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Trace Formula for Orthogonal Polynomials with Asymptotically 2-Periodic Recurrence Coefficients
Journal of Approximation Theory AT3133 Journal of Approximation Theory 92, 442462 (1998) Article No. AT973133
Trace Formula for Orthogonal Polynomials with Asymptotically 2-Periodic Recurrence Coefficients Boris P. Osilenker Department of Mathematics, Moscow State Civil Engineering University, 2-oj Krestovskii pereulok Dom 4, Apt. 55, 129041 Moscow, Russia Communicated by Walter Van Assche Received March 4, 1996; accepted in revised form March 15, 1997
1. INTRODUCTION Suppose + is a positive probability measure on a compact set on the real line, that is, + is a positive Borel measure with d+=1. Then there is a unique sequence of polynomials p n(x)=k n x n + } } } ,
k n >0
(n # Z + =[0, 1, ...])
such that
|p
m
(x) p n(x) d+(x)=$ mn
(m, n # Z + ).
These orthonormal polynomials satisfy a three-term recurrence relation xp n(x)=a n+1 p n+1(x)+b n p n(x)+a n p n&1(x)
(n # Z + )
(1.1)
with the initial conditions p &1(x)=0,
p 0(x)=1,
a 0 =0,
(1.2)
where a n+1 =k n k n+1 >0 and b n # R. Conversely, by Favard's Theorem, if the polynomials p n(x) are given by the recurrence formula (1.1) with a n+1 >0 and b n # R, then there exists a positive Boreal measure + such that [ p n ] (n # Z + ) is an orthonormal polynomial system with respect to the measure +. If a n and b n are bounded, then the measure + is unique and the support of + is compact. 442 0021-904598 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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443
TRACE FORMULA
The following result (Trace Formula) establishes the connection between Jacobi matrices
J=
\
b0 a1 0 }}}
a1 b1 a2 }}}
0 a2 b2 }}}
0 0 a3 }}}
0 0 0 }}}
}}} }}} }}} }}}
+
and their spectral measures. Theorem 1 [8, 14] (see also [16]). If Supp(+)=[&1, 1] and if the recursion coefficients [a n+1 ] and [b n ] satisfy lim a n = 12 ,
n
lim b n =0
(1.3)
n
and
: ( |a n+1 &a n | + |b n+1 &b n | )< ,
(1.4)
n=0
then
: [(a 2n+1 &a 2n ) p 2n(x)+a n(b n &b n&1 ) p n&1(x) p n(x)]= n=0
1 - 1&x 2 2? +$(x)
holds uniformly on all compact sets in (&1, 1). In addition, the measure + is absolutely continuous in the open interval (&1, 1), +$(x)>0 for all x # (&1, 1), and +$(x) is continuous in (&1, 1). Given two periodic sequences [a 0n+1 ] (a 0n+1 >0) and [b 0n ] (b 0n # R) (n # Z + ) with period N 1, the Jacobi matrix J is called asymptotically N-periodic (J # AP N ), if lim [ |a n &a 0n | + |b n &b 0n | ]=0.
n
(1.5)
In a survey [16] P. Nevai has posed the following problem: Extend the Trace Formula to asymptotically N-periodic Jacobi matrices. We investigate this problem for the case N=2.
2. REPRESENTATIONS OF THE KERNELS We assume that two periodic sequences a 0n+1 >0 and b 0n (n # Z + ) are given such that a 0n+2 =a 0n
(n=1, 2, ...),
b 0n+2 =b 0n
(n=0, 1, 2, ...)
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444
BORIS P. OSILENKER
(i.e., period N=2), and that the recurrence coefficients a n+1 and b n satisfy (1.5). We will write J # AP 2 . Denote the orthonormal polynomials with periodic recurrence coefficients a 0n+1 and b 0n by q n(x). Then xq n(x)=a 0n+1 q n+1(x)+b 0n q n(x)+a 0n q n&1(x)
(n # Z + )
q 0(x)=1.
q &1(x)=0, Let
T(x)=
a0 1 q 2(x)& 20 . 2 a1
_
&
(2.1)
The essential spectrum of the polynomials q n(x), resp. p n(x), consists of two intervals E, where E=[x # R, &1 T(x) 1]. The set E is of the form
_
:,
;&: :+; :+; ;&: & t _ +t ,; 2 2 2 2
& _
&
for some 0 t<1,
or in other words, if &1, 1 is the smallest and largest boundary point of E (this can be obtained easily by a linear transformation of q n(x) resp. p n(x)) then E is of the form E=[&1, &t] _ [t, 1]
for some 0 t<1.
These facts follow from [20, 21]. The special case when the intervals touch each other (the set [T 2(x)=1] consists of the endpoints of the intervals) leads to sieved orthogonal polynomials (see details in [1, 911, 21]). Lemma 1.
If J # AP 2 , then the following recurrence relation is valid,
s(x) p n(x)=: n+2 p n+2(x)+; n+1 p n+1(x)+# n p n(x) +; n p n&1(x)+: n p n&2(x)
(n # Z + )
(2.2)
with boundary conditions p &2(x)=0,
p &1(x)=0,
: 0 =: 1 =0,
; 0 =0,
(2.3)
where s(x)=2a 01 a 02 T(x)=(x&b 00 )(x&b 01 )&(a 01 ) 2 &(a 02 ) 2 : n+2 =a n+1 a n+2 , 2 n
# n =a +a
2 n+1
; n =a n(b n&1 +b n &b 00 &b 01 ) 0 2 1
0 2 2
0 0
0 1
&(a ) &(a ) +(b n &b )(b n &b )
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(2.4)
445
TRACE FORMULA
with lim : n =a 01 a 02 ,
n
lim ; n =0,
n
lim # n =0.
n
(2.5)
In fact, Lemma 1 follows from the definition of AP 2 and the five-term recurrence relation x 2p n(x)=a n+1 a n+2 p n+2(x)+a n+1(b n +b n+1 ) p n+1(x) +(a 2n +a 2n+1 +b 2n ) p n(x)+a n(b n&1 +b n ) p n&1(x) +a n a n&1 p n&2(x) by a direct computation. Lemma 2.
Let J # AP 2 . Then for the Dirichlet kernel n
D n(t, x)= : p k (t) p k (x) k=0
the representation [s(t)&s(x)] D n(t, x)=: n+2[ p n+2(t) p n(x)& p n(t) p n+2(x)] +: n+1[ p n+1(t) p n&1(x)& p n&1(t) p n+1(x)] +; n+1[ p n+1(t) p n(x)& p n(t) p n+1(x)]
(2.6)
holds. Here s(x), : n , ; n are defined by (2.4). In fact, formula (2.6) follows from (2.2) and (2.3) by straightforward calculations. Lemma 3. If J # AP 2 , then the following representation of the Fejer kernel, F n(t, x)=
n 1 : D k(t, x), n+1 k=0
is valid (n+1)[s(t)&s(x)] 2 F n(t, x) =! n(t, x)+! n(x, t)+G n(t, x)+G n(x, t),
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(2.7)
446
BORIS P. OSILENKER
where G n(t, x)=: 2n+2[ p n(t) p n(x)& p n+2(t) p n+2(x)] &2[: 2n p n(t) p n(x)+: 2n+1 p n+1(t) p n+1(x)] +: n+2 : n+4 p n+4(t) p n(x)+2: n+1 : n+3 p n+3(t) p n&1(x) +: n : n+2 p n+2(t) p n&2(x)+: n+2(# n+2 &# n ) p n+2(t) p n(x) &; 2n+1 p n+1(t) p n+1(x)+(: n+2 ; n+3 +: n+3 ; n+1 ) p n+3(t) p n(x) +2: n+1 ; n+2 p n+2(t) p n&1(x)+; n+1 ; n+2 p n+2(t) p n(x) +(: n+2 ; n+2 &2: n+1 ; n ) p n+1(t) p n(x) &: n+2 ; n+1 p n+2 (t) p n+1 (x)&: n+2 ; n+1 p n+1(t) p n+2(x) &: n+1 ; n p n(t) p n+1(x)
(2.8)
and n&1
n
! n(t, x)=2 : (: 2k+2 &: 2k ) p k(t) p k(x)+ : ( ; 2k+1 &; 2k ) p k(t) p k(x) k=0
k=0
n&1
+2 : : k+2(# k+2 &# k ) p k+2(t) p k(x) k=0 n&2
+ : (: k+2 ; k+3 &: k+3 ; k+1 ) p k+3(t) p k(x) k=0 n&1
+2 : (: k+2 ; k+2 &: k+1 ; k ) p k+1(t) p k(x) k=0 n
+ : ; k+1(# k+1 &# k ) p k+1(t) p k(x) k=0 n&1
+ : (: k+2 ; k+2 &: k+1 ; k ) p k(t) p k+1(x),
(2.9)
k=0
where : n , ; n , # n are defined by (2.4). Proof.
By formula (2.6) one has (n+1)[s(t)&s(x)] 2 F n(t, x)=` n(t, x)+` n(x, t),
(2.10)
where n
` n(t, x)= : % k(t, x) k=0
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(2.11)
447
TRACE FORMULA
with % k(t, x)=: k+2 s(t) p k+2(t) p k(x)+: k+1 s(t) p k+1(t) p k&1(x) +; k+1 s(t) p k+1(t) p k(x)&: k+2 p k+2(t) s(x) p k(x) &: k+1 p k+1(t) s(x) p k&1(x)&; k+1 p k+1(t) s(x) p k(x). The recurrence relation (2.2) yields % k(t, x)=: k+2 : k+4 p k+4(t) p k(x)+: k+2 ; k+3 p k+3(t) p k(x) +: k+2 # k+2 p k+2(t) p k(x)+: k+2 ; k+2 p k+1(t) p k(x) +: 2k+2 p k(t) p k(x)+: k+1 : k+3 p k+3(t) p k&1(x) +: k+1 ; k+2 p k+2(t) p k&1(x)+: k+1 # k+1 p k+1(t) p k&1(x) +: k+1 ; k+1 p k(t) p k&1(x)+: k+3 ; k+1 p k+3(t) p k(x) +; k+1 ; k+2 p k+2(t) p k(x)+; k+1 # k+1 p k+1(t) p k(x) +; 2k+1 p k(t) p k(x)+: k+1 ; k+1 p k&1(t) p k(x) &: 2k+2 p k+2(t) p k+2(x)&: k+2 ; k+1 p k+2(t) p k+1(x) &: k+2 # k p k+2(t) p k(x)&: k+2 ; k p k+2(t) p k&1(x) &: k+2 ; k+1 p k+1(t) p k+2(x)&; 2k+1 p k+1(t) p k+1(x) &; k+1 # k p k+1(t) p k(x)&: k+2 : k p k+2(t) p k&2(x) &: 2k+1 p k+1(t) p k+1(x)&: k&1 : k+1 p k+1(t) p k&3(x) &; k+1 ; k p k+1(t) p k&1(x)&: k ; k+1 p k+1(t) p k&2(x) &: k+1 ; k p k+1(t) p k(x)&: k+1 # k&1 p k+1(t) p k&1(x) &: k+1 ; k+1 p k+1(t) p k&2(x)+: 2k+1 p k&1(t) p k&1(x). For the calculation of ` n(t, x) we regroup similar terms of the last relation. Then ` n(t, x) can be representated in the form 6
` n(t, x)= : _ i (t, x), i=1
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(2.12)
448
BORIS P. OSILENKER
where n
n
_ 1(t, x)= : : 2k+2 p k(t) p k(x)+ : : 2k+1 p k&1(t) p k&1(x) k=0
k=0 n
n
& : : 2k+2 p k+2(t) p k+2(x)& : : 2k+1 p k+1(t) p k+1(x) k=0 n
k=0 n
+ : ; 2k+1 p k(t) p k(x)& : ; 2k+1 p k+1(t) p k+1(x); k=0 n
k=0 n
_ 2(t, x)= : : k+2 : k+4 p k+4(t) p k(x)& : : k : k+2 p k+2(t) p k&2(x) k=0
k=0 n
n
+ : : k+1 : k+3 p k+3(t) p k&1(x)& : : k&1 : k p k+1(t) p k&3(x); k=0 n
k=0 n
_ 3(t, x)= : : k+2 ; k+3 p k+3(t) p k(x)& : : k+2 ; k p k+2(t) p k&1(x) k=0
k=0 n
n
+ : : k+1 ; k+2 p k+2(t) p k&1(x)& : : k+1 ; k&1 p k+1(t) p k&2(x) k=0
k=0
n
n
+ : : k+3 ; k+1 p k+3(t) p k(x)& : : k ; k+1 p k+1(t) p k&2(x); k=0 n
k=0 n
_ 4(t, x)= : : k+2 # k+2 p k+2(t) p k(x)+ : : k+1 # k+1 p k+1(t) p k&1(x) k=0
k=0 n
n
+ : ; k+1 ; k+2 p k+2(t) p k(x)& : : k+2 # k p k+2(t) p k(x) k=0
k=0
n
n
& : ; k ; k+1 p k+1(t) p k&1(x)& : : k+1 # k&1 p k+1(t) p k&1(x); k=0 n
k=0 n
_ 5(t, x)= : : k+2 ; k+2 p k+1(t) p k(x)+ : : k+1 ; k+1 p k(t) p k&1(x) k=0
k=0 n
n
+ : ; k+1 # k+1 p k+1(t) p k(x)& : : k+2 ; k+1 p k+2(t) p k+1(x) k=0 n
k=0 n
& : ; k+1 # k p k+1(t) p k(x)& : : k+1 ; k p k+1(t) p k(x); k=0 n
k=0 n
_ 6(t, x)= : : k+1 ; k+1 p k&1(t) p k(x)& : : k+2 ; k+1 p k+1(t) p k+2(x). k=0
k=0
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TRACE FORMULA
449
Using Abel's summation by parts and the initial conditions (2.3), it is not difficult to show that the following formulas are valid _ 1(t, x)=: 2n+2 [ p n(t) p n(x)& p n+2(t) p n+2(x)] &2[: 2n p n(t) p n(x)+: 2n+1 p n+1(t) p n+1(x)] n&1
&; 2n+1 p n+1(t) p n+1(x)+2 : (: 2k+2 &: 2k ) p k(t) p k(x) k=0 n
+ : (; 2k+1 &; 2k ) p k(t) p k(x); k=0
_ 2(t, x)=: n+2 : n+4 p n+4(t) p n(x)+2: n+1 : n+3 p n+3(t) p n&1(x) +: n : n+2 p n+2(t) p n&2 (x); _ 3(t, x)=(: n+2 ; n+3 +: n+3 ; n+1 ) p n+3(t) p n(x) n&2
+2: n+1 ; n+2 p n+2(t) p n&1(x)+ : [: k+2( ; k+3 &; k+1 ) k=0
+(: k+2 &: k+3 ) ; k+1 ] p k+3(t) p k(x); _ 4(t, x)=: n+2(# n+2 &# n ) p n+2(t) p n(x)+; n+1 ; n+2 p n+2(t) p n(x) n&1
+2 : : k+2(# k+2 &# k ) p k+2(t) p k(x); k=0
_ 5(t, x)=(: n+2 ; n+2 &2: n+1 ; n ) p n+1(t) p n(x) n&1
&: n+2 ; n+1 p n+2(t) p n+1(x)+2 : [(: k+2 &: k+1 ) ; k+2 k=0
+(; k+2 &; k ) : k+1 ] p k+1(t) p k(x) n
+ : ; k+1(# k+1 &# k ) p k+1(t) p k(x); k=0
_ 6(t, x)= &: n+2 ; n+1 p n+1(t) p n+2(x)&: n+1 ; n p n(t) p n+1(x) n&1
+ : [: k+2( ; k+2 &; k )+(: k+2 &: k+1 ) ; k ] p k(t) p k+1(x). k=0
The representation (2.7)(2.9) follows from (2.10)(2.12) and the last six relations. Lemma 3 is completely proved. Remark. For N=1 the representation of F n(t, x) was given in [17, 18] (see also [19] for N=2). Fejer's kernel plays an important role in some problems of summability of Fourier series in orthogonal polynomials [17, 18]. The following assertion can be inferred from Lemma 3, if we put t=x.
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450
BORIS P. OSILENKER
Corollary 1. If J # AP 2 , then n&1
n
2 : (: 2k+2 &: 2k ) p 2k(x)+ : (; 2k+1 &; 2k ) p 2k(x) k=0
k=0 n&1
+2 : : k+2(# k+2 &# k ) p k(x) p k+2(x) k=0 n&1
+3 : (: k+2 ; k+2 &: k+1 ; k ) p k(x) p k+1(x) k=0 n&2
+ : (: k+2 ; k+3 &: k+3 ; k+1 ) p k(x) p k+3(x) k=0 n
+ : ; k+1(# k+1 &# k ) p k(x) p k+1(x)=G n(x), k=0
where G n(x)=(2: 2n &: 2n+2 ) p 2n(x)+(2: 2n+1 +; 2n+1 ) p 2n+1(x)+: 2n+2 p 2n+2(x) &: n+2 : n+4 p n(x) p n+4(x)&2: n+1 : n+3 p n&1(x) p n+3(x) &: n : n+2 p n&2(x) p n+2(x)&: n+2(# n+2 &# n ) p n(x) p n+2(x) &; n+1 ; n+2 p n(x) p n+2(x)&2: n+1 ; n+2 p n&1(x) p n+2(x) &(: n+2 ; n+3 +: n+3 ; n+1 ) p n(x) p n+3(x) +2: n+2 ; n+1 p n+1(x) p n+2(x) +(3: n+1 ; n &: n+2 ; n+2 ) p n(x) p n+1(x),
(2.13)
where : n , ; n , and # n are defined by (2.4). Remark. For N=1 formula (2.13) was obtained by J. Dombrowski [7].
3. TRACE FORMULA If [ p n ] is a system of orthonormal polynomials, satisfying (1.1), (1.2), then one can introduce the associated polynomials [ p (m) n (x)] of order m (m # Z + ) by the shifted recurrence formula (m) (m) (m) xp (m) n (x)=a n+m+1 p n+1(x)+b n+m p n (x)+a n+m p n&1(x)
(n # Z) (3.1)
with boundary conditions p (m) &1 (x)=0,
p (m) 0 (x)=1.
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(3.2)
451
TRACE FORMULA
We need the following result Lemma 4 [11]. Assume that the Jacobi matrix J # AP 2 . Then for every continuous function f and for all integers k and j one has
| f (x) p
lim
n
=
2n+ j
(x) p 2n+k(x) d+(x)
1 4?a
0 j+1
a
0 k+1
|
f (x) Sign[T$(x)]
E
- 1&T 2(x)
( j+1) 0 (k+1) _[a 0k+1 q k& j+1(x)+a j+1 q j&k+1(x)] dx,
(3.3)
where 0 (m+k+1) (x) a 0m+k+1 q (m) k (x)=&a m q &k&2
(k<0).
(3.4)
The next assertion gives the weak type asymptotics. Lemma 5. Suppose that the Jacobi matrix J # AP 2 . Then for every continuous function f and for all integers k one has lim
n
|
f (x) G 2n+k(x) d+(x)
E
= =
4(a 01 a 02 ) 2 ? 1 ?
f (x) |T $(x)| - 1&T 2(x) dx E
|
f (x) |2x&b 0k &b 0k+1 |
[(a 0k+2 +a 0k+1 ) 2 &(x&b 0k )(x&b 0k+1 )]_ dx. [(x&b 0k )(x&b 0k+1 )&(a 0k+2 &a 0k+1 ) 2 ]
E
_ Proof.
|
Using the definition of the class AP 2 and (2.5), we have
I (2) k := lim
|
= lim
|
n
n
f (x) G 2n+k(x) d+(x) E
E
f (x)[: 22n+k+2 p 22n+k+2(x)+2: 22n+k p 22n+k(x)
&: 22n+k+2 p 22n+k(x)+2: 22n+k+1 p 22n+k+1(x) &: 2n+k+2 : 2n+k+4 p 2n+k(x) p 2n+k+4(x) &2: 2n+k+1 : 2n+k+3 p 2n+k&1(x) p 2n+k+3(x) &: 2n+k : 2n+k+2 p 2n+k&2(x) p 2n+k+2(x)] d+(x).
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(3.5)
452
BORIS P. OSILENKER
By (1.5), (3.3), and (3.4) 1 4?
0 0 2 I (2) k =(a k+1 a k+2 )
_
{a
& & =
2 0 k+1
&
- 1&T 2(x)
E
4 (k+2) 2 (x)+ 0 q (k+1)(x) 0 q1 ak a k+1 1
1 2 (k+5) [q (k+1) (x)+q &3 (x)]& 0 [q (k) (x)+q (k+4) (x)] 5 &3 a 0k+1 ak 5 1 a
0 k+1
{a
&
f (x) Sign[T$(x)]
q (k+3) (x)+ 1
=
(k+3) [q (k&1) (x)+q &3 (x)] dx 5
(a 0k+1 a 0k+2 ) 2 4? _
|
2 0 k+1
|
f (x) Sign[T $(x)] - 1&T 2(x)
E
q (k+3) (x)+ 1
4 (k+2) 2 q (x)+ 0 q (k+1)(x) a 0k 1 a k+1 1
2 1 [q (k+1) (x)&q 1(k+3)(x)]& 0 [q (k) (x)&q (k+2) (x)] 5 1 a 0k+1 ak 5 1 a
0 k+1
=
[q (k&1) (x)&q 1(k+1)(x)] dx. 5
Since (see [11]) (m) (m) 2 (m) q (m) 5 (x)=2T(x) q 3 (x)&q 1 (x)=[4T (x)&1] q 1 (x),
then I (2) k =
(a 0k+1 a 0k+2 ) 2 4?
{_a +
3 0 k+1
|
f (x) Sign[T $(x)]
E
- 1&T 2(x)
q (k+3) (x)+ 1
6 (k+2) 4 q (x)+ 0 q (k+1)(x) a 0k 1 a k+1 1
2 (k) 1 q 1 (x)+ 0 q (k&1)(x) 0 ak a k+1 1
&4T 2(x)
_a
1 0 k+1
q (k+1) (x)+ 1
&
2 (k) 1 q (x)+ 0 q (k&1) (x) a 0k 1 a k+1 1
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&= dx.
TRACE FORMULA
453
It follows from the definition of the class AP 2 and (3.1), (3.2), that q (m) 1 (x)=
1 a
0 m+1
(x&b 0m ),
q 1(m+2)(x)=q (m) 1 (x).
So 1 a
0 k+1
q 1(k+1)(x)+
2 (k) 1 2 q 1 (x)+ 0 q 1(k&1)(x)= 0 (2x&b 0k &b 0k+1 ) 0 ak a k+1 a k+1 a 0k+2
and 3 a
0 k+1
q (k+3) (x)+ 1 +
6 (k+2) 4 q1 (x)+ 0 q (k+1)(x) 0 ak a k+1 1
1 8 2 (k) q 1 (x)+ 0 q 1(k&1)(x)= 0 (2x&b 0k &b 0k+1 ). 0 ak a k+1 a k+1 a 0k+2
Consequently, 2 0 0 I (2) k = a k+1 a k+2 ?
|
E
f (x)(2x&b 0k &b 0k+1 ) Sign[T $(x)] - 1&T 2(x) dx.
One can calculate the integrand. By (2.1) and (3.1) (for m=0) one obtains 2T(x)= 2T $(x)= 1&T 2(x)=
a
1 [(x&b 0k )(x&b 0k+1 )&(a 0k+1 ) 2 &(a 0k+2 ) 2 ], a 0k+2
0 k+1
2x&b 0k &b 0k+1 , a 0k+1 a 0k+2 (2a
1 [[(a 0k+1 +a 0k+2 ) 2 &(x&b 0k )(x&b 0k+1 )] a 0k+2 ) 2
0 k+1
_[(x&b 0k )(x&b 0k+1 )&(a 0k+1 &a 0k+2 ) 2 ]]. Lemma 5 is completely proved. Corollary 2. Assume that the recurrence coefficients [a n+1 ] and [b n ] belong to the class AP 2 and that they satisfy (1.3). Then for every continuous function f and for all integers k one has lim
n
|
E
f (x) G 2n+k(x) d+(x)=
2 ?
|
f(x) x 2 - 1&x 2 dx.
E
The main result is the following analog of Theorem 1.
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454
BORIS P. OSILENKER
Theorem 2. a n , b n satisfy
Let the Jacobi matrix J # AP 2 and the recursion coefficients
: ( |a n+2 &a n | + |b n+2 &b n | )<
(3.6)
n=0
and
: ( |a n+1 &a n | + |b n+1 &b n | )( |e n | + |e n+1 | )< ,
(3.7)
n=0
where e n =b n&1 +b n &b 00 &b 01 , lim n e n =0. Then uniformly on every closed subset of E 0 :=E"[T 2(x)=1] the Trace Formula 1 2 : [(: 2n+2 &: 2n )+ (; 2n+1 &; 2n )] p 2n(x) 2 n=0
+ : [ ; n+1(# n+1 &# n )+3(: n+2 ; n+2 &: n+1 ; n )] p n(x) p n+1(x) n=0
+2 : : n+2(# n+2 &# n ) p n(x) p n+2(x) n=0
+ : (: n+2 ; n+3 &: n+3 ; n+1 ) p n(x) p n+3(x) n=0
|T $(x)| - 1&T 2(x) 4 . = (a 01 a 02 ) 2 ? w(x) holds, where : n , ; n , and # n are defined by (2.4). Recall, that in our case, the measure + is absolutely continuous in E 0 , and +$(x)=w(x) is strictly positive and continuous on E 0 (see [11]). Proof of Theorem 2. We adapt the methods of [11] to the present situation. It is not difficult to see that from (2.4), (3.6), and (3.7) one obtains
: [|: 2n+2 &: 2n | + | ; 2n+1 &; 2n | + | ; n+1(# n+1 &# n )| n=0
+ |: n+2 ; n+2 &: n+1 ; n | + |: n+2(# n+2 &# n )| + |: n+2 ; n+3 &: n+3 ; n+1 | ]< .
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455
TRACE FORMULA
So, the series on the left side of our assertion converges uniformly on the closed subset K from E 0 . If we denote its sum by (x), then lim G n(x)=(x)
n
uniformly on K; consequently, for every continuous function f and for all integers k one has lim
n
|
f (x) G 2n+k(x) d+(x)=
E
|
f (x) (x) w(x) dx.
E
On the other hand, using periodicity of [a 0n+1 ] and [b 0n ], and by Lemma 5 one obtains lim
n
4 f (x) G 2n+k(x) w(x) dx= (a 01 a 02 ) 2 ?
|
E
|
f (x) |T$(x)| - 1&T 2(x) dx.
E
This means that for x # E 0 4 (x) w(x)= (a 01 a 02 ) 2 |T$(x)| - 1&T 2(x) ? from which the Trace Formula follows, if we use the end of the proof of Lemma 5. This completes the proof of Theorem 2. Corollary 3. If the recurrence coefficients satisfy 1 lim a n = , n 2
: |a n+2 &a n | < ,
b n =0
(n # Z + ),
n=0
then the Trace Formula
: (a 2n+1 a 2n+2 &a 2n&1 a 2n ) p 2n(x) n=0
+ : a n+1 a n+2(a 2n+3 +a 2n+2 &a 2n+1 &a 2n ) p n(x) p n+2(x) n=0
=
1 x 2 - 1&x 2 ? w(x)
holds uniformly on every compact subset of E 0 . In fact, in this case : n =a n&1 a n ,
; n =0,
# n =a 2n +a 2n+1 &(a 01 ) 2 &(a 02 ) 2.
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456
BORIS P. OSILENKER
Remarks. 1. In view of the assumption on the recurrence coefficients the relation E 0 =(&1, 1) holds, and Corollary 3 gives a Trace Formula for the single interval case. 2. Another Trace Formula is obtained in [11, 23].
4. EXAMPLES 1. The sieved Pollaczek polynomials. (a) The symmetric sieved Pollaczek polynomials [12]. Let [C *n(x; a; 2) (a>0, *>0)] be the symmetric sieved Pollaczek polynomials of the first kind. Then C *0(x; a; 2)=1, C *1(x; a; 2)=
x(*+a) , *
1 1 xC *2n+1(x; a; 2)= C *2n+2(x; a; 2)+ C *2n(x; a; 2), 2 2 xC *2n(x; a; 2)=
1 n+2* 1 n C *2n+1(x; a; 2)+ C * (x; a; 2). 2 n+a+* 2 n+a+* 2n&1
If [C *n(x; a; 2)] is the corresponding orthonormal system, then C *n(x; a; 2)=
{
?1(2*) &2*+1 2 a+*
=
12
* 12 *n(x; a; 2) n C
with * 2n =
n ! (a+*) , (n+a+*)(2*) n
* 2n+1 =
n ! (a+*) , (2*) n+1
where, as usual, (a n )=
1(n+a) . 1(a)
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457
TRACE FORMULA
We have n+1
1 2
n+a+*+1 C (x; a; 2) n+2* 1 + C (x; a; 2) 2 n+a+* n+2* 1 C (x; a; 2)= (x; a; 2) 2 n+a+* 1 n + (x; a; 2). C 2 n+a+*
xC *2n+1(x; a; 2)=
* 2n+2
* 2n
xC *2n
* 2n+1
* 2n&1
So a 2n =
1 2
n
n+a+* ,
a 2n+1 =
1 2
n+2*
n+a+* ,
b n =0.
(4.1)
From (4.1) one easily finds 1 a 2n &a 2n+1 rC , n which means that (1.4) does not hold and hence that Theorem 1 cannot be applied. On the other hand, a 2(n+1) &a 2n =O
1
\n + , 2
a 2n+1 &a 2n+3 =O
1
\n + , 2
which means that Corollary 3 can be applied. Note that when a=0 then C *n(x; a; 2)'s reduce to the C *n(x; 2)'s of Al-Salam et al. [1], for which Theorem 1 cannot be applied, and Corollary 3 can be applied. (b) The nonsymmetric sieved Pollaczek polynomials ([4]; see also [16]). Given a, b, c, * # R, we define the 2-sieved 4-parameter Pollaczek polynomials as the characteristic polynomials associated with Jacobi matrix J(2; a, b, c; *), where an =
A n&1 D n , n&1 B n
B
bn =
Cn Dn
(n=1, 2, ...).
Here A 2n =n+c+2*,
A 2n+1 =1,
B 2n =2(n+a+c+*),
B 2n+1 =2,
C 2n = &2b,
C 2n+1 =0,
D 2n =n+c+2*&1,
D 2n+1 =1.
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458
BORIS P. OSILENKER
For the 4-parameter Pollaczek polynomials see Chihara's book [5, p. 185], whereas for the sieving process see the works [1, 4, 12]. The recurrence coefficients associated with the above orthogonal polynomials a 2n =
1 2
n+c+2*&1
n+a+c+* ,
b 2n = &
b , n+a+c+*
a 2n+1 =
1 2
n+c+2*
n+a+c+* ,
b 2n+1 =0
are not bounded variation, but it is not difficult to see that the conditions of (3.6) and (3.7) of Theorem 2 are satisfied. 2. Orthonormal polynomial system defined by the recurrence relation (1.1) with 1 1 (&1) n D an = + +O p+1 , p 2 n n
\ +
bn=
1 (&1) n R +O p+1 p n n
\ +
( p 1), (4.2)
where D and R are constants independent of n, satisfy conditions (3.6) and (3.7). 3. Polynomials orthogonal on two intervals [2, 3, 6, 20]. Denote E ! =[&1, &!] _ [!, 1] (a)
(0 !<1).
Let w 0(x)=
x+!
1&x
x&! 1+x
(x # E ! ).
In the paper [3] the polynomials \^ n(x; !)=k n x n + } } } ,
k n >0
(n # Z + ; x # E ! )
were introduced; they are orthonormal with respect to the weight w 0(x) on E ! . They satisfy the following three-term recurrence relation a n+1 \^ n+1(x; !)&(x+b n ) \^ n(x; !)+a n \^ n&1(x; !)=0
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459
TRACE FORMULA
with an =
1+(&1) n ! (n=1, 2, ...), 2
b n =0
(n=1, 2, ...),
b 0 =&
1&! . 2
Obviously, the conditions of Theorem 2 are satisfied. (b)
Denote
w
( p, q)
(x)=
{
\
2 1&! 2
+
p+q
(x+:)(x 2 &! 2 ) p (1&x 2 ) q
for &1 x &! p+q 2 (x+:)(x 2 &! 2 ) p (1&x 2 ) q & 1&! 2
\
0
+
for for
! x 1 x E! ,
where p> &1, q> &1, &! : !. Let [ \ (np, q)(x)] (n # Z + ) be an orthogonal polynomial system on E ! with respect to the weight w ( p, q)(x), where the leading coefficients of \ (np, q)(x) are equal to 1. In [2] G. I. Barkov has shown that \ (2np, q)(x)= p, q) (x)= \ (2n+1
m 2n =
\
1&! 2 2
n
+\
2 1&! 2
+
p+q
I (np, q)
\
2x 2 &! 2 &1 , 1&! 2
+
p, q) \ (2n+2 (x)&m 2n \ (2np, q)(x) , x+: p, q) (:) \ (2n+2 , ( p, q) \ 2n (:)
where I (np, q)(x) are Jacobi polynomials with leading coefficient 1 orthogonal with respect to the weight (1+x) p (1&x) q (&1 x 1). The corresponding orthonormal polynomials [ \^ (np, q) ] can be represented in the form \^ (2np, q)(x)=(1&! 2 ) &(2n+1)2 2 &( p+q)2 _
_
n! 1(n+ p+1) 1(n+q+1) 1(n+ p+q+1) 1(2n+ p+q+1) 1(2n+ p+q+2)
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&
12
\ (2np, q)(x)
460
BORIS P. OSILENKER
and p, q) \^ (2n+1 (x)=(1&! 2 ) &(2n+1)2 2 &( p+q)2
_
_
n ! 1(n+ p+1) 1(n+q+1) 1(n+p+q+1) 1(2n+ p+q+1) 1(2n+ p+q+2)
&
12
p, q) _[&m 2n ] &12 \ (2n+1 (x),
and satisfy the following recurrence relation p, q) ( p, q) p, q) x\^ (np, q)(x)=a (n+1 \^ n+1 (x)+b (np, q) \^ (np, q)(x)+a (np, q) \^ (n&1 (x)
(n # Z + ) with p, q) =- &m 2n , a (2n+1 p, q) a (2n+2 =
1&! 2 - &m 2n
{
(n+1)(n+ p+1)(n+q+1)(n+ p+q+1) (2n+ p+q+1)(2n+ p+q+2) 2 (2n+ p+q+3)
=
12
,
b (np, q) =(&1) n :. It is not difficult to see that the corresponding Jacobi matrix belongs to the class AP 2 . In the case :=\!, i.e., w(x)= |x\!| (x 2 &! 2 ) p (1&x 2 )q (2(1&! 2)) p+q one obtains
p, q) a (2n+1
(n+q+1)(n+1) (2n+ p+q+3)(2n+ p+q+2)
_ & (n+ p+q+1)(n+ p+1) =- 1&! _(2n+ p+q+1)(2n+ p+q+2)&
p, q) =- 1&! 2 a (2n+2
2
12
12
(4.3)
b n =(&1) n !. These sequences of recurrence coefficients do not satisfy (1.4), but they satisfy the conditions of Theorem 2. In particular, in the case :=!=0, i.e., w(x)= |x| 2 p+1 (1&x 2 ) q we have the generalized Chebychev polynomials, for which [13, 15] (see (4.2) with p=1): 1 C 1 a (np, q) = + p, q (&1) n +O 2 , 2 n n
\ +
b n =0.
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TRACE FORMULA
461
ACKNOWLEDGMENTS I thank P. Nevai for his attention and the referee for his very useful remarks.
REFERENCES 1. W. Al-Salam, W. Allaway, and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 284 (1984), 3955. 2. G. I. Barkov, On some systems of polynomials orthogonal on two symmetric intervals, Izv. Vyssch. Uchebn. Zaved. Mat. 4 (1960), 316. [Russian] 3. V. F. Brjechka, On a certain class of polynomials orthogonal on two finite symmetric intervals, Zap. NII Mat. Mech. Kharkov State Univ. and Kharkov Math. Soc. XYII (1940), 7598. [Russian] 4. J. Charris and M. E. H. Ismail, On sieved orthogonal polynomials. Y. Sieved Pollaczek polynomials, SIAM J. Math. Anal. 18 (1987), 11771218. 5. T. S. Chihara, ``An Introduction to Orthogonal Polynomials,'' Gordon 6 Breach, New York, 1978. 6. T. S. Chihara, On kernel polynomials and related systems, Boll. Un. Mat. Ital. (3) 19 (1964), 451459. 7. J. Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, I, II, Pacific J. Math. 114 (1984), 325334; Pacific J. Math. 120 (1985), 4753. 8. J. Dombrowski and P. Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17 (1986), 752759. 9. J. S. Geronimo and W. Van Assche, Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986), 251283. 10. J. S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), 559581. 11. J. S. Geronimo and W. Van Assche, Approximating the weight function for the orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991), 341371. 12. M. E. H. Ismail, On sieved orthogonal polynomials, I. Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 10931113. 13. K. V. Laschenov, On a certain class of orthogonal polynomials, Uchen. Zap. Leningrad State Pedagog. Inst. 89 (1953), 169189. [Russian] 14. A. Mate and P. Nevai, Orthogonal polynomials and absolutely continuous measures, in ``Approximation IV'' (C. K. Chiu et al., Eds.), pp. 611617, Academic Press, New York, 1983. 15. P. Nevai, ``Orthogonal Polynomials,'' Mem. Amer. Math. Soc., Vol. 213, Am. Math. Soc., Providence, RI, 1979. 16. P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, in ``Progress in Approx. Theory'' (A. A. Gonchar and E. B. Saff, Eds.), pp. 79104, Springer-Verlag, BerlinNew York, 1992. 17. B. P. Osilenker, On summation of polynomial Fourier series expansions of functions of the class L +p (1 p ), Dokl. Akad. Nauk SSSR 202 (1972), 529531; English translation in Soviet Math. Dokl. 13 (1972), 140142. 18. B. P. Osilenker, Convergence and summability of Fourier expansions in orthogonal polynomials, associated with difference operator of second order, Sibirsk. Mat. J. 15 (1974), 892908; English translation in Siberian Math. J. 15 (1974), 623643. 19. B. P. Osilenker, On functional systems associated with the five-term recursion, I. Representations of the bilinear kernels, preprint. [Russian]
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462
BORIS P. OSILENKER
20. F. Peherstorfer, Orthogonal and Chebyshev polynomials on two intervals, Acta Math. Hungar. 55 (1990), 256278. 21. F. Peherstorfer, On BernsteinSzego orthogonal polynomials on several intervals, II. Orthogonal polynomials with periodic recurrence coefficients, J. Approx. Theory 64 (1991), 123161. 22. W. Van Assche, Asymptotics for orthogonal polynomials and three term recurrences, in ``Orthogonal Polynomials: Theory and Practice'' (P. Nevai, Ed.), NATO ASI Ser., Vol. C294, pp. 435462, Kluwer, Dordrecht, 1990. 23. W. Van Assche, Christoffel functions and Turan determinants on several intervals, J. Comput. Appl. Math. 48 (1993), 207223.
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Trace Formula for Orthogonal Polynomials with Asymptotically 2-Periodic Recurrence Coefficients Boris P. Osilenker Department of Mathematics, Moscow State Civil Engineering University, 2-oj Krestovskii pereulok Dom 4, Apt. 55, 129041 Moscow, Russia Communicated by Walter Van Assche Received March 4, 1996; accepted in revised form March 15, 1997
1. INTRODUCTION Suppose + is a positive probability measure on a compact set on the real line, that is, + is a positive Borel measure with d+=1. Then there is a unique sequence of polynomials p n(x)=k n x n + } } } ,
k n >0
(n # Z + =[0, 1, ...])
such that
|p
m
(x) p n(x) d+(x)=$ mn
(m, n # Z + ).
These orthonormal polynomials satisfy a three-term recurrence relation xp n(x)=a n+1 p n+1(x)+b n p n(x)+a n p n&1(x)
(n # Z + )
(1.1)
with the initial conditions p &1(x)=0,
p 0(x)=1,
a 0 =0,
(1.2)
where a n+1 =k n k n+1 >0 and b n # R. Conversely, by Favard's Theorem, if the polynomials p n(x) are given by the recurrence formula (1.1) with a n+1 >0 and b n # R, then there exists a positive Boreal measure + such that [ p n ] (n # Z + ) is an orthonormal polynomial system with respect to the measure +. If a n and b n are bounded, then the measure + is unique and the support of + is compact. 442 0021-904598 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
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TRACE FORMULA
The following result (Trace Formula) establishes the connection between Jacobi matrices
J=
\
b0 a1 0 }}}
a1 b1 a2 }}}
0 a2 b2 }}}
0 0 a3 }}}
0 0 0 }}}
}}} }}} }}} }}}
+
and their spectral measures. Theorem 1 [8, 14] (see also [16]). If Supp(+)=[&1, 1] and if the recursion coefficients [a n+1 ] and [b n ] satisfy lim a n = 12 ,
n
lim b n =0
(1.3)
n
and
: ( |a n+1 &a n | + |b n+1 &b n | )< ,
(1.4)
n=0
then
: [(a 2n+1 &a 2n ) p 2n(x)+a n(b n &b n&1 ) p n&1(x) p n(x)]= n=0
1 - 1&x 2 2? +$(x)
holds uniformly on all compact sets in (&1, 1). In addition, the measure + is absolutely continuous in the open interval (&1, 1), +$(x)>0 for all x # (&1, 1), and +$(x) is continuous in (&1, 1). Given two periodic sequences [a 0n+1 ] (a 0n+1 >0) and [b 0n ] (b 0n # R) (n # Z + ) with period N 1, the Jacobi matrix J is called asymptotically N-periodic (J # AP N ), if lim [ |a n &a 0n | + |b n &b 0n | ]=0.
n
(1.5)
In a survey [16] P. Nevai has posed the following problem: Extend the Trace Formula to asymptotically N-periodic Jacobi matrices. We investigate this problem for the case N=2.
2. REPRESENTATIONS OF THE KERNELS We assume that two periodic sequences a 0n+1 >0 and b 0n (n # Z + ) are given such that a 0n+2 =a 0n
(n=1, 2, ...),
b 0n+2 =b 0n
(n=0, 1, 2, ...)
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444
BORIS P. OSILENKER
(i.e., period N=2), and that the recurrence coefficients a n+1 and b n satisfy (1.5). We will write J # AP 2 . Denote the orthonormal polynomials with periodic recurrence coefficients a 0n+1 and b 0n by q n(x). Then xq n(x)=a 0n+1 q n+1(x)+b 0n q n(x)+a 0n q n&1(x)
(n # Z + )
q 0(x)=1.
q &1(x)=0, Let
T(x)=
a0 1 q 2(x)& 20 . 2 a1
_
&
(2.1)
The essential spectrum of the polynomials q n(x), resp. p n(x), consists of two intervals E, where E=[x # R, &1 T(x) 1]. The set E is of the form
_
:,
;&: :+; :+; ;&: & t _ +t ,; 2 2 2 2
& _
&
for some 0 t<1,
or in other words, if &1, 1 is the smallest and largest boundary point of E (this can be obtained easily by a linear transformation of q n(x) resp. p n(x)) then E is of the form E=[&1, &t] _ [t, 1]
for some 0 t<1.
These facts follow from [20, 21]. The special case when the intervals touch each other (the set [T 2(x)=1] consists of the endpoints of the intervals) leads to sieved orthogonal polynomials (see details in [1, 911, 21]). Lemma 1.
If J # AP 2 , then the following recurrence relation is valid,
s(x) p n(x)=: n+2 p n+2(x)+; n+1 p n+1(x)+# n p n(x) +; n p n&1(x)+: n p n&2(x)
(n # Z + )
(2.2)
with boundary conditions p &2(x)=0,
p &1(x)=0,
: 0 =: 1 =0,
; 0 =0,
(2.3)
where s(x)=2a 01 a 02 T(x)=(x&b 00 )(x&b 01 )&(a 01 ) 2 &(a 02 ) 2 : n+2 =a n+1 a n+2 , 2 n
# n =a +a
2 n+1
; n =a n(b n&1 +b n &b 00 &b 01 ) 0 2 1
0 2 2
0 0
0 1
&(a ) &(a ) +(b n &b )(b n &b )
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(2.4)
445
TRACE FORMULA
with lim : n =a 01 a 02 ,
n
lim ; n =0,
n
lim # n =0.
n
(2.5)
In fact, Lemma 1 follows from the definition of AP 2 and the five-term recurrence relation x 2p n(x)=a n+1 a n+2 p n+2(x)+a n+1(b n +b n+1 ) p n+1(x) +(a 2n +a 2n+1 +b 2n ) p n(x)+a n(b n&1 +b n ) p n&1(x) +a n a n&1 p n&2(x) by a direct computation. Lemma 2.
Let J # AP 2 . Then for the Dirichlet kernel n
D n(t, x)= : p k (t) p k (x) k=0
the representation [s(t)&s(x)] D n(t, x)=: n+2[ p n+2(t) p n(x)& p n(t) p n+2(x)] +: n+1[ p n+1(t) p n&1(x)& p n&1(t) p n+1(x)] +; n+1[ p n+1(t) p n(x)& p n(t) p n+1(x)]
(2.6)
holds. Here s(x), : n , ; n are defined by (2.4). In fact, formula (2.6) follows from (2.2) and (2.3) by straightforward calculations. Lemma 3. If J # AP 2 , then the following representation of the Fejer kernel, F n(t, x)=
n 1 : D k(t, x), n+1 k=0
is valid (n+1)[s(t)&s(x)] 2 F n(t, x) =! n(t, x)+! n(x, t)+G n(t, x)+G n(x, t),
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(2.7)
446
BORIS P. OSILENKER
where G n(t, x)=: 2n+2[ p n(t) p n(x)& p n+2(t) p n+2(x)] &2[: 2n p n(t) p n(x)+: 2n+1 p n+1(t) p n+1(x)] +: n+2 : n+4 p n+4(t) p n(x)+2: n+1 : n+3 p n+3(t) p n&1(x) +: n : n+2 p n+2(t) p n&2(x)+: n+2(# n+2 &# n ) p n+2(t) p n(x) &; 2n+1 p n+1(t) p n+1(x)+(: n+2 ; n+3 +: n+3 ; n+1 ) p n+3(t) p n(x) +2: n+1 ; n+2 p n+2(t) p n&1(x)+; n+1 ; n+2 p n+2(t) p n(x) +(: n+2 ; n+2 &2: n+1 ; n ) p n+1(t) p n(x) &: n+2 ; n+1 p n+2 (t) p n+1 (x)&: n+2 ; n+1 p n+1(t) p n+2(x) &: n+1 ; n p n(t) p n+1(x)
(2.8)
and n&1
n
! n(t, x)=2 : (: 2k+2 &: 2k ) p k(t) p k(x)+ : ( ; 2k+1 &; 2k ) p k(t) p k(x) k=0
k=0
n&1
+2 : : k+2(# k+2 &# k ) p k+2(t) p k(x) k=0 n&2
+ : (: k+2 ; k+3 &: k+3 ; k+1 ) p k+3(t) p k(x) k=0 n&1
+2 : (: k+2 ; k+2 &: k+1 ; k ) p k+1(t) p k(x) k=0 n
+ : ; k+1(# k+1 &# k ) p k+1(t) p k(x) k=0 n&1
+ : (: k+2 ; k+2 &: k+1 ; k ) p k(t) p k+1(x),
(2.9)
k=0
where : n , ; n , # n are defined by (2.4). Proof.
By formula (2.6) one has (n+1)[s(t)&s(x)] 2 F n(t, x)=` n(t, x)+` n(x, t),
(2.10)
where n
` n(t, x)= : % k(t, x) k=0
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(2.11)
447
TRACE FORMULA
with % k(t, x)=: k+2 s(t) p k+2(t) p k(x)+: k+1 s(t) p k+1(t) p k&1(x) +; k+1 s(t) p k+1(t) p k(x)&: k+2 p k+2(t) s(x) p k(x) &: k+1 p k+1(t) s(x) p k&1(x)&; k+1 p k+1(t) s(x) p k(x). The recurrence relation (2.2) yields % k(t, x)=: k+2 : k+4 p k+4(t) p k(x)+: k+2 ; k+3 p k+3(t) p k(x) +: k+2 # k+2 p k+2(t) p k(x)+: k+2 ; k+2 p k+1(t) p k(x) +: 2k+2 p k(t) p k(x)+: k+1 : k+3 p k+3(t) p k&1(x) +: k+1 ; k+2 p k+2(t) p k&1(x)+: k+1 # k+1 p k+1(t) p k&1(x) +: k+1 ; k+1 p k(t) p k&1(x)+: k+3 ; k+1 p k+3(t) p k(x) +; k+1 ; k+2 p k+2(t) p k(x)+; k+1 # k+1 p k+1(t) p k(x) +; 2k+1 p k(t) p k(x)+: k+1 ; k+1 p k&1(t) p k(x) &: 2k+2 p k+2(t) p k+2(x)&: k+2 ; k+1 p k+2(t) p k+1(x) &: k+2 # k p k+2(t) p k(x)&: k+2 ; k p k+2(t) p k&1(x) &: k+2 ; k+1 p k+1(t) p k+2(x)&; 2k+1 p k+1(t) p k+1(x) &; k+1 # k p k+1(t) p k(x)&: k+2 : k p k+2(t) p k&2(x) &: 2k+1 p k+1(t) p k+1(x)&: k&1 : k+1 p k+1(t) p k&3(x) &; k+1 ; k p k+1(t) p k&1(x)&: k ; k+1 p k+1(t) p k&2(x) &: k+1 ; k p k+1(t) p k(x)&: k+1 # k&1 p k+1(t) p k&1(x) &: k+1 ; k+1 p k+1(t) p k&2(x)+: 2k+1 p k&1(t) p k&1(x). For the calculation of ` n(t, x) we regroup similar terms of the last relation. Then ` n(t, x) can be representated in the form 6
` n(t, x)= : _ i (t, x), i=1
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(2.12)
448
BORIS P. OSILENKER
where n
n
_ 1(t, x)= : : 2k+2 p k(t) p k(x)+ : : 2k+1 p k&1(t) p k&1(x) k=0
k=0 n
n
& : : 2k+2 p k+2(t) p k+2(x)& : : 2k+1 p k+1(t) p k+1(x) k=0 n
k=0 n
+ : ; 2k+1 p k(t) p k(x)& : ; 2k+1 p k+1(t) p k+1(x); k=0 n
k=0 n
_ 2(t, x)= : : k+2 : k+4 p k+4(t) p k(x)& : : k : k+2 p k+2(t) p k&2(x) k=0
k=0 n
n
+ : : k+1 : k+3 p k+3(t) p k&1(x)& : : k&1 : k p k+1(t) p k&3(x); k=0 n
k=0 n
_ 3(t, x)= : : k+2 ; k+3 p k+3(t) p k(x)& : : k+2 ; k p k+2(t) p k&1(x) k=0
k=0 n
n
+ : : k+1 ; k+2 p k+2(t) p k&1(x)& : : k+1 ; k&1 p k+1(t) p k&2(x) k=0
k=0
n
n
+ : : k+3 ; k+1 p k+3(t) p k(x)& : : k ; k+1 p k+1(t) p k&2(x); k=0 n
k=0 n
_ 4(t, x)= : : k+2 # k+2 p k+2(t) p k(x)+ : : k+1 # k+1 p k+1(t) p k&1(x) k=0
k=0 n
n
+ : ; k+1 ; k+2 p k+2(t) p k(x)& : : k+2 # k p k+2(t) p k(x) k=0
k=0
n
n
& : ; k ; k+1 p k+1(t) p k&1(x)& : : k+1 # k&1 p k+1(t) p k&1(x); k=0 n
k=0 n
_ 5(t, x)= : : k+2 ; k+2 p k+1(t) p k(x)+ : : k+1 ; k+1 p k(t) p k&1(x) k=0
k=0 n
n
+ : ; k+1 # k+1 p k+1(t) p k(x)& : : k+2 ; k+1 p k+2(t) p k+1(x) k=0 n
k=0 n
& : ; k+1 # k p k+1(t) p k(x)& : : k+1 ; k p k+1(t) p k(x); k=0 n
k=0 n
_ 6(t, x)= : : k+1 ; k+1 p k&1(t) p k(x)& : : k+2 ; k+1 p k+1(t) p k+2(x). k=0
k=0
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TRACE FORMULA
449
Using Abel's summation by parts and the initial conditions (2.3), it is not difficult to show that the following formulas are valid _ 1(t, x)=: 2n+2 [ p n(t) p n(x)& p n+2(t) p n+2(x)] &2[: 2n p n(t) p n(x)+: 2n+1 p n+1(t) p n+1(x)] n&1
&; 2n+1 p n+1(t) p n+1(x)+2 : (: 2k+2 &: 2k ) p k(t) p k(x) k=0 n
+ : (; 2k+1 &; 2k ) p k(t) p k(x); k=0
_ 2(t, x)=: n+2 : n+4 p n+4(t) p n(x)+2: n+1 : n+3 p n+3(t) p n&1(x) +: n : n+2 p n+2(t) p n&2 (x); _ 3(t, x)=(: n+2 ; n+3 +: n+3 ; n+1 ) p n+3(t) p n(x) n&2
+2: n+1 ; n+2 p n+2(t) p n&1(x)+ : [: k+2( ; k+3 &; k+1 ) k=0
+(: k+2 &: k+3 ) ; k+1 ] p k+3(t) p k(x); _ 4(t, x)=: n+2(# n+2 &# n ) p n+2(t) p n(x)+; n+1 ; n+2 p n+2(t) p n(x) n&1
+2 : : k+2(# k+2 &# k ) p k+2(t) p k(x); k=0
_ 5(t, x)=(: n+2 ; n+2 &2: n+1 ; n ) p n+1(t) p n(x) n&1
&: n+2 ; n+1 p n+2(t) p n+1(x)+2 : [(: k+2 &: k+1 ) ; k+2 k=0
+(; k+2 &; k ) : k+1 ] p k+1(t) p k(x) n
+ : ; k+1(# k+1 &# k ) p k+1(t) p k(x); k=0
_ 6(t, x)= &: n+2 ; n+1 p n+1(t) p n+2(x)&: n+1 ; n p n(t) p n+1(x) n&1
+ : [: k+2( ; k+2 &; k )+(: k+2 &: k+1 ) ; k ] p k(t) p k+1(x). k=0
The representation (2.7)(2.9) follows from (2.10)(2.12) and the last six relations. Lemma 3 is completely proved. Remark. For N=1 the representation of F n(t, x) was given in [17, 18] (see also [19] for N=2). Fejer's kernel plays an important role in some problems of summability of Fourier series in orthogonal polynomials [17, 18]. The following assertion can be inferred from Lemma 3, if we put t=x.
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450
BORIS P. OSILENKER
Corollary 1. If J # AP 2 , then n&1
n
2 : (: 2k+2 &: 2k ) p 2k(x)+ : (; 2k+1 &; 2k ) p 2k(x) k=0
k=0 n&1
+2 : : k+2(# k+2 &# k ) p k(x) p k+2(x) k=0 n&1
+3 : (: k+2 ; k+2 &: k+1 ; k ) p k(x) p k+1(x) k=0 n&2
+ : (: k+2 ; k+3 &: k+3 ; k+1 ) p k(x) p k+3(x) k=0 n
+ : ; k+1(# k+1 &# k ) p k(x) p k+1(x)=G n(x), k=0
where G n(x)=(2: 2n &: 2n+2 ) p 2n(x)+(2: 2n+1 +; 2n+1 ) p 2n+1(x)+: 2n+2 p 2n+2(x) &: n+2 : n+4 p n(x) p n+4(x)&2: n+1 : n+3 p n&1(x) p n+3(x) &: n : n+2 p n&2(x) p n+2(x)&: n+2(# n+2 &# n ) p n(x) p n+2(x) &; n+1 ; n+2 p n(x) p n+2(x)&2: n+1 ; n+2 p n&1(x) p n+2(x) &(: n+2 ; n+3 +: n+3 ; n+1 ) p n(x) p n+3(x) +2: n+2 ; n+1 p n+1(x) p n+2(x) +(3: n+1 ; n &: n+2 ; n+2 ) p n(x) p n+1(x),
(2.13)
where : n , ; n , and # n are defined by (2.4). Remark. For N=1 formula (2.13) was obtained by J. Dombrowski [7].
3. TRACE FORMULA If [ p n ] is a system of orthonormal polynomials, satisfying (1.1), (1.2), then one can introduce the associated polynomials [ p (m) n (x)] of order m (m # Z + ) by the shifted recurrence formula (m) (m) (m) xp (m) n (x)=a n+m+1 p n+1(x)+b n+m p n (x)+a n+m p n&1(x)
(n # Z) (3.1)
with boundary conditions p (m) &1 (x)=0,
p (m) 0 (x)=1.
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(3.2)
451
TRACE FORMULA
We need the following result Lemma 4 [11]. Assume that the Jacobi matrix J # AP 2 . Then for every continuous function f and for all integers k and j one has
| f (x) p
lim
n
=
2n+ j
(x) p 2n+k(x) d+(x)
1 4?a
0 j+1
a
0 k+1
|
f (x) Sign[T$(x)]
E
- 1&T 2(x)
( j+1) 0 (k+1) _[a 0k+1 q k& j+1(x)+a j+1 q j&k+1(x)] dx,
(3.3)
where 0 (m+k+1) (x) a 0m+k+1 q (m) k (x)=&a m q &k&2
(k<0).
(3.4)
The next assertion gives the weak type asymptotics. Lemma 5. Suppose that the Jacobi matrix J # AP 2 . Then for every continuous function f and for all integers k one has lim
n
|
f (x) G 2n+k(x) d+(x)
E
= =
4(a 01 a 02 ) 2 ? 1 ?
f (x) |T $(x)| - 1&T 2(x) dx E
|
f (x) |2x&b 0k &b 0k+1 |
[(a 0k+2 +a 0k+1 ) 2 &(x&b 0k )(x&b 0k+1 )]_ dx. [(x&b 0k )(x&b 0k+1 )&(a 0k+2 &a 0k+1 ) 2 ]
E
_ Proof.
|
Using the definition of the class AP 2 and (2.5), we have
I (2) k := lim
|
= lim
|
n
n
f (x) G 2n+k(x) d+(x) E
E
f (x)[: 22n+k+2 p 22n+k+2(x)+2: 22n+k p 22n+k(x)
&: 22n+k+2 p 22n+k(x)+2: 22n+k+1 p 22n+k+1(x) &: 2n+k+2 : 2n+k+4 p 2n+k(x) p 2n+k+4(x) &2: 2n+k+1 : 2n+k+3 p 2n+k&1(x) p 2n+k+3(x) &: 2n+k : 2n+k+2 p 2n+k&2(x) p 2n+k+2(x)] d+(x).
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(3.5)
452
BORIS P. OSILENKER
By (1.5), (3.3), and (3.4) 1 4?
0 0 2 I (2) k =(a k+1 a k+2 )
_
{a
& & =
2 0 k+1
&
- 1&T 2(x)
E
4 (k+2) 2 (x)+ 0 q (k+1)(x) 0 q1 ak a k+1 1
1 2 (k+5) [q (k+1) (x)+q &3 (x)]& 0 [q (k) (x)+q (k+4) (x)] 5 &3 a 0k+1 ak 5 1 a
0 k+1
{a
&
f (x) Sign[T$(x)]
q (k+3) (x)+ 1
=
(k+3) [q (k&1) (x)+q &3 (x)] dx 5
(a 0k+1 a 0k+2 ) 2 4? _
|
2 0 k+1
|
f (x) Sign[T $(x)] - 1&T 2(x)
E
q (k+3) (x)+ 1
4 (k+2) 2 q (x)+ 0 q (k+1)(x) a 0k 1 a k+1 1
2 1 [q (k+1) (x)&q 1(k+3)(x)]& 0 [q (k) (x)&q (k+2) (x)] 5 1 a 0k+1 ak 5 1 a
0 k+1
=
[q (k&1) (x)&q 1(k+1)(x)] dx. 5
Since (see [11]) (m) (m) 2 (m) q (m) 5 (x)=2T(x) q 3 (x)&q 1 (x)=[4T (x)&1] q 1 (x),
then I (2) k =
(a 0k+1 a 0k+2 ) 2 4?
{_a +
3 0 k+1
|
f (x) Sign[T $(x)]
E
- 1&T 2(x)
q (k+3) (x)+ 1
6 (k+2) 4 q (x)+ 0 q (k+1)(x) a 0k 1 a k+1 1
2 (k) 1 q 1 (x)+ 0 q (k&1)(x) 0 ak a k+1 1
&4T 2(x)
_a
1 0 k+1
q (k+1) (x)+ 1
&
2 (k) 1 q (x)+ 0 q (k&1) (x) a 0k 1 a k+1 1
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&= dx.
TRACE FORMULA
453
It follows from the definition of the class AP 2 and (3.1), (3.2), that q (m) 1 (x)=
1 a
0 m+1
(x&b 0m ),
q 1(m+2)(x)=q (m) 1 (x).
So 1 a
0 k+1
q 1(k+1)(x)+
2 (k) 1 2 q 1 (x)+ 0 q 1(k&1)(x)= 0 (2x&b 0k &b 0k+1 ) 0 ak a k+1 a k+1 a 0k+2
and 3 a
0 k+1
q (k+3) (x)+ 1 +
6 (k+2) 4 q1 (x)+ 0 q (k+1)(x) 0 ak a k+1 1
1 8 2 (k) q 1 (x)+ 0 q 1(k&1)(x)= 0 (2x&b 0k &b 0k+1 ). 0 ak a k+1 a k+1 a 0k+2
Consequently, 2 0 0 I (2) k = a k+1 a k+2 ?
|
E
f (x)(2x&b 0k &b 0k+1 ) Sign[T $(x)] - 1&T 2(x) dx.
One can calculate the integrand. By (2.1) and (3.1) (for m=0) one obtains 2T(x)= 2T $(x)= 1&T 2(x)=
a
1 [(x&b 0k )(x&b 0k+1 )&(a 0k+1 ) 2 &(a 0k+2 ) 2 ], a 0k+2
0 k+1
2x&b 0k &b 0k+1 , a 0k+1 a 0k+2 (2a
1 [[(a 0k+1 +a 0k+2 ) 2 &(x&b 0k )(x&b 0k+1 )] a 0k+2 ) 2
0 k+1
_[(x&b 0k )(x&b 0k+1 )&(a 0k+1 &a 0k+2 ) 2 ]]. Lemma 5 is completely proved. Corollary 2. Assume that the recurrence coefficients [a n+1 ] and [b n ] belong to the class AP 2 and that they satisfy (1.3). Then for every continuous function f and for all integers k one has lim
n
|
E
f (x) G 2n+k(x) d+(x)=
2 ?
|
f(x) x 2 - 1&x 2 dx.
E
The main result is the following analog of Theorem 1.
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454
BORIS P. OSILENKER
Theorem 2. a n , b n satisfy
Let the Jacobi matrix J # AP 2 and the recursion coefficients
: ( |a n+2 &a n | + |b n+2 &b n | )<
(3.6)
n=0
and
: ( |a n+1 &a n | + |b n+1 &b n | )( |e n | + |e n+1 | )< ,
(3.7)
n=0
where e n =b n&1 +b n &b 00 &b 01 , lim n e n =0. Then uniformly on every closed subset of E 0 :=E"[T 2(x)=1] the Trace Formula 1 2 : [(: 2n+2 &: 2n )+ (; 2n+1 &; 2n )] p 2n(x) 2 n=0
+ : [ ; n+1(# n+1 &# n )+3(: n+2 ; n+2 &: n+1 ; n )] p n(x) p n+1(x) n=0
+2 : : n+2(# n+2 &# n ) p n(x) p n+2(x) n=0
+ : (: n+2 ; n+3 &: n+3 ; n+1 ) p n(x) p n+3(x) n=0
|T $(x)| - 1&T 2(x) 4 . = (a 01 a 02 ) 2 ? w(x) holds, where : n , ; n , and # n are defined by (2.4). Recall, that in our case, the measure + is absolutely continuous in E 0 , and +$(x)=w(x) is strictly positive and continuous on E 0 (see [11]). Proof of Theorem 2. We adapt the methods of [11] to the present situation. It is not difficult to see that from (2.4), (3.6), and (3.7) one obtains
: [|: 2n+2 &: 2n | + | ; 2n+1 &; 2n | + | ; n+1(# n+1 &# n )| n=0
+ |: n+2 ; n+2 &: n+1 ; n | + |: n+2(# n+2 &# n )| + |: n+2 ; n+3 &: n+3 ; n+1 | ]< .
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455
TRACE FORMULA
So, the series on the left side of our assertion converges uniformly on the closed subset K from E 0 . If we denote its sum by (x), then lim G n(x)=(x)
n
uniformly on K; consequently, for every continuous function f and for all integers k one has lim
n
|
f (x) G 2n+k(x) d+(x)=
E
|
f (x) (x) w(x) dx.
E
On the other hand, using periodicity of [a 0n+1 ] and [b 0n ], and by Lemma 5 one obtains lim
n
4 f (x) G 2n+k(x) w(x) dx= (a 01 a 02 ) 2 ?
|
E
|
f (x) |T$(x)| - 1&T 2(x) dx.
E
This means that for x # E 0 4 (x) w(x)= (a 01 a 02 ) 2 |T$(x)| - 1&T 2(x) ? from which the Trace Formula follows, if we use the end of the proof of Lemma 5. This completes the proof of Theorem 2. Corollary 3. If the recurrence coefficients satisfy 1 lim a n = , n 2
: |a n+2 &a n | < ,
b n =0
(n # Z + ),
n=0
then the Trace Formula
: (a 2n+1 a 2n+2 &a 2n&1 a 2n ) p 2n(x) n=0
+ : a n+1 a n+2(a 2n+3 +a 2n+2 &a 2n+1 &a 2n ) p n(x) p n+2(x) n=0
=
1 x 2 - 1&x 2 ? w(x)
holds uniformly on every compact subset of E 0 . In fact, in this case : n =a n&1 a n ,
; n =0,
# n =a 2n +a 2n+1 &(a 01 ) 2 &(a 02 ) 2.
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BORIS P. OSILENKER
Remarks. 1. In view of the assumption on the recurrence coefficients the relation E 0 =(&1, 1) holds, and Corollary 3 gives a Trace Formula for the single interval case. 2. Another Trace Formula is obtained in [11, 23].
4. EXAMPLES 1. The sieved Pollaczek polynomials. (a) The symmetric sieved Pollaczek polynomials [12]. Let [C *n(x; a; 2) (a>0, *>0)] be the symmetric sieved Pollaczek polynomials of the first kind. Then C *0(x; a; 2)=1, C *1(x; a; 2)=
x(*+a) , *
1 1 xC *2n+1(x; a; 2)= C *2n+2(x; a; 2)+ C *2n(x; a; 2), 2 2 xC *2n(x; a; 2)=
1 n+2* 1 n C *2n+1(x; a; 2)+ C * (x; a; 2). 2 n+a+* 2 n+a+* 2n&1
If [C *n(x; a; 2)] is the corresponding orthonormal system, then C *n(x; a; 2)=
{
?1(2*) &2*+1 2 a+*
=
12
* 12 *n(x; a; 2) n C
with * 2n =
n ! (a+*) , (n+a+*)(2*) n
* 2n+1 =
n ! (a+*) , (2*) n+1
where, as usual, (a n )=
1(n+a) . 1(a)
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TRACE FORMULA
We have n+1
1 2
n+a+*+1 C (x; a; 2) n+2* 1 + C (x; a; 2) 2 n+a+* n+2* 1 C (x; a; 2)= (x; a; 2) 2 n+a+* 1 n + (x; a; 2). C 2 n+a+*
xC *2n+1(x; a; 2)=
* 2n+2
* 2n
xC *2n
* 2n+1
* 2n&1
So a 2n =
1 2
n
n+a+* ,
a 2n+1 =
1 2
n+2*
n+a+* ,
b n =0.
(4.1)
From (4.1) one easily finds 1 a 2n &a 2n+1 rC , n which means that (1.4) does not hold and hence that Theorem 1 cannot be applied. On the other hand, a 2(n+1) &a 2n =O
1
\n + , 2
a 2n+1 &a 2n+3 =O
1
\n + , 2
which means that Corollary 3 can be applied. Note that when a=0 then C *n(x; a; 2)'s reduce to the C *n(x; 2)'s of Al-Salam et al. [1], for which Theorem 1 cannot be applied, and Corollary 3 can be applied. (b) The nonsymmetric sieved Pollaczek polynomials ([4]; see also [16]). Given a, b, c, * # R, we define the 2-sieved 4-parameter Pollaczek polynomials as the characteristic polynomials associated with Jacobi matrix J(2; a, b, c; *), where an =
A n&1 D n , n&1 B n
B
bn =
Cn Dn
(n=1, 2, ...).
Here A 2n =n+c+2*,
A 2n+1 =1,
B 2n =2(n+a+c+*),
B 2n+1 =2,
C 2n = &2b,
C 2n+1 =0,
D 2n =n+c+2*&1,
D 2n+1 =1.
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BORIS P. OSILENKER
For the 4-parameter Pollaczek polynomials see Chihara's book [5, p. 185], whereas for the sieving process see the works [1, 4, 12]. The recurrence coefficients associated with the above orthogonal polynomials a 2n =
1 2
n+c+2*&1
n+a+c+* ,
b 2n = &
b , n+a+c+*
a 2n+1 =
1 2
n+c+2*
n+a+c+* ,
b 2n+1 =0
are not bounded variation, but it is not difficult to see that the conditions of (3.6) and (3.7) of Theorem 2 are satisfied. 2. Orthonormal polynomial system defined by the recurrence relation (1.1) with 1 1 (&1) n D an = + +O p+1 , p 2 n n
\ +
bn=
1 (&1) n R +O p+1 p n n
\ +
( p 1), (4.2)
where D and R are constants independent of n, satisfy conditions (3.6) and (3.7). 3. Polynomials orthogonal on two intervals [2, 3, 6, 20]. Denote E ! =[&1, &!] _ [!, 1] (a)
(0 !<1).
Let w 0(x)=
x+!
1&x
x&! 1+x
(x # E ! ).
In the paper [3] the polynomials \^ n(x; !)=k n x n + } } } ,
k n >0
(n # Z + ; x # E ! )
were introduced; they are orthonormal with respect to the weight w 0(x) on E ! . They satisfy the following three-term recurrence relation a n+1 \^ n+1(x; !)&(x+b n ) \^ n(x; !)+a n \^ n&1(x; !)=0
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459
TRACE FORMULA
with an =
1+(&1) n ! (n=1, 2, ...), 2
b n =0
(n=1, 2, ...),
b 0 =&
1&! . 2
Obviously, the conditions of Theorem 2 are satisfied. (b)
Denote
w
( p, q)
(x)=
{
\
2 1&! 2
+
p+q
(x+:)(x 2 &! 2 ) p (1&x 2 ) q
for &1 x &! p+q 2 (x+:)(x 2 &! 2 ) p (1&x 2 ) q & 1&! 2
\
0
+
for for
! x 1 x E! ,
where p> &1, q> &1, &! : !. Let [ \ (np, q)(x)] (n # Z + ) be an orthogonal polynomial system on E ! with respect to the weight w ( p, q)(x), where the leading coefficients of \ (np, q)(x) are equal to 1. In [2] G. I. Barkov has shown that \ (2np, q)(x)= p, q) (x)= \ (2n+1
m 2n =
\
1&! 2 2
n
+\
2 1&! 2
+
p+q
I (np, q)
\
2x 2 &! 2 &1 , 1&! 2
+
p, q) \ (2n+2 (x)&m 2n \ (2np, q)(x) , x+: p, q) (:) \ (2n+2 , ( p, q) \ 2n (:)
where I (np, q)(x) are Jacobi polynomials with leading coefficient 1 orthogonal with respect to the weight (1+x) p (1&x) q (&1 x 1). The corresponding orthonormal polynomials [ \^ (np, q) ] can be represented in the form \^ (2np, q)(x)=(1&! 2 ) &(2n+1)2 2 &( p+q)2 _
_
n! 1(n+ p+1) 1(n+q+1) 1(n+ p+q+1) 1(2n+ p+q+1) 1(2n+ p+q+2)
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&
12
\ (2np, q)(x)
460
BORIS P. OSILENKER
and p, q) \^ (2n+1 (x)=(1&! 2 ) &(2n+1)2 2 &( p+q)2
_
_
n ! 1(n+ p+1) 1(n+q+1) 1(n+p+q+1) 1(2n+ p+q+1) 1(2n+ p+q+2)
&
12
p, q) _[&m 2n ] &12 \ (2n+1 (x),
and satisfy the following recurrence relation p, q) ( p, q) p, q) x\^ (np, q)(x)=a (n+1 \^ n+1 (x)+b (np, q) \^ (np, q)(x)+a (np, q) \^ (n&1 (x)
(n # Z + ) with p, q) =- &m 2n , a (2n+1 p, q) a (2n+2 =
1&! 2 - &m 2n
{
(n+1)(n+ p+1)(n+q+1)(n+ p+q+1) (2n+ p+q+1)(2n+ p+q+2) 2 (2n+ p+q+3)
=
12
,
b (np, q) =(&1) n :. It is not difficult to see that the corresponding Jacobi matrix belongs to the class AP 2 . In the case :=\!, i.e., w(x)= |x\!| (x 2 &! 2 ) p (1&x 2 )q (2(1&! 2)) p+q one obtains
p, q) a (2n+1
(n+q+1)(n+1) (2n+ p+q+3)(2n+ p+q+2)
_ & (n+ p+q+1)(n+ p+1) =- 1&! _(2n+ p+q+1)(2n+ p+q+2)&
p, q) =- 1&! 2 a (2n+2
2
12
12
(4.3)
b n =(&1) n !. These sequences of recurrence coefficients do not satisfy (1.4), but they satisfy the conditions of Theorem 2. In particular, in the case :=!=0, i.e., w(x)= |x| 2 p+1 (1&x 2 ) q we have the generalized Chebychev polynomials, for which [13, 15] (see (4.2) with p=1): 1 C 1 a (np, q) = + p, q (&1) n +O 2 , 2 n n
\ +
b n =0.
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461
ACKNOWLEDGMENTS I thank P. Nevai for his attention and the referee for his very useful remarks.
REFERENCES 1. W. Al-Salam, W. Allaway, and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 284 (1984), 3955. 2. G. I. Barkov, On some systems of polynomials orthogonal on two symmetric intervals, Izv. Vyssch. Uchebn. Zaved. Mat. 4 (1960), 316. [Russian] 3. V. F. Brjechka, On a certain class of polynomials orthogonal on two finite symmetric intervals, Zap. NII Mat. Mech. Kharkov State Univ. and Kharkov Math. Soc. XYII (1940), 7598. [Russian] 4. J. Charris and M. E. H. Ismail, On sieved orthogonal polynomials. Y. Sieved Pollaczek polynomials, SIAM J. Math. Anal. 18 (1987), 11771218. 5. T. S. Chihara, ``An Introduction to Orthogonal Polynomials,'' Gordon 6 Breach, New York, 1978. 6. T. S. Chihara, On kernel polynomials and related systems, Boll. Un. Mat. Ital. (3) 19 (1964), 451459. 7. J. Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, I, II, Pacific J. Math. 114 (1984), 325334; Pacific J. Math. 120 (1985), 4753. 8. J. Dombrowski and P. Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17 (1986), 752759. 9. J. S. Geronimo and W. Van Assche, Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986), 251283. 10. J. S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), 559581. 11. J. S. Geronimo and W. Van Assche, Approximating the weight function for the orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991), 341371. 12. M. E. H. Ismail, On sieved orthogonal polynomials, I. Symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 10931113. 13. K. V. Laschenov, On a certain class of orthogonal polynomials, Uchen. Zap. Leningrad State Pedagog. Inst. 89 (1953), 169189. [Russian] 14. A. Mate and P. Nevai, Orthogonal polynomials and absolutely continuous measures, in ``Approximation IV'' (C. K. Chiu et al., Eds.), pp. 611617, Academic Press, New York, 1983. 15. P. Nevai, ``Orthogonal Polynomials,'' Mem. Amer. Math. Soc., Vol. 213, Am. Math. Soc., Providence, RI, 1979. 16. P. Nevai, Orthogonal polynomials, recurrences, Jacobi matrices, and measures, in ``Progress in Approx. Theory'' (A. A. Gonchar and E. B. Saff, Eds.), pp. 79104, Springer-Verlag, BerlinNew York, 1992. 17. B. P. Osilenker, On summation of polynomial Fourier series expansions of functions of the class L +p (1 p ), Dokl. Akad. Nauk SSSR 202 (1972), 529531; English translation in Soviet Math. Dokl. 13 (1972), 140142. 18. B. P. Osilenker, Convergence and summability of Fourier expansions in orthogonal polynomials, associated with difference operator of second order, Sibirsk. Mat. J. 15 (1974), 892908; English translation in Siberian Math. J. 15 (1974), 623643. 19. B. P. Osilenker, On functional systems associated with the five-term recursion, I. Representations of the bilinear kernels, preprint. [Russian]
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20. F. Peherstorfer, Orthogonal and Chebyshev polynomials on two intervals, Acta Math. Hungar. 55 (1990), 256278. 21. F. Peherstorfer, On BernsteinSzego orthogonal polynomials on several intervals, II. Orthogonal polynomials with periodic recurrence coefficients, J. Approx. Theory 64 (1991), 123161. 22. W. Van Assche, Asymptotics for orthogonal polynomials and three term recurrences, in ``Orthogonal Polynomials: Theory and Practice'' (P. Nevai, Ed.), NATO ASI Ser., Vol. C294, pp. 435462, Kluwer, Dordrecht, 1990. 23. W. Van Assche, Christoffel functions and Turan determinants on several intervals, J. Comput. Appl. Math. 48 (1993), 207223.
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