# weighted means

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##### 1: 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 $\lim_{r\to 0}M(r)=G.$
##### 2: 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,
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: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
where the infinite sum means convergence in norm,thus generalizing the inner product of (1.18.9). When $\alpha$ is absolutely continuous, i.e. $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x$ , see §1.4(v), where the nonnegative weight function $w(x)$ is Lebesgue measurable on $X$ . In this section we will only consider the special case $w(x)=1$ , so $\,\mathrm{d}\alpha(x)=\,\mathrm{d}x$ ; in which case $L^{2}\left(X\right)\equiv L^{2}\left(X,\,\mathrm{d}x\right)$ .where the limit has to be understood in the sense of $L^{2}$ convergence in the mean:
##### 7: 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 …
##### 8: 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 $(a+k_{1},\dots,a+k_{m})$. …