weight functions
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1: 18.32 OP’s with Respect to Freud Weights
§18.32 OP’s with Respect to Freud Weights
►A Freud weight is a weight function of the form ►
18.32.1
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►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions
see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999).
2: 12.15 Generalized Parabolic Cylinder Functions
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►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function.
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3: 18.31 Bernstein–Szegő Polynomials
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►The Bernstein–Szegő polynomials
, , are orthogonal on with respect to three types of weight function: , , .
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4: 18.3 Definitions
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5: 18.2 General Orthogonal Polynomials
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►A system (or set) of polynomials , , is said to be orthogonal on
with respect to the weight function
() if
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18.2.1
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18.2.5
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18.2.6
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6: 3.5 Quadrature
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►An interpolatory quadrature rule
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Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
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Gauss Formula for a Logarithmic Weight Function
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7: 18.36 Miscellaneous Polynomials
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►These are OP’s on the interval with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at and to the weight function for the Jacobi polynomials.
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8: 18.19 Hahn Class: Definitions
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Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, normalizations, and parameter constraints.
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18.19.2
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18.19.3
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18.19.7
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9: 18.33 Polynomials Orthogonal on the Unit Circle
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►A system of polynomials , , where is of proper degree , is orthonormal on the unit circle with respect
to the weight function
() if
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18.33.1
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►Let and , , be OP’s with weight functions
and , respectively, on .
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18.33.16
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