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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 w ( x ) = exp ( - Q ( x ) ) , - < x < ,
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).
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: 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 …
1.2.22 M ( r ) = 0 , r < 0 and a 1 a 2 a n = 0 .
M ( 1 ) = A ,
M ( - 1 ) = H ,
4: 18.36 Miscellaneous Polynomials
These are OP’s on the interval ( - 1 , 1 ) with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at - 1 and 1 to the weight function for the Jacobi polynomials. …
5: 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. …
6: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) ifThen a system of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on X with respect to the weights w x if …
18.2.4 x X x 2 n w x < , n = 0 , 1 , ,
18.2.5 h n = a b ( p n ( x ) ) 2 w ( x ) d x  or  x X ( p n ( x ) ) 2 w x ,
18.2.6 h ~ n = a b x ( p n ( x ) ) 2 w ( x ) d x  or  x X x ( p n ( x ) ) 2 w x ,
7: 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 . …
8: 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 ,
9: 18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
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 Normalizations: 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