weighted means

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1: 1.2 Elementary Algebra

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1.2.21 $$M(r)={(p_1{a}_1^r+p_2{a}_2^r+\mathrm{\cdots }+p_n{a}_n^r)}^{1/r},$$

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Defines:

$M(r)$: weighted mean (locally)

Symbols:

$n$: nonnegative integer and $p_j$; positive numbers

Permalink:

http://dlmf.nist.gov/1.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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1.2.22 $$M(r)=0,$$ and $a_1{a}_2\mathrm{\dots }{a}_n=0$.

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.1.15

Permalink:

http://dlmf.nist.gov/1.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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$M(1)=A,$

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$M(-1)=H,$

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1.2.26 $$\underset{r\to 0}{lim}M(r)=G.$$

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Symbols:

$G$: geometric mean and $M(r)$: weighted mean

A&S Ref:

3.1.18

Permalink:

http://dlmf.nist.gov/1.2.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §1.2(iv), §1.2 and Ch.1

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2: 1.7 Inequalities

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1.7.8 $$\min (a_1,a_2,\mathrm{\dots },a_n)\le M(r)\le \max (a_1,a_2,\mathrm{\dots },a_n),$$

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Symbols:

$n$: nonnegative integer and $M(r)$: weighted mean

A&S Ref:

3.2.2

Permalink:

http://dlmf.nist.gov/1.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

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1.7.9 $$M(r)\le M(s),$$ ,

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Symbols:

$M(r)$: weighted mean

A&S Ref:

3.2.4

Permalink:

http://dlmf.nist.gov/1.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.7(iii), §1.7 and Ch.1

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3: 3.5 Quadrature

… ►These can be found by means of the recursion … ►An interpolatory quadrature rule … ►The 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

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On the zeros of confluent hypergeometric functions. III. Characterization by means of nonlinear equations. Lett. Nuovo Cimento (2) 29 (11), pp. 353–358.

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External Links:

ISSN 0024-1318, MathReview (James M. Horner)

Referenced by:

§13.9(i).

Encodings:

BibTeX

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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.

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External Links:

MathReview

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, $\pi$, and the Ladies Diary. Amer. Math. Monthly 95 (7), pp. 585–608.

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External Links:

ISSN 0002-9890, FullText, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.9(i).

Encodings:

BibTeX

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A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.

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Notes:

For corrigendum see same journal v. 90 (1998), no. 2, p. 265

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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Szegő Type Asymptotics for the Reproducing Kernel in Spaces of Full-Plane Weighted Polynomials. Comm. Math. Phys. 398 (3), pp. 1291–1348.

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External Links:

ISSN 0010-3616, Document, Link, MathReview Entry

Referenced by:

§8.11(iii), §8.11(iii), Erratum (V1.1.10) for Section 8.11(iii).

Encodings:

BibTeX

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5: Bibliography G

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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.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

Encodings:

BibTeX

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Orthogonality of certain functions with respect to complex valued weights. Canad. J. Math. 33 (5), pp. 1261–1270.

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External Links:

ISSN 0008-414X, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(v).

Encodings:

BibTeX

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A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

BibTeX

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On mean convergence of extended Lagrange interpolation. J. Comput. Appl. Math. 43 (1-2), pp. 19–35.

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Notes:

Orthogonal polynomials and numerical methods

External Links:

ISSN 0377-0427, Document, Link, MathReview (Mircea Ivan)

Referenced by:

Erratum (V1.0.28) for Table 18.3.1.

Encodings:

BibTeX

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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. $d\alpha (x)=w(x)dx$ , 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 $d\alpha (x)=dx$ ; in which case $L^2\left(X\right)\equiv L^2(X,dx)$ . … ►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

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§35.4(i) Definitions

… ►Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. … ►

Mean-Value

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