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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: 3.5 Quadrature
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Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.
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Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.
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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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►Then the weights are given by
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Gauss Formula for a Logarithmic Weight Function
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3: 1.2 Elementary Algebra
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§1.2(iv) Means
… ►If is a nonzero real number, then the weighted mean of nonnegative numbers , and positive numbers with … ►
1.2.22
and .
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4: 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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5: 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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6: 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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►Then a system of polynomials , , is said to be orthogonal on with respect to the weights
if
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18.2.4
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18.2.5
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18.2.6
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7: 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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8: 1.7 Inequalities
9: 18.3 Definitions
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