weighted means

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11—13 of 13 matching pages

11: 18.2 General Orthogonal Polynomials

… ►For OP’s

\{p_{n}(x)\}

\mathbb{R}

with respect to an even weight function

w(x)

we have … ►More generally, §18.30 defines the recurrence relation of the

c

th associated monic OP by means of a similar shift by

c

in (18.2.11_5). … ►Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). … ►

Monotonic Weight Functions

… ► …

12: 1.4 Calculus of One Variable

… ►

Mean Value Theorem

… ►For

\alpha(x)

nondecreasing on the closure

I

of an interval

(a,b)

, the measure

\,\mathrm{d}\alpha

is absolutely continuous if

\alpha(x)

is continuous and there exists a weight function

w(x)\geq 0

, Riemann (or Lebesgue) integrable on finite subintervals of

I

, such that … ►

1.4.23_2

\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=\int_{a}^{b}f(x)w(x)\,\mathrm{d}x,

f

integrable with respect to

\,\mathrm{d}\alpha

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $w(x)$ : weight function
Referenced by:: §1.4(v)
Permalink:: http://dlmf.nist.gov/1.4.E23_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►

First Mean Value Theorem

… ►

Second Mean Value Theorem

…

13: 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. …Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. … ►Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2. …