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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). …
18.32.2 w ( x ) = | x | α exp ( Q ( x ) ) , x ,  α > 1 ,
2: 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. …
3: 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 . …
4: 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 . …
5: 3.5 Quadrature
An interpolatory quadrature rule
Gauss Formula for a Logarithmic Weight Function
Table 3.5.17: Nodes and weights for the 20-point Gauss formula for the logarithmic weight function.
x k w k
Table 3.5.18: Nodes and weights for the 5-point complex Gauss quadrature formula with s = 1 .
ζ k w k
Table 3.5.21: Cubature formulas for disk and square.
Diagram ( x j , y j ) w j R
6: 18.39 Applications in the Physical Sciences
§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. …
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. … In the attractive case (18.35.6_4) for the discrete parts of the weight function where with x k < 1 , are also simplified: …
7: 18.33 Polynomials Orthogonal on the Unit Circle
A system of polynomials { ϕ n ( z ) } , n = 0 , 1 , , where ϕ n ( z ) is of proper degree n , is orthonormal on the unit circle with respect to the weight function w ( z ) ( 0 ) if … Instead of (18.33.9) one might take monic OP’s { q n ( x ) } with weight function ( 1 + x ) w 1 ( x ) , and then express q n ( 1 2 ( z + z 1 ) ) in terms of ϕ 2 n ( z ± 1 ) or ϕ 2 n + 1 ( z ± 1 ) . …
18.33.19 d μ ( z ) = 1 2 π i w ( z ) d z z
for some weight function w ( z ) ( 0 ) then (18.33.17) (see also (18.33.1)) takes the form … For w ( z ) as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure μ in (18.33.17)) and with α n the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …
8: 18.2 General Orthogonal Polynomials
A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , where p n ( x ) has degree n as in §18.1(i), is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) ifFor OP’s { p n ( x ) } on with respect to an even weight function w ( x ) we have … Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). …
Monotonic Weight Functions
9: 18.40 Methods of Computation
18.40.4 lim N F N ( z ) = F ( z ) 1 μ 0 a b w ( x ) d x z x , z \ [ a , b ] ,
18.40.6 lim ε 0 + a b w ( x ) d x x + i ε x d x = a b w ( x ) d x x x i π w ( x ) ,
See accompanying text
Figure 18.40.1: Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over [ 1 , x ) , see Figure 18.39.2 for an exact result, for Z = + 1 , shown for N = 12 and N = 120 . Magnify
18.40.8 w ( x i , N ) w i , N d x ( j , N ) d j | j = i .
The example chosen is inversion from the α n , β n for the weight function for the repulsive Coulomb–Pollaczek, RCP, polynomials of (18.39.50). …
10: 18.19 Hahn Class: Definitions
Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.
p n ( x ) X w x h n
18.19.2 w ( z ; a , b , a ¯ , b ¯ ) = Γ ( a + i z ) Γ ( b + i z ) Γ ( a ¯ i z ) Γ ( b ¯ i z ) ,
18.19.3 w ( x ) = w ( x ; a , b , a ¯ , b ¯ ) = | Γ ( a + i x ) Γ ( b + i x ) | 2 ,
18.19.7 w ( λ ) ( z ; ϕ ) = Γ ( λ + i z ) Γ ( λ i z ) e ( 2 ϕ π ) z ,