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).
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18.32.2
, ,
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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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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints.
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►For a finite system of Jacobi polynomials is orthogonal on with weight function
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Name | Constraints | ||||||
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5: 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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6: 18.39 Applications in the Physical Sciences
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Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applicationsa
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►Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2.
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►In the attractive case (18.35.6_4) for the discrete parts of the weight function where with , are also simplified:
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§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
… ►Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. … ►Name of OP System | Notation | Applications | ||
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7: 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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►Instead of (18.33.9) one might take monic OP’s with weight function
, and then express in terms of or .
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18.33.19
►for some weight function
() then (18.33.17) (see also (18.33.1)) takes the form
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►For as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure in (18.33.17)) and with the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that
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8: 18.2 General Orthogonal Polynomials
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►A system (or set) of polynomials , , where has degree as in §18.1(i), is said to be orthogonal on
with respect to the weight function
() if
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►For OP’s on with respect to an even weight function
we have
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►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2).
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Monotonic Weight Functions
… ► …9: 18.40 Methods of Computation
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18.40.4
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18.40.6
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18.40.8
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►The example chosen is inversion from the for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50).
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