Volume 8 (2007), Issue 1, Article 3, 6 pp.
LOCAL ESTIMATES FOR JACOBI POLYNOMIALS
MICHAEL FELTEN
FACULTY OF M ATHEMATICS AND I NFORMATICS
U NIVERSITY OF H AGEN
58084 H AGEN , G ERMANY
michael.felten@fernuni-hagen.de
Received 12 October, 2006; accepted 02 January, 2007
Communicated by A. Lupaş
(α,β)
A BSTRACT. It is shown that if α, β ≥ − 12 , then the orthonormal Jacobi polynomials pn
fulfill the local estimate
C(α, β)
|p(α,β)
(t)| ≤ √
√
1
n
1 α+ 12
( 1 − x + n)
( 1 + x + n1 )β+ 2
for all t ∈ Un (x) and each x ∈ [−1, 1], where Un (x) are subintervals of [−1, 1] defined by
√
Un (x) = [x− ϕnn(x) , x+ ϕnn(x) ]∩[−1, 1] for n ∈ N and x ∈ [−1, 1] with ϕn (x) = 1 − x2 + n1 .
Applications of the local estimate are given at the end of the paper.
Key words and phrases: Jacobi polynomials, Jacobi weights, Local estimates.
2000 Mathematics Subject Classification. 33C45, 42C05.
1. I NTRODUCTION
Let w(α,β) (x) = (1 − x)α (1 + x)β , x ∈ [−1, 1], be a Jacobi weight with α, β > −1.
(α,β)
(α,β)
Let pn (x) = pn (x) = γn xn + . . ., n ∈ N0 , denote the unique Jacobi polynomials of
(α,β)
precise
degree n, with leading coefficients γn
> 0, fulfilling the orthonormal condition
R1
(α,β)
p
(x)p
(x)w
(x)
dx
=
δ
,
n,
m
∈
N
.
m
n,m
0
−1 n
This paper is concerned with local estimates of Jacobi polynomials by means of modified
Jacobi weights. By the modified Jacobi weights we understand the functions
2α 2β
√
√
1
1
(α,β)
(1.1)
wn (x) :=
1−x+
1+x+
, x ∈ [−1, 1], n ∈ N.
n
n
(α,β)
We observe that all modified Jacobi weights wn
are finite and positive. This is in contrast
(α,β)
to the fact that the Jacobi weight w
may have singularities and roots in ±1, depending on
whether α and β are negative or positive. The Jacobi polynomials can be estimated by means
005-07
2
M ICHAEL F ELTEN
of modified Jacobi weights as follows (see [3] and Theorem 2.1 below):
1
|p(α,β)
(x)| ≤ C α 1 β 1
n
( + , + )
wn2 4 2 4 (x)
for all x ∈ [−1, 1]. If α, β ≥ − 12 , then we will show that this estimate can be further extended,
namely
1
|p(α,β)
(t)| ≤ C α 1 β 1
n
( + , + )
wn2 4 2 4 (x)
for all t ∈ Un (x) and each x ∈ [−1, 1], where Un (x) are subintervals of [−1, 1] defined by
ϕn (x)
(1.2)
Un (x) := t ∈ [−1, 1] |t − x| ≤
n
ϕn (x)
ϕn (x)
= x−
,x +
∩ [−1, 1]
n
n
for n ∈ N and x ∈ [−1, 1] with
√
1
.
n
Thus Un (x) is located around x and is small, i.e., |Un (x)| = O(1/n). In our case of Jacobi
weights on [−1, 1] we need intervals around x with radius ϕnn(x) instead of n1 . In this case the
radius varies together with x and becomes smaller if x tends to 1 or −1.
(1.3)
ϕn (x) :=
1 − x2 +
2. T HEOREMS
The following theorem provides a useful local estimate of the orthonormal Jacobi polynomials by means of the modified weights wn . The estimate can also be found in the paper [3] by
Lubinsky and Totik. Here we will give an explicit proof. The proof is essentially based on an
estimate taken from Szegö [4].
Theorem 2.1. Let α, β > −1 and n ∈ N. Then
(2.1)
|p(α,β)
(x)| ≤ C
n
1
(α
+ 14 , β2 + 14 )
2
wn
(x)
for all x ∈ [−1, 1] with a positive constant C = C(α, β) being independent of n and x.
(α,β)
=
Proof. First let x ∈ [0, 1], and let t ∈ [0, π2 ] such that x = cos t. Moreover, let Pn = Pn
(α,β) 1 (α,β)
(α,β) 1
(hn ) 2 pn (x), n ∈ N, be the polynomials normalized by the factor (hn ) 2 , namely
(α,β)
(α,β) 1 (α,β)
Pn
= (hn ) 2 pn (x), as can be found in Szegö [4, eq. (4.3.4)]. According to Szegö’s
book [4, Theorem 7.32.2] the estimate
( α
n ,
if 0 ≤ t ≤ nc
(α,β)
(2.2)
|Pn (cos t)| ≤ C
1
1
t−(α+ 2 ) n− 2 , if nc ≤ t ≤ π2
is valid, where c and C are fixed positive constants being independent of n and t. We substitute
1
(α,β)
(α,β) 1 (α,β)
(α,β)
t = arccos x ∈ [0, π2 ] and Pn (x) = (hn ) 2 pn (x) in (2.2) and obtain, using (hn )− 2 ≤
1
C̃ · n 2 (resulting from [4, eq. (4.3.4)]),
( α+ 1
n 2,
if 0 ≤ arccos x ≤ nc
(α,β)
(2.3)
|pn (x)| ≤ C1
1
(arccos x)−(α+ 2 ) , if nc ≤ arccos x ≤ π2
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E STIMATES FOR JACOBI P OLYNOMIALS
3
with C1 = C1 (α, β) > 0 independent of n and x. Below we will make use of the estimates
r
π√
t
1−x
π
π
= √ sin
1−x= √
2
2
2
2
2
π
2 t
(2.4)
≥√
·√
= t = arccos x
2 π
2
and
r
√ √
1−x
t
t
(2.5)
2 1−x=2
= 2 sin ≤ 2 · = t = arccos x.
2
2
2
The cases −1 < α ≤ − 21 and α > − 12 are considered separately in the following.
Case −1 < α ≤ − 12 : In this case it follows that − α + 12 ≥ 0. If 0 ≤ arccos x ≤ nc , then
−(α+ 12 )
−(α+ 12 )
(α,β) (2.3)
√
1
1
1
α+
pn (x) ≤ C1 n 2 = C1
≤ C1
1−x+
.
n
n
If
c
n
≤ arccos x ≤ π2 , then
(2.4)
(α,β) (2.3)
√
pn (x) ≤ C1 (arccos x)−(α+ 12 ) ≤ C2 ( 1 − x)−(α+ 12 )
− α+ 1
1 ( 2)
≤ C2
1−x+
.
n
Case α > − 21 : In this case we obtain − α + 21 < 0. If 0 ≤ arccos ≤ nc , then from (2.5) we
√ √
obtain nc ≥ 2 1 − x and hence
−(α+ 12 )
(α,β) (2.3)
pn (x) ≤ C1 nα+ 12 = C2 c + c
n n
−(α+ 12 )
√
1
≤ C3
1−x+
.
n
If
c
n
√
≤ arccos x ≤ π2 , then
(α,β) (2.3)
pn (x) ≤ C1 (arccos x)−(α+ 12 ) = C4 (arccos x + arccos x)−(α+ 12 )
| {z }
≥ nc
(2.5)
≤ C5
√
1
1−x+
n
−(α+ 12 )
.
With both previous cases we have proved
−(α+ 12 ) −(β+ 12 )
(α,β) √
√
1
1
pn (x) ≤ C6 (α, β)
1−x+
·
1+x+
n
n
(α,β)
(β,α)
for all x ∈ [0, 1], n ∈ N and α, β > −1. Since pn (x) = (−1)n pn (−x), we obtain
−(β+ 12 ) −(α+ 12 )
(α,β) √
√
1
1
pn (x) ≤ C6 (β, α)
1+x+
·
1−x+
n
n
for all x ∈ [−1, 0), n ∈ N and α, β > −1. This furnishes the validity of (2.1).
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 3, 6 pp.
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4
M ICHAEL F ELTEN
Estimate (2.1) of Theorem 2.1 cannot hold true for n = 0 since the modified weight wn is
not defined for n = 0. However, if n = 0, then
1
(α,β) (2.6)
,
p0 (x) ≤ C(α, β) α 1 β 1
( 2 +4, 2 +4)
w1
(x)
(α+1,β+1)
(α,β)
since p0 (x) is a constant and C1 (α, β) ≤ w1 2 4 2 4 (x) ≤ C2 (α, β) with positive constants
C1 (α, β) and C2 (α, β).
Next, we
2.1
will see that the local estimate of Theorem
can be further extended. We will
(α,β) (α,β) show that pn (x) in (2.1) can be replaced by pn (t), whenever t is not too far away from
i
h
x, namely if t is in the interval Un (x) = x − ϕnn(x) , x + ϕnn(x) ∩ [−1, 1]. However, for this
estimate we will need the assumption α, β ≥ − 21 . The result is stated in the following
Theorem 2.2. Let α, β ≥ − 12 and n ∈ N. Then
(2.7)
|p(α,β)
(t)| ≤ C
n
1
(
wn
α
+ 41 , β2 + 14
2
)
(x)
for all t ∈ Un (x) and each x ∈ [−1, 1], where the interval Un (x) has been given in (1.2) and
C = C(α, β) is a positive constant independent of n, t and x.
It must be mentioned that Theorem 2.2 cannot be extended to hold true even for all α, β >
( α + 14 , β2 + 14 )
−1. This is due to the fact that 1/wn2
x = 1 or x = −1 and α2 + 14 < 0 or β2 +
First, we need an auxiliary lemma.
1
4
(x) → 0 as n → ∞, if x is a boundary point
< 0 respectively.
Lemma 2.3. Let a, b ≤ 0, n ∈ N and x ∈ [−1, 1]. Then
(2.8)
wn(a,b) (t) ≤ 16−(a+b) wn(a,b) (x)
for all t ∈ Un (x).
Proof. First, let a ≤ 0. We will prove that
2a 2a
√
√
1
1
a
(2.9)
16
1−t+
≤
1−x+
n
n
holds true for all t ∈ Un (x) with x ∈ [−1, 1] and n ∈ N. There is nothing to prove for a = 0.
Let a < 0. Then inequality (2.9) is equivalent to
√
√
1
1
4
1−t+
≥ 1−x+
n
n
and
√
√
3
4 1−t≥ 1−x−
n
respectively.
In order
to prove (2.10) for t ∈ Un (x) we will discuss below the cases x ∈
1 − n92 , 1 and x ∈ −1, 1 − n92 separately. We must note that the latter interval is empty for
n = 1, 2, 3.
√
Case x ∈ 1 − n92 , 1 : In this case we obtain 1 − x − n3 ≤ n3 − n3 = 0, which immediately
gives (2.10).
(2.10)
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 3, 6 pp.
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E STIMATES FOR JACOBI P OLYNOMIALS
5
√
Case x ∈ −1, 1 − n92 : In this case we obtain 1 − x − n3 > 0. Therefore inequality (2.10) is
equivalent to (squaring both sides of (2.10))
9
6√
16(1 − t) ≥ 1 − x −
1−x+ 2
n
n
or, rewritten,
6√
9
1 − x − 2 ≥ 16t.
(2.11)
15 + x +
n
n
h
i
ϕn (x)
ϕn (x)
Since t ∈ Un (x) ⊂ x − n , x + n , we obtain
9
2√
1
4√
10
6√
1−x− 2 = x+
1−x+ 2 +
1−x− 2
x+
n
n
n
n
n
n
√
ϕn (x) 4
10
≥x+
+
1−x− 2
n
n
n
4√
10
≥t+
1 − x − 2.
n
n
Hence, inequality (2.11) holds true if
4√
10
15 + | 1{z− x} − 2 ≥ 15t
n
n
3
≥n
or if
2
≥ 15t.
n2
Since t ≤ 1, inequality (2.12) is fulfilled. Hence inequality (2.10) is also proved. This completes
the proof of (2.9) for all x ∈ [−1, 1] and t ∈ Un (x).
Now, let b ≤ 0, x ∈ [−1, 1] and t ∈ Un (x). Then −t ∈ Un (−x). From (2.9) we obtain
2b
2b
p
√
1
1
b
b
= 16
16
1+t+
1 − (−t) +
n
n
2b
2b
(2.9)
p
√
1
1
≤
1 − (−x) +
=
1+x+
,
n
n
15 +
(2.12)
which proves the validity of (2.8).
Proof of Theorem 2.2. Since α, β ≥ − 12 , it follows that α2 + 14 , β2 +
apply Lemma 2.3 with a = − α2 − 14 and b = − β2 − 14 , obtaining
1
( α + 14 , β2 + 14 )
wn2
(− α
− 14 ,− β2 − 14 )
2
= wn
≤
(t)
≥ 0. Therefore we can
4α+β+1
Lem. 2.3
(t)
1
4
( α + 14 , β2 + 14 )
wn2
(x)
for all t ∈ Un (x). Application of Theorem 2.1 therefore yields inequality (2.2) for all t ∈ Un (x)
as claimed.
3. A PPLICATIONS
In this section we will give some applications of the local estimates of the Jacobi polynomials.
We apply Theorem 2.2 and obtain
Z
Z
(α,β) 2 (α,β)
1
pn (t) w
(t) dt ≤ C
w(α,β) (t) dt.
(α+ 12 ,β+ 12 )
Un (x)
Un (x)
wn
(x)
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6
M ICHAEL F ELTEN
Using
Z
w(α,β) (t) dt ≤ C
Un (x)
1 (α+ 12 ,β+ 12 )
(x)
wn
n
(see [2]) we find that
Z
(3.1)
(α,β) 2 (α,β)
1
pn (t) w
(t) dt ≤ C(α, β) ,
n
Un (x)
x ∈ [−1, 1],
is valid for all n ∈ N with α, β ≥ − 12 . Estimate (3.1) shows that the intervals Un (x) are
appropriate for measuring the growth of the orthonormal polynomials on subintervals of [−1, 1]:
Un (x) is located around x, |Un (x)| = O(1/n), the radius ϕnn(x) varies together with x and
(α,β)
becomes smaller if x tends to 1 or −1 and the weighted integration of (pn (t))2 on Un (x) is
O(1/n), whereas the weighted integral on [−1, 1] equals 1, i.e.,
Z 1
(α,β) 2 (α,β)
pn (t) w
(t) dt = 1, x ∈ [−1, 1].
−1
Let a, b > − 21 and C1 , C2 > 0. Let m : [1, ∞) → R be a differentiable function fulfilling the
Hormander conditions
and |m0 (t)| ≤ C2 t−1
0 ≤ m(t) ≤ C1
for t ≥ 1. It was proved in [1] that
n
X
m(k)
(3.2)
(a,b)
(x)
k=1 wk
≤C
n
(a,b)
wn (x)
for all x ∈ [−1, 1] and n ∈ N with a positive constant C = C(a, b, C1 , C2 ) being independent
of n and x.
Let α, β ≥ − 21 . Now, we will apply Theorem 2.2 and the above estimate (3.2) with a =
α + 12 ≥ 0 and b = β + 21 ≥ 0, to obtain
(3.3)
n
X
(α,β)
m(k) (pk
(t))2
Theorem 2.2
≤
C
wn
(3.2)
k=1
n
(α+ 12 ,β+ 12 )
(x)
for all t ∈ Un (x) and each x ∈ [−1, 1] with a constant C = C(α, β, C1 , C2 ) > 0 being
independent of n and x.
In particular, if we let m(k) = 1, then estimate (3.3) shows that the Christoffel function,
defined by
( n
)−1
X (α,β)
λ(α,β)
(t) :=
(pk (t))2
,
n
k=1
fulfills the estimate
(λ(α,β)
(t))−1 ≤ C(α, β)
n
n
(α+ 12 ,β+ 12 )
wn
(x)
for t ∈ Un (x) and x ∈ [−1, 1] and n ∈ N.
R EFERENCES
[1] M. FELTEN, Multiplier theorems for finite sums of Jacobi polynomials, submitted, 1–9.
[2] M. FELTEN, Uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms. II
(Some necessary estimations), accepted for publication in Acta Math. Hung.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 3, 6 pp.
http://jipam.vu.edu.au/
E STIMATES FOR JACOBI P OLYNOMIALS
7
[3] D.S. LUBINSKY AND V. TOTIK, Best weighted polynomial approximation via Jacobi expansions,
SIAM Journal on Mathematical Analysis, 25(2) (1994), 555–570.
[4] G. SZEGŐ, Orthogonal Polynomials, 4th Ed., American Mathematical Society, Providence, R.I.,
1975, American Mathematical Society, Colloquium Publications, Vol. XXIII.
J. Inequal. Pure and Appl. Math., 8(1) (2007), Art. 3, 6 pp.
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