# weight of

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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 $[0,\infty)$ see also Levin and Lubinsky (2005).
###### Gauss Formula for a Logarithmic Weight Function
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 $\{p_{n}(x)\}$, $n=0,1,\dots$, are orthogonal on $(-1,1)$ with respect to three types of weight function: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. …
##### 6: 18.3 Definitions
For $-1-\beta>\alpha>-1$ a finite system of Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$ is orthogonal on $(1,\infty)$ with weight function $w(x)=(x-1)^{\alpha}(x+1)^{\beta}$. …
##### 7: 1.7 Inequalities
1.7.8 $\min(a_{1},a_{2},\dots,a_{n})\leq M(r)\leq\max(a_{1},a_{2},\dots,a_{n}),$
1.7.9 $M(r)\leq M(s),$ $r,
##### 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 $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with …
1.2.22 $M(r)=0,$ $r<0$ and $a_{1}a_{2}\dots a_{n}=0$.
$M(1)=A,$
$M(-1)=H,$
##### 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. … Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2. …
##### 10: 18.25 Wilson Class: Definitions
If $\alpha+1=-N$, 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 $\int_{0}^{\infty}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=h_{n}\delta_{n,m}.$
18.25.4 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$