On estimates for the weights in Gaussian quadrature in the ultraspherical case
Authors: Förster Klaus-Jürgen , Petras Knut 
Published: June 2010

In this paper the Christoffel numbers

a

v
,
n

(
λ
)
G

a_{v,n}^{(\lambda )G}

for ultraspherical weight functions

w
λ

{w_\lambda }

,

w
λ

(
x
)
=

(
1
−

x
2

)

λ
−
1

/

2

{w_\lambda }(x) = {(1 - {x^2})^{\lambda - 1/2}}

, are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by

θ

v
,
n

(
λ
)

\theta _{v,n}^{(\lambda )}

the trigonometric representation of the Gaussian nodes, we obtain for

λ
∈
[
0
,
1
]

\lambda \in [0,1]

the inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartFraction pi Over n plus lamda EndFraction sine Superscript 2 lamda Baseline theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline left-brace 1 minus StartFraction lamda left-parenthesis 1 minus lamda right-parenthesis Over 2 left-parenthesis n plus lamda right-parenthesis squared sine squared theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline EndFraction right-brace 2nd Row less-than-or-equal-to a Subscript v comma n Superscript left-parenthesis lamda right-parenthesis upper G Baseline less-than-or-equal-to StartFraction pi Over n plus lamda EndFraction sine Superscript 2 lamda Baseline theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis EndLayout">

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

{

1
−

λ
(
1
−
λ
)

2

(
n
+
λ
)

2

sin

2

θ

v
,
n

(
λ
)

}

≤

a

v
,
n

(
λ
)
G

≤

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

\begin {array}{*{20}{c}} {\frac {\pi }{{n + \lambda }}{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}}} \right \}} \\ { \leq a_{v,n}^{(\lambda )G} \leq \frac {\pi }{{n + \lambda }}\;{{\sin }^{2\lambda }}\theta _{v,n}^{(\lambda )}} \\ \end {array}

\]
and similar results for

λ
∉
(
0
,
1
)

\lambda \notin (0,1)

. Furthermore, assuming that

θ

v
,
n

(
λ
)

\theta _{v,n}^{(\lambda )}

remains in a fixed closed interval, lying in the interior of

(
0
,
π
)

(0,\pi )

as

n
→
∞

n \to \infty

, we show that, for every fixed

λ
>
−
1

/

2

\lambda > - 1/2

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript v comma n Superscript left-parenthesis lamda right-parenthesis upper G Baseline equals StartFraction pi Over n plus lamda EndFraction sine Superscript 2 lamda Baseline theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline left-brace 1 minus StartFraction lamda left-parenthesis 1 minus lamda right-parenthesis Over 2 left-parenthesis n plus lamda right-parenthesis squared sine squared theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline EndFraction minus StartFraction lamda left-parenthesis 1 minus lamda right-parenthesis left-bracket 3 left-parenthesis lamda plus 1 right-parenthesis left-parenthesis lamda minus 2 right-parenthesis plus 4 sine squared theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline right-bracket Over 8 left-parenthesis n plus lamda right-parenthesis Superscript 4 Baseline sine Superscript 4 Baseline theta Subscript v comma n Superscript left-parenthesis lamda right-parenthesis Baseline EndFraction right-brace plus upper O left-parenthesis n Superscript negative 7 Baseline right-parenthesis period">

a

v
,
n

(
λ
)
G

=

π

n
+
λ

sin

2
λ

θ

v
,
n

(
λ
)

{

1
−

λ
(
1
−
λ
)

2

(
n
+
λ
)

2

sin

2

θ

v
,
n

(
λ
)

−

λ
(
1
−
λ
)

[
3
(
λ
+
1
)
(
λ
−
2
)
+
4

sin

2

θ

v
,
n

(
λ
)

]

8

(
n
+
λ
)

4

sin

4

θ

v
,
n

(
λ
)

}

+
O
(

n

−
7

)
.

a_{v,n}^{(\lambda )G} = \frac {\pi }{{n + \lambda }}\;{\sin ^{2\lambda }}\theta _{v,n}^{(\lambda )}\left \{ {1 - \frac {{\lambda (1 - \lambda )}}{{2{{(n + \lambda )}^2}{{\sin }^2}\theta _{v,n}^{(\lambda )}}} - \frac {{\lambda (1 - \lambda )\;[3(\lambda + 1)(\lambda - 2) + 4{{\sin }^2}\theta _{v,n}^{(\lambda )}]}}{{8{{(n + \lambda )}^4}{{\sin }^4}\theta _{v,n}^{(\lambda )}}}} \right \} + O({n^{ - 7}}).

\]