Favard theorem

SearchAdvancedHelp

(0.001 seconds)

3 matching pages

1: 18.2 General Orthogonal Polynomials

… ►If polynomials $p_n(x)$ are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the $p_n(x)$ are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function $\alpha (x)$ on $(a,b)$ yielding the orthogonality realtion (18.2.4_5) is guaranteed. … ►

§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

►If a system of polynomials $\{p_n(x)\}$ satisfies any of the formula pairs (recurrence relation and coefficient inequality) (18.2.8), (18.2.9_5) or (18.2.10), (18.2.11_2) or (18.2.11_5), (18.2.11_6) or (18.2.11_8), (18.2.11_6) then $\{p_n(x)\}$ is orthogonal with respect to some positive measure on $ℝ$ (Favard’s theorem). …

2: 18.36 Miscellaneous Polynomials

… ►Orthogonality of the the classical OP’s with respect to a positive weight function, as in Table 18.3.1 requires, via Favard’s theorem, $A_n{A}_{n-1}{C}_n>0$ for $n\ge 1$ as per (18.2.9_5). …

3: 18.35 Pollaczek Polynomials

… ►

18.35.6_2

ⓘ

Symbols:

$n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, (5.5.8)); Case (iii) is overlooked in the given reference.(proved)

Proof sketch:

By Favard’s theorem (see §18.2(viii)) the necessary and sufficient condition for orthogonality is that in (18.35.2_4), for $c=0$, the coefficient of $Q_n^{(\lambda )}(x;a,b,0)$ is real for $n\ge 0$ and the coefficient of $Q_{n-1}^{(\lambda )}(x;a,b,0)$ is positive for $n\ge 1$.

Referenced by:

§18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

…