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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 … ►For asymptotic approximations to OP’s that correspond to Freud weights with more general functions $Q(x)$ see Deift et al. (1999a, b), Bleher and Its (1999), and Kriecherbauer and McLaughlin (1999). ►Generalized Freud weights have the form … ►For (generalized) Freud weights on a subinterval of $[0,\mathrm{\infty })$ see also Levin and Lubinsky (2005).

2: 32.15 Orthogonal Polynomials

… ►See also Freud (1976), Brézin et al. (1978), Fokas et al. (1992), and Magnus (1995). …

3: 18.39 Applications in the Physical Sciences

… ►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. … ►

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applications^a

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Name of OP System	$w(x)$	$[a,b]$	Notation	Applications
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Quartic Freud	$\exp \left(-x^4/4-z{x}^2\right)$	$(-\mathrm{\infty },\mathrm{\infty })$	$p_n(x)$	§32.15 and application refs. therein: Quantum Gravity and Graph Theory Combinatorics
Half-Freud Druvesteyn	$\exp \left(-x^4\right)$	$[0,\mathrm{\infty })$	$D_n(x)$	Electron Transport in Plasmasd
Half-Freud Gaussian	$\exp \left(-{(x-x_0)}^2\right)$	$[0,\mathrm{\infty })$	$G_n(x)$	Fokker–Planck DVRe
…

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4: Bibliography F

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

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5: Bibliography N

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Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48 (1), pp. 3–167.

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ISSN 0021-9045, Document, MathReview (T. S. Chihara), ZentralBlatt

Referenced by:

§18.32.

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6: Bibliography K

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

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Strong asymptotics of polynomials orthogonal with respect to Freud weights. Internat. Math. Res. Notices 1999 (6), pp. 299–333.

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ISSN 1073-7928, Document, MathReview (Heinrich Begehr), ZentralBlatt

Referenced by:

§18.32.

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7: 18.38 Mathematical Applications

… ►Hermite polynomials (and their Freud-weight analogs (§18.32)) play an important role in random matrix theory. …

8: Bibliography C

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Properties of generalized Freud polynomials. J. Approx. Theory 225, pp. 148–175.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Maria das Neves Rebocho)

Referenced by:

§18.32.

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BibTeX

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