# 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
18.32.1 ${w(x)=\exp\left(-Q(x)\right)},$ $-\infty,
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).
###### Gauss Formula for a Logarithmic Weight Function
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 $a_{1},a_{2},\dots,a_{n}$, and $n$ positive numbers $p_{1},p_{2},\dots,p_{n}$ with …
1.2.22 $M(r)=0,$ $r<0$ and $a_{1}a_{2}\dots a_{n}=0$.
$M(1)=A,$
$M(-1)=H,$
##### 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 $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ($\geq 0$) ifThen a system of polynomials $\{p_{n}(x)\}$, $n=0,1,2,\ldots$, is said to be orthogonal on $X$ with respect to the weights $w_{x}$ if …
##### 7: 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: $(1-x^{2})^{-\frac{1}{2}}(\rho(x))^{-1}$, $(1-x^{2})^{\frac{1}{2}}(\rho(x))^{-1}$, $(1-x)^{\frac{1}{2}}(1+x)^{-\frac{1}{2}}(\rho(x))^{-1}$. …
##### 8: 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,
##### 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 Normalizations: 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 $w(y^{2})=\frac{1}{2y}\left|\frac{\prod_{j}\Gamma\left(a_{j}+iy\right)}{\Gamma% \left(2iy\right)}\right|^{2},$