linear operators
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11—20 of 23 matching pages
11: 18.27 -Hahn Class
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βΊThe
-hypergeometric OP’s comprise the -Hahn class (or -linear lattice class) OP’s and the Askey–Wilson class (or -quadratic lattice class) OP’s (§18.28).
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βΊThe -Hahn class OP’s comprise systems of OP’s , , or , that are eigenfunctions of a second order -difference operator.
…In the -Hahn class OP’s the role of the operator
in the Jacobi, Laguerre, and Hermite cases is played by the -derivative , as defined in (17.2.41).
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12: 3.6 Linear Difference Equations
§3.6 Linear Difference Equations
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3.6.2
βΊwhere , , and .
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§3.6(vii) Linear Difference Equations of Other Orders
…13: Bibliography T
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Numerical Linear Algebra.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Dunkl shift operators and Bannai-Ito polynomials.
Adv. Math. 229 (4), pp. 2123–2158.
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Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades.
J. Appl. Math. Mech. 23, pp. 1549–1565.
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14: 1.16 Distributions
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βΊThe linear space of all test functions with the above definition of convergence is called a test function space.
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βΊA mapping is a linear functional if
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βΊA tempered distribution is a continuous linear functional on .
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βΊA distribution in is a continuous linear functional on .
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βΊTempered distributions are continuous linear functionals on this space of test functions.
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15: Bibliography H
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An elementary approach to the hypergeometric shift operators of Opdam.
Invent. Math. 103 (2), pp. 341–350.
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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach.
2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.
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Two Parametric Eigenvalue Problems of Differential Equations.
In Spectral Theory of Differential Operators (Birmingham, AL,
1981),
North-Holland Math. Stud., Vol. 55, pp. 233–241.
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16: Bibliography K
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Quasi-linear Stokes phenomenon for the Painlevé first equation.
J. Phys. A 37 (46), pp. 11149–11167.
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Lowering and Raising Operators for Some Special Orthogonal Polynomials.
In Jack, Hall-Littlewood and Macdonald Polynomials,
Contemp. Math., Vol. 417, pp. 227–238.
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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators.
SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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17: Philip J. Davis
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βΊAt that time John Todd was Chief of the Numerical Analysis Section of the Applied Mathematics Division and head of the Computation Laboratory that co-developed, with the NBS Electronic Computer Laboratory, the Standards Eastern Automatic Computer (SEAC), the first fully operational stored-program electronic digital computer in the United States.
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βΊThis immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms.
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18: Bibliography G
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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The Computation of Special Functions by Linear Difference Equations.
In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. GyΕri, and G. Ladas (Eds.),
pp. 213–243.
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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs.
SIAM J. Numer. Anal. 41 (3), pp. 827–855.
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Linear Differential Equations and Group Theory from Riemann to Poincaré.
2nd edition, Birkhäuser Boston Inc., Boston, MA.
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Kinetic theory of linear shear flow.
Phys. Fluids 1 (3), pp. 215–224.
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19: Bibliography R
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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials.
Canad. J. Math. 33 (4), pp. 915–928.
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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators.
Academic Press, New York.
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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters.
Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
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On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators.
Z. Angew. Math. Mech. 78 (1), pp. 66–72.
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On the foundations of combinatorial theory. VIII. Finite operator calculus.
J. Math. Anal. Appl. 42, pp. 684–760.
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20: 3.11 Approximation Techniques
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βΊAlso, in cases where satisfies a linear ordinary differential equation with polynomial coefficients, the expansion (3.11.11) can be substituted in the differential equation to yield a recurrence relation satisfied by the .
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βΊWith , the last equations give as the solution of a system of linear equations.
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βΊ(3.11.29) is a system of
linear equations for the coefficients .
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βΊMore generally, let be approximated by a linear combination
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βΊIn consequence of this structure the number of operations can be reduced to
operations.
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