# weighted means

(0.002 seconds)

## 1—10 of 13 matching pages

##### 1: 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,
##### 2: 1.2 Elementary Algebra
1.2.21 $M(r)=(p_{1}a_{1}^{r}+p_{2}a_{2}^{r}+\dots+p_{n}a_{n}^{r})^{1/r},$
1.2.22 $M(r)=0,$ $r<0$ and $a_{1}a_{2}\dots a_{n}=0$.
$M(1)=A,$
$M(-1)=H,$
1.2.26 $\lim_{r\to 0}M(r)=G.$
These can be found by means of the recursion … An interpolatory quadrature ruleThe nodes $x_{k}$ and weights $w_{k}$ are known explicitly: …
###### Gauss Formula for a Logarithmic Weight Function
Then the weights are given by …
##### 4: Bibliography
• S. Ahmed and M. E. Muldoon (1980) On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.
• G. Allasia and R. Besenghi (1989) Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
• G. Almkvist and B. Berndt (1988) Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.
• H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.
• Y. Ameur and J. Cronvall (2023) Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.
• ##### 5: Bibliography G
• L. Gårding (1947) The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.
• G. Gasper (1981) Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.
• W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
• W. Gautschi (1992) On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.
• ##### 6: Bibliography D
• P. Deift, T. Kriecherbauer, K. T. McLaughlin, S. Venakides, and X. Zhou (1999a) Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52 (12), pp. 1491–1552.
• P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
• J. Deltour (1968) The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.
• A. J. Durán (1993) Functions with given moments and weight functions for orthogonal polynomials. Rocky Mountain J. Math. 23, pp. 87–104.
• ##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
where the infinite sum means convergence in norm, … $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x$, see §1.4(v), where the nonnegative weight function $w(x)$ is Lebesgue measurable on $X$. … where the limit has to be understood in the sense of $L^{2}$ convergence in the mean: …
##### 8: 3.11 Approximation Techniques
Here the single prime on the summation symbol means that the first term is to be halved. …
###### §3.11(iii) Minimax Rational Approximations
Then the minimax (or best uniform) rational approximation … Then (3.11.29) is replaced by …
##### 9: 18.30 Associated OP’s
with weight function … For other cases there may also be, in addition to a possible integral as in (18.30.10), a finite sum of discrete weights on the negative real $x$-axis each multiplied by the polynomial product evaluated at the corresponding values of $x$, as in (18.2.3). … with weight function … with weight function … Ismail (2009, §2.3) discusses the meaning of linearly independent in this situation. …
##### 10: 35.4 Partitions and Zonal Polynomials
###### §35.4(i) Definitions
Also, $|\kappa|$ denotes $k_{1}+\dots+k_{m}$, the weight of $\kappa$; $\ell(\kappa)$ denotes the number of nonzero $k_{j}$; $a+\kappa$ denotes the vector $(a+k_{1},\dots,a+k_{m})$. …