LOCAL ESTIMATES FOR JACOBI POLYNOMIALS
MICHAEL FELTEN
Estimates for Jacobi
Polynomials
Faculty of Mathematics and Informatics
University of Hagen
58084 Hagen, Germany.
EMail: michael.felten@fernuni-hagen.de
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Received:
12 October, 2006
Accepted:
02 January, 2007
Communicated by:
A. Lupaş
2000 AMS Sub. Class.:
Contents
JJ
II
33C45, 42C05.
J
I
Key words:
Jacobi polynomials, Jacobi weights, Local estimates.
Page 1 of 14
Abstract:
It is shown that if α, β ≥ − 12 , then the orthonormal Jacobi polynomials pn
fulfill the local estimate
(α,β)
|p(α,β)
(t)| ≤ √
n
( 1−x+
C(α, β)
√
1 α+ 1
) 2( 1+x+
n
Go Back
Full Screen
β+ 1
2
1
)
n
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.
Close
Contents
1
Introduction
3
2
Theorems
5
3
Applications
12
Estimates for Jacobi
Polynomials
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
JJ
II
J
I
Page 2 of 14
Go Back
Full Screen
Close
1.
Introduction
Let w(α,β) (x) = (1 − x)α (1 + x)β , x ∈ [−1, 1], be a Jacobi weight with α, β >
(α,β)
(α,β)
−1. Let pn (x) = pn (x) = γ n xn + . . ., n ∈ N0 , denote the unique Jacobi
(α,β)
polynomials of precise Rdegree n, with leading coefficients γ n
> 0, fulfilling the
1
(α,β)
orthonormal condition −1 pn (x)pm (x)w
(x) dx = δ n,m , n, m ∈ N0 .
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)
, x ∈ [−1, 1], n ∈ N.
(1.1) wn (x) :=
1+x+
1−x+
n
n
Estimates for Jacobi
Polynomials
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
(α,β)
We observe that all modified Jacobi weights wn
are finite and positive. This
is in contrast to the fact that the Jacobi weight w(α,β) 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 of modified Jacobi weights as follows (see
[3] and Theorem 2.1 below):
|p(α,β)
(x)| ≤ C
n
1
(α
+ 14 , β2 + 14 )
2
wn
for all x ∈ [−1, 1]. If α, β ≥
extended, namely
− 21 ,
JJ
II
J
I
Page 3 of 14
Go Back
(x)
then we will show that this estimate can be further
|p(α,β)
(t)| ≤ C
n
Contents
1
(α+1,β+1)
wn2 4 2 4 (x)
for all t ∈ Un (x) and each x ∈ [−1, 1], where Un (x) are subintervals of [−1, 1]
Full Screen
Close
defined by
(1.2)
ϕn (x)
Un (x) := t ∈ [−1, 1] |t − x| ≤
n
ϕ (x)
ϕ (x)
= x − n ,x + n
∩ [−1, 1]
n
n
Estimates for Jacobi
Polynomials
for n ∈ N and x ∈ [−1, 1] with
(1.3)
ϕn (x) :=
√
1
1 − x2 + .
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.
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
JJ
II
J
I
Page 4 of 14
Go Back
Full Screen
Close
2.
Theorems
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].
Estimates for Jacobi
Polynomials
Theorem 2.1. Let α, β > −1 and n ∈ N. Then
(2.1)
|p(α,β)
(x)| ≤ C
n
1
+ 14 , β2 + 14 )
(α
2
wn
Michael Felten
(x)
for all x ∈ [−1, 1] with a positive constant C = C(α, β) being independent of n and
x.
Proof. First let x ∈ [0, 1], and let t ∈ [0, π2 ] such that x = cos t. Moreover, let
(α,β)
(α,β) 1 (α,β)
P n = Pn
= (hn ) 2 pn (x), n ∈ N, be the polynomials normalized by the
(α,β)
(α,β) 1 (α,β)
(α,β) 1
factor (hn ) 2 , namely 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.
(α,β)
(α,β) 1 (α,β)
We substitute t = arccos x ∈ [0, π2 ] and Pn (x) = (hn ) 2 pn (x) in (2.2) and
1
1
(α,β)
obtain, using (hn )− 2 ≤ 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
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
JJ
II
J
I
Page 5 of 14
Go Back
Full Screen
Close
with C1 = C1 (α, β) > 0 independent of n and x. Below we will make use of the
estimates
r
π
π√
π
1−x
t
1−x= √
= √ sin
2
2
2
2
2
π
2 t
≥√
(2.4)
·√
= t = arccos x
2 π
2
and
Michael Felten
(2.5)
vol. 8, iss. 1, art. 3, 2007
r
√ √
t
t
1−x
2 1−x=2
= 2 sin ≤ 2 · = t = arccos x.
2
2
2
The cases −1 < α ≤ − 12 and α > − 12 are considered separately in the following.
Case −1 < α ≤ − 21 : In this case it follows that − α + 12 ≥ 0. If 0 ≤ arccos x ≤
c
, then
n
(α,β) (2.3)
pn (x) ≤ C1 nα+ 12 = C1
If
c
n
Estimates for Jacobi
Polynomials
− α+ 1
−(α+ 12 )
√
1
1 ( 2)
≤ C1
.
1−x+
n
n
Title Page
Contents
JJ
II
J
I
Page 6 of 14
Go Back
≤ arccos x ≤ π2 , then
Full Screen
(2.4)
(α,β) (2.3)
√
pn (x) ≤ C1 (arccos x)−(α+ 12 ) ≤ C2 ( 1 − x)−(α+ 12 )
≤ C2
√
1
1−x+
n
Close
−(α+ 12 )
Case α > − 12 : In this case we obtain − α +
.
1
2
< 0. If 0 ≤ arccos ≤
c
,
n
then
from (2.5) we obtain
c
n
≥
√ √
2 1 − x and hence
−(α+ 12 )
(α,β) (2.3)
pn (x) ≤ C1 nα+ 12 = C2 c + c
n n
− α+ 1
√
1 ( 2)
≤ C3
1−x+
.
n
If
c
n
Estimates for Jacobi
Polynomials
≤ arccos x ≤ π2 , then
Michael Felten
vol. 8, iss. 1, art. 3, 2007
(α,β) (2.3)
pn (x) ≤ C1 (arccos x)−(α+ 12 ) = C4 (arccos x + arccos x)−(α+ 12 )
| {z }
≥ nc
(2.5)
≤ C5
√
1−x+
1
n
Title Page
−(α+ 12 )
Contents
.
With both previous cases we have proved
(α,β) pn (x) ≤ C6 (α, β)
− α+ 1 − β+ 1
√
1 ( 2) √
1 ( 2)
1−x+
·
1+x+
n
n
(α,β)
pn (x)
(β,α)
(−1)n pn (−x),
for all x ∈ [0, 1], n ∈ N and α, β > −1. Since
=
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).
JJ
II
J
I
Page 7 of 14
Go Back
we
Full Screen
Close
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)
p
(x)
,
0
≤ 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 will see
that the local estimate of Theorem 2.1 can be
further extended.
(α,β) (α,β) We will show that pn (x) in (2.1) can be replaced by pn (t), whenever t is not
i
h
too far away from 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 ≥
result is stated in the following
− 12 .
(2.7)
≤C
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
The
Theorem 2.2. Let α, β ≥ − 12 and n ∈ N. Then
|p(α,β)
(t)|
n
Estimates for Jacobi
Polynomials
1
(α+1,β+1)
wn 2 4 2 4 (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
(α
+ 14 , β2 + 14 )
2
all α, β > −1. This is due to the fact that 1/wn
(x) → 0 as n → ∞, if x is
β
α
1
a boundary point x = 1 or x = −1 and 2 + 4 < 0 or 2 + 14 < 0 respectively.
First, we need an auxiliary lemma.
Contents
JJ
II
J
I
Page 8 of 14
Go Back
Full Screen
Close
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
(2.10)
√
√
3
4 1−t≥ 1−x−
n
respectively.
In order to prove
(2.10) fort ∈ 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).
√
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))
16(1 − t) ≥ 1 − x −
9
6√
1−x+ 2
n
n
Estimates for Jacobi
Polynomials
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
JJ
II
J
I
Page 9 of 14
Go Back
Full Screen
Close
or, rewritten,
6√
9
1 − x − 2 ≥ 16t.
n
n
h
i
Since t ∈ Un (x) ⊂ x − ϕnn(x) , x + ϕnn(x) , we obtain
(2.11)
15 + x +
6√
9
x+
1−x− 2 =
n
n
4√
2√
1
10
x+
1−x+ 2 +
1−x− 2
n
n
n
n
10
ϕ (x) 4 √
1−x− 2
≥x+ n
+
n
n
n
10
4√
1 − x − 2.
≥t+
n
n
Estimates for Jacobi
Polynomials
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
Hence, inequality (2.11) holds true if
15 +
4√
10
1
−
x
−
≥ 15t
n | {z } n2
3
≥n
or if
(2.12)
JJ
II
J
I
Page 10 of 14
Go Back
2
15 + 2 ≥ 15t.
n
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
Full Screen
Close
obtain
b
16
√
1
1+t+
n
2b
2b
p
1
= 16
1 − (−t) +
n
2b
2b
(2.9)
p
√
1
1
≤
1 − (−x) +
=
1+x+
,
n
n
b
Estimates for Jacobi
Polynomials
which proves the validity of (2.8).
Proof of Theorem 2.2. Since α, β ≥ − 12 , it follows that α2 + 14 , β2 + 14 ≥ 0. Therefore
we can apply Lemma 2.3 with a = − α2 − 14 and b = − β2 − 14 , obtaining
1
+ 14 , β2 + 14 )
(α
2
wn
(− α
− 14 ,− β2 − 14 )
2
= wn
(t)
4α+β+1
Lem. 2.3
(t)
≤
+ 14 , β2 + 14 )
(α
2
wn
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
(x)
for all t ∈ Un (x). Application of Theorem 2.1 therefore yields inequality (2.2) for
all t ∈ Un (x) as claimed.
JJ
II
J
I
Page 11 of 14
Go Back
Full Screen
Close
3.
Applications
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.
1
1
(α+ ,β+ )
Un (x)
wn 2 2 (x) Un (x)
Estimates for Jacobi
Polynomials
Michael Felten
Using
Z
w(α,β) (t) dt ≤ C
Un (x)
1 (α+ 12 ,β+ 12 )
(x)
wn
n
(see [2]) we find that
Z
(α,β) 2 (α,β)
1
pn (t) w
(3.1)
(t) dt ≤ C(α, β) ,
n
Un (x)
Title Page
−1
Let a, b > − 12 and C1 , C2 > 0. Let m : [1, ∞) → R be a differentiable function
fulfilling the Hormander conditions
and
Contents
x ∈ [−1, 1],
is valid for all n ∈ N with α, β ≥ − 21 . 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 becomes smaller if x tends to 1 or −1 and the weighted
(α,β)
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].
0 ≤ m(t) ≤ C1
vol. 8, iss. 1, art. 3, 2007
|m0 (t)| ≤ C2 t−1
JJ
II
J
I
Page 12 of 14
Go Back
Full Screen
Close
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 ≥ − 12 . Now, we will apply Theorem 2.2 and the above estimate (3.2)
with a = α + 12 ≥ 0 and b = β + 12 ≥ 0, to obtain
(3.3)
n
X
(α,β)
m(k) (pk
Theorem 2.2
(t))2
≤
C
Michael Felten
vol. 8, iss. 1, art. 3, 2007
n
(α+ 12 ,β+ 12 )
wn
(3.2)
k=1
Estimates for Jacobi
Polynomials
(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)
(α,β)
2
λn (t) :=
(pk (t))
,
Title Page
Contents
JJ
II
J
I
Page 13 of 14
Go Back
k=1
Full Screen
fulfills the estimate
(λ(α,β)
(t))−1 ≤ C(α, β)
n
(α+ 12 ,β+ 12 )
wn
for t ∈ Un (x) and x ∈ [−1, 1] and n ∈ N.
Close
n
(x)
References
[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.
[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.
Estimates for Jacobi
Polynomials
Michael Felten
vol. 8, iss. 1, art. 3, 2007
Title Page
Contents
JJ
II
J
I
Page 14 of 14
Go Back
Full Screen
Close