discrete weights
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1: 18.39 Applications in the Physical Sciences
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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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2: 18.30 Associated OP’s
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►For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real -axis each multiplied by the polynomial product evaluated at the corresponding values of , as in (18.2.3).
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3: 18.19 Hahn Class: Definitions
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4: 18.28 Askey–Wilson Class
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►The Askey–Wilson polynomials form a system of OP’s , , that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set.
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5: 18.25 Wilson Class: Definitions
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§18.25(iii) Weights and Normalizations: Discrete Cases
…6: 18.38 Mathematical Applications
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►The basic ideas of Gaussian quadrature, and their extensions to non-classical weight functions, and the computation of the corresponding quadrature abscissas and weights, have led to discrete variable representations, or DVRs, of Sturm–Liouville and other differential operators.
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7: 3.11 Approximation Techniques
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►Now suppose that when , that is, the functions
are orthogonal with respect to weighted summation on the
discrete set
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8: 18.35 Pollaczek Polynomials
9: 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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►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials , , are orthogonal on the discrete point set comprising the zeros , of :
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►For another version of the discrete orthogonality property of the polynomials see (3.11.9).
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►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007).
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►For a finite system of Jacobi polynomials is orthogonal on with weight function .
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