►A Freud weight is a weight function of the form
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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).
►Generalized Freud weights have the form
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►For (generalized) Freud weights on a subinterval of see also Levin and Lubinsky (2005).
A numerical approach to the recursion coefficients and quadrature abscissas and weights
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►These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to , as will be considered in the following paragraphs.
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►The quadrature abscissas and weights
then follow from the discussion of §3.5(vi).
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►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain .
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►The quadrature points and weights can be put to a more direct and efficient use.
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§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
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►For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge.
Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i.
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Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applicationsa►