 0036-1399, 1/2014, Volume 74, Issue 2, pp. 454 - 476

Several recent numerical schemes for high frequency scattering simulations are based on the extraction of known phase functions from an oscillatory solution....

Circles | Wave diffraction | Degrees of freedom | Approximation | Numerical methods | Differential equations | Boundary conditions | Textual collocation | High frequencies | Boundary points | Asymptotics | High frequency | Scattering problems | WAVE | MATHEMATICS, APPLIED | OSCILLATORY INTEGRALS | high frequency | asymptotics | scattering problems | SURFACE | DIFFRACTION | GEOMETRICAL-THEORY | EQUATION | Shadows | Asymptotic properties | Mathematical analysis | Scattering | Mathematical models | Extraction | Boundaries

Circles | Wave diffraction | Degrees of freedom | Approximation | Numerical methods | Differential equations | Boundary conditions | Textual collocation | High frequencies | Boundary points | Asymptotics | High frequency | Scattering problems | WAVE | MATHEMATICS, APPLIED | OSCILLATORY INTEGRALS | high frequency | asymptotics | scattering problems | SURFACE | DIFFRACTION | GEOMETRICAL-THEORY | EQUATION | Shadows | Asymptotic properties | Mathematical analysis | Scattering | Mathematical models | Extraction | Boundaries

Journal Article

IMA Journal of Numerical Analysis, ISSN 0272-4979, 10/2013, Volume 33, Issue 4, pp. 1322 - 1341

The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect...

integral transforms | numerical integration | orthogonal polynomials | MATHEMATICS, APPLIED | DERIVATIVES | Asymptotic properties

integral transforms | numerical integration | orthogonal polynomials | MATHEMATICS, APPLIED | DERIVATIVES | Asymptotic properties

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 2010, Volume 229, Issue 14, pp. 5357 - 5372

We consider the solution of high-frequency scattering problems in two dimensions, modeled by an integral equation on the boundary of a smooth scattering...

Helmholtz problem | Oscillatory problems | Integral equations | Filon-type method | Boundary layers | ACOUSTIC SCATTERING | GAUSSIAN QUADRATURE-RULES | SURFACE RADIATION CONDITION | IMPLEMENTATION | ITERATIVE SOLUTION | PHYSICS, MATHEMATICAL | HIGH-ORDER ALGORITHM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OSCILLATORY INTEGRALS | FORMULATIONS | DIFFRACTION | EQUATION | Boundary layer | Computation | Scattering | Differential equations | Mathematical models | Boundaries | High frequencies | Two dimensional

Helmholtz problem | Oscillatory problems | Integral equations | Filon-type method | Boundary layers | ACOUSTIC SCATTERING | GAUSSIAN QUADRATURE-RULES | SURFACE RADIATION CONDITION | IMPLEMENTATION | ITERATIVE SOLUTION | PHYSICS, MATHEMATICAL | HIGH-ORDER ALGORITHM | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OSCILLATORY INTEGRALS | FORMULATIONS | DIFFRACTION | EQUATION | Boundary layer | Computation | Scattering | Differential equations | Mathematical models | Boundaries | High frequencies | Two dimensional

Journal Article

Foundations of Computational Mathematics, ISSN 1615-3375, 8/2014, Volume 14, Issue 4, pp. 635 - 687

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed...

Economics general | Stability | 65T40 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Fourier series | Convergence | Numerical Analysis | 42A10 | Equispaced data | Applications of Mathematics | Math Applications in Computer Science | Fourier extension | Computer Science, general | 42C15 | MATHEMATICS | MATHEMATICS, APPLIED | APPROXIMATION | RUNGE PHENOMENON | GIBBS PHENOMENON | HIGH-ORDER | COMPUTER SCIENCE, THEORY & METHODS | Series | Fourier transformations | Research | Mathematical research | Applied mathematics | Fourier analysis | Linear systems | Approximation | Computation | Mathematical analysis | Mathematical models

Economics general | Stability | 65T40 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Fourier series | Convergence | Numerical Analysis | 42A10 | Equispaced data | Applications of Mathematics | Math Applications in Computer Science | Fourier extension | Computer Science, general | 42C15 | MATHEMATICS | MATHEMATICS, APPLIED | APPROXIMATION | RUNGE PHENOMENON | GIBBS PHENOMENON | HIGH-ORDER | COMPUTER SCIENCE, THEORY & METHODS | Series | Fourier transformations | Research | Mathematical research | Applied mathematics | Fourier analysis | Linear systems | Approximation | Computation | Mathematical analysis | Mathematical models

Journal Article

Journal of Sound and Vibration, ISSN 0022-460X, 07/2014, Volume 333, Issue 15, pp. 3367 - 3381

Sonic crystals can be used as acoustic lenses in certain frequencies and the design of such systems by creating vacancies and using genetic algorithms has been...

ACOUSTICS | NEGATIVE-REFRACTION | SOUND | MECHANICS | RADIATION | ENGINEERING, MECHANICAL | Computer science | Genetic research | Usage | Algorithms | Analysis | Methods

ACOUSTICS | NEGATIVE-REFRACTION | SOUND | MECHANICS | RADIATION | ENGINEERING, MECHANICAL | Computer science | Genetic research | Usage | Algorithms | Analysis | Methods

Journal Article

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