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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 , ) see also Levin and Lubinsky (2005).
2: 3.5 Quadrature
Gauss Formula for a Logarithmic Weight Function
Table 3.5.14: Nodes and weights for the 5-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.15: Nodes and weights for the 10-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
x k w k
Then the weights are given by …
3: 18.40 Methods of Computation
A numerical approach to the recursion coefficients and quadrature abscissas and weights
These quadrature weights and abscissas will then allow construction of a convergent sequence of approximations to w ( x ) , as will be considered in the following paragraphs. … The quadrature abscissas x n and weights w n then follow from the discussion of §3.5(vi). … 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 w ( x ) . … The quadrature points and weights can be put to a more direct and efficient use. …
4: 12.15 Generalized Parabolic Cylinder Functions
This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function. …
5: 18.31 Bernstein–Szegő Polynomials
The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( 1 , 1 ) with respect to three types of weight function: ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x ) 1 2 ( 1 + x ) 1 2 ( ρ ( x ) ) 1 . …
6: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
For 1 β > α > 1 a finite system of Jacobi polynomials P n ( α , β ) ( x ) is orthogonal on ( 1 , ) with weight function w ( x ) = ( x 1 ) α ( x + 1 ) β . …
7: 1.7 Inequalities
1.7.8 min ( a 1 , a 2 , , a n ) M ( r ) max ( a 1 , a 2 , , a n ) ,
1.7.9 M ( r ) M ( s ) , r < s ,
8: 1.2 Elementary Algebra
§1.2(iv) Means
If r is a nonzero real number, then the weighted mean M ( r ) of n nonnegative numbers a 1 , a 2 , , a n , and n positive numbers p 1 , p 2 , , p n with …
1.2.21 M ( r ) = ( p 1 a 1 r + p 2 a 2 r + + p n a n r ) 1 / r ,
M ( 1 ) = A ,
M ( 1 ) = H ,
9: 18.39 Applications in the Physical Sciences
§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
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. …
Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applicationsa
Name of OP System w ( x ) [ a , b ] Notation Applications
Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2. …
10: 18.25 Wilson Class: Definitions
If α + 1 = N , then the weights will be positive iff one of the following eight sets of inequalities holds: …
§18.25(ii) Weights and Standardizations: Continuous Cases
18.25.2 0 p n ( x ) p m ( x ) w ( x ) d x = h n δ n , m .
18.25.4 w ( y 2 ) = 1 2 y | j Γ ( a j + i y ) Γ ( 2 i y ) | 2 ,
§18.25(iii) Weights and Normalizations: Discrete Cases