Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 299–310
Some Muntz orthogonal systems1
Gradimir V. Milovanovic a; ∗ , Bratislav Dankovic b , Snezana Lj. Rancic b
a
Department Mathematics, University of Nis, Faculty of Electronic Engineering, P.O. Box 73, 18000 Nis,
Serbia, Yugoslavia
b
Department Automatics, University of Nis, Faculty of Electronic Engineering, P.O. Box 73, 18000 Nis,
Serbia, Yugoslavia
Received 30 October 1997; received in revised form 20 March 1998
Abstract
Let = {0 ; 1 ; : : :} be a given
of complex P
numbers such that |i |¿1 and Re(i j − 1)¿0 for each i and j.
Psequence
n
m
i
For Muntz polynomials P(x) =
p
x
and
Q(x)
=
q xj we de ne an inner product [P; Q] by
i
i=0
j=0 j
[P; Q] =
Z
1
(P ⊙ Q)(x)
0
dx
;
x2
Pn Pm
where (P ⊙ Q)(x) =
p q xi j . We obtain the Muntz polynomials (Qn (x)) orthogonal with respect to this inner
i=0
j=0 i j
product and connect them with the Malmquist rational functions
Qn−1
=0
Wn (z) = Q
n
(z − 1= )
=0
(z − )
orthogonal on the unit circle with respect to the inner product
(u; v) =
1
2i
I
|z|=1
u(z)v(z)
dz
:
z
Several interesting properties of polynomials Qn (x) are given.
Keywords: Muntz systems; Muntz polynomials; Malmquist systems; Rational functions; Orthogonality; Inner product;
Recurrence formulas; Zeros
∗
1
Corresponding author. E-mail: grade@elfak.ni.ac.yu.
This work was supported in part by the Serbian Ministry of Science and Technology, grant number 04M03.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 4 - 2
300
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
1. Introduction
This paper is devoted to some classes of orthogonal Muntz polynomials, as well as to the associated
Malmquist systems of orthogonal rational functions. The rst papers for orthogonal rational functions
on the unit circle whose poles are xed were given by Djrbashian [7, 8]. These rational functions
generalise the orthogonal polynomials of Szeg˝o [19, pp. 287–295]. Recently, the paper of Djrbashian
[8], which originally appeared in two parts, has been translated to English by Muller and Bultheel
[14]. A survey on the theory of such orthogonal systems and some open problems was written also
by Djrbashian [9]. Several papers in this direction have been appeared in the last period (cf. [4–6,
10, 15–18]).
The orthogonal Muntz systems were considered rst by Armenian mathematicians Badalyan [1]
and Taslakyan [20]. Recently, it was investigated by McCarthy et al. [11] and more completely by
Borwein et al. [3] (see also the recent book [2]).
In this paper we consider a class of Muntz polynomials orthogonal on (0; 1) with respect to an
inner product introduced in an unusual way. The paper is organised as follows. Section 2 is devoted
to one class of orthogonal Muntz systems on (0; 1). The Malmquist systems of orthogonal rational
functions and a connection with the orthogonal Muntz systems are considered in Section 3. Some
recurrence formulae of orthogonal Muntz polynomials are given in Section 4. Finally, real zeros of
the real Muntz polynomials are analysed in Section 5.
2. Orthogonal Muntz systems
Let = {0 ; 1 ; : : :} be a given sequence of complex numbers. Taking the following de nition
for x :
x = e log x ;
x ∈ (0; ∞); ∈ C;
and the value at x = 0 is de ned to be the limit of x as x → 0 from (0; ∞) whenever the limits
exists, we will consider orthogonal Muntz polynomials as linear combinations of the Muntz system
{x0 ; x1 ; : : : ; xn } (see [2, 3]). The set of all such polynomials we will denote by Mn (), i.e.,
Mn () = span{x0 ; x1 ; : : : ; xn };
where the linear span is over the complex numbers C in general. The union of all Mn () is denoted
by M ().
For realPnumbers 0 6 0 ¡1 ¡ · · · → ∞, it is well P
known that the real Muntz polynomials of
n
−1
k
2
the form k=0 ak x are dense in L [0; 1] if and only if ∞
k=1 k = + ∞. In addition, if 0 = 0 this
condition also characterises the denseness of the Muntz polynomials in C[0; 1] in the uniform norm.
The rst considerations of orthogonal Muntz systems were made by Badalyan [1] and Taslakyan
[20]. Recently, it was investigated by McCarthy et al. [11] and more completely by Borwein et al. [3].
Supposing that Re(k )¿ − 21 (k = 0; 1; : : :) they introduced the Muntz-Legendre polynomials on
(0; 1] as (see [3, 20]):
1
Pn (0 ; : : : ; n ; x) =
2i
where the simple contour
I n−1
Y s + + 1 xs ds
=0
s −
s − n
(n = 0; 1; : : :);
surrounds all the zeros of the denominator in the integrand.
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
301
The polynomials Pn (x) ≡ Pn (0 ; : : : ; n ; x) satisfy an orthogonality relation on (0; 1)
Z
1
Pn (x)Pm (x) dx = n; m =(1 + n + n );
0
for every n; m = 0; 1; : : : .
De nition 1. For ; ∈ C we have
x ⊙ x = x
(x ∈ (0; ∞)):
(1)
Using (1) we can introduce an external operation for the Muntz polynomials from M ().
De nition 2. For polynomials P ∈ Mn () and Q ∈ Mm (), i.e.,
P(x) =
n
X
pi xi
and
Q(x) =
i=0
m
X
q j x j ;
(2)
j=0
we have
(P ⊙ Q)(x) =
n X
m
X
p i q j x i j :
(3)
i=0 j=0
Under restrictions that for each i and j we have
|i |¿1;
Re(i j − 1)¿0;
(4)
then we can introduce a new inner product for Muntz polynomials.
De nition 3. Let the conditions (4) be satis ed for the Muntz polynomials P(x) and Q(x) given by
(2). Their inner product [P; Q] is de ned by
[P; Q] =
Z
0
1
(P ⊙ Q)(x)
dx
;
x2
(5)
where (P ⊙ Q)(x) is determined by (3).
It is not clear immediately that (5) represents an inner product. Therefore, we prove the following
result:
Theorem 4. Let = {0 ; 1 ; : : :} be a sequence of the complex numbers such that the conditions
(4) hold. Then
(i) [P; P]¿0;
(ii) [P; P] = 0 ⇔ P(x) ≡ 0;
(iii) [P + Q; R] = [P; R] + [Q; R];
(iv) [cP; Q] = c[P; Q];
(v) [P; Q] = [Q; P]
for each P; Q; R ∈ M () and each c ∈ C.
302
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. Let P(x) and Q(x) be given by (2). Using (5) and (3) we have
[P; Q] =
Z
1
(P ⊙ Q)(x)
0
Z 1
m
n X
dx X
xi j −2 dx:
q
−
p
i
j
2
x
0
i=0 j=0
Because of (4), we nd that
[P; Q] =
m
n X
X
i=0 j=0
R1
0
xi j −2 dx = 1=(i j − 1) for each i and j, so that we get
pi q j
:
i j − 1
(6)
In order to prove (i) and (ii) it is enough to conclude that the quadratic form
[P; P] =
n X
n
X
i=0
1
pi p j ;
j = 0 i j − 1
i.e., its matrix Hn = [1=(i j −1)]ni; j=0 , is positive de nite. Therefore, we use the Sylvester’s necessary
and sucient conditions (cf. [12, p. 214])
Dk = det Hk = det[1=(i j − 1)]ki; j=0 ¿0
(k = 0; 1; : : : ; n):
In order to evaluate the determinants Dk , we use Cauchy’s formula (see [13, p. 345])
det
"
1
ai + bj
#k
=
i; j = 0
Qk
i¿j=0
(ai − aj )(bi − bj )
Qk
i; j=0
(ai + bj )
with ai = i and bj = − 1= j . Thus, we obtain
Dk = Qk
1
j=0
= Qk
1
j=0
i.e.,
j
j
det
·
"
Qk
1
i − 1= j
i¿j=0
#k
i; j=0
(i − j )((−1= i ) + (1= j ))
;
Qk
(i − (1= j ))
i; j=0
2
i¿j=0 |i − j | = i j
· Qk
:
j=0 j
i; j=0 (i j − 1)= j
Dk = Qk
1
Qk
Since D0 = 1=(|0 |2 − 1)¿0 (because of (4) and
Dk =
Y |k − i |2
Dk−1 k−1
;
|k |2 − 1 i=0 |i k − 1|2
by induction we conclude that Dk ¿0 for all k.
The properties (iii)–(v) follow directly from (5) or (6).
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
303
Now we are ready to de ne the Muntz polynomials
Qn (x) ≡ Qn (0 ; 1 ; : : : ; n ; x)
(n = 0; 1; : : :)
(7)
orthogonal with respect to the inner product (5).
De nition 5. Let = {0 ; 1 ; : : :} be a sequence of the complex numbers such that the conditions
(4) hold and let
Qn−1
(s − 1= )
Wn (s) = Q=0
:
n
=0
(8)
(s − )
The nth Muntz polynomial Qn (x), associated to the rational function (8), is de ned by
Qn (x) =
1
2i
I
Wn (s)xs ds;
where the simple contour
(9)
surrounds all the points ( = 0; 1; : : : ; n).
Using (9), (16), (17) and Cauchy’s residue theorem we get a representation of (7) in the form
Qn (x) =
n
X
An; k xk ;
(10)
k=0
where
Qn−1
=0 (k − 1= )
An; k = Q
n
=0
6=k
(k − )
(k = 0; 1; : : : ; n);
(11)
and where we assumed that i 6= j (i 6= j).
The following theorem gives an orthogonality relation for polynomials Qn (x).
Theorem 6. Under conditions (4) on the sequence ; the Muntz polynomials Qn (x); n = 0; 1; : : : ;
de ned by (9), are orthogonal with respect to the inner product (5), i.e.,
[Qn ; Qm ] =
1
n; m :
(|n |2 − 1)|0 1 · · · n−1 |2
(12)
The proof of this theorem will be given in the next section using the orthogonality of a Malmquist
system of rational functions.
One particular result of Theorem 6, when → for each , can be interesting:
Corollary 7. Let Qn (x) be de ned by (9) and let 0 = 1 = · · · = n = . Then
log x);
Qn (x) = x Ln (−( − 1= )
(13)
where Ln (x) is the Laguerre polynomial orthogonal with respect to e−x on [0; ∞) and such that
Ln (0) = 1.
304
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. For 0 = 1 = · · · = n = , (9) reduces to
Qn (x) =
1
2i
n
(s − 1= )
xs ds;
(s − )n+1
I
where ∈ int . Since
#
"
n
dn
1
(s − 1= )
s
n xs ];
lim
=
x
[(s − 1= )
Res
z=
(s − )n+1
n! z→ dsn
we obtain by Cauchy’s residue theorem
Qn (x) =
n
n
1 X
k x (log x)k
n(n − 1) · · · (k + 1)( − 1= )
n! k = 0 k
=x
n
X
1 n
k!
k =0
k
k (log x)k ;
( − 1= )
which gives (13).
3. Orthogonal Malmquist systems and a connection with an orthogonal Muntz system
Let A = {a0 ; a1 ; : : :} be an arbitrary sequence of complex numbers in the unit circle (|a |¡1). The
Malmquist system of rational functions (see [7–9],[21]) is de ned in the following way:
n (s) =
Y a − s |a |
(1 − |an |2 )1=2 n−1
·
1 − an s =0 1 − a s a
(n = 0; 1; : : :);
(14)
where for a = 0 we put |a |=a = a =|a | = − 1. In the case n = 0, an empty product should be taken
to be equal 1. Such system of functions was intensively investigated in several papers by Djrbashian
[7–9], Bultheel et al. [4–6], Pan [15–18], etc.
In this section we want to prove some auxiliary results in order to connect this system of orthogonal
functions with some Muntz orthogonal system of polynomials.
Excluding the normalisation constants, the system (14) can be represented in the form
Qn−1
(s − a )
Wn (s) = Q=0
;
n
∗
=0 (s
(15)
− a )
where a∗ = 1= a . For a = 0 we put only s instead of (s − a )=(s − a∗ ).
Suppose now that a 6= a for 6= . Then (15) can be written in the form
Wn (s) =
n
X
An; k
k=0
s − a∗k
(16)
;
where
Qn−1
(a∗k − a )
∗
∗
=0 (a − a )
k
An; k = Q=0
n
(k = 0; 1; : : : ; n):
6=k
The case when a = a can be considered as a limiting process a → a .
(17)
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
305
Alternatively, for |a |¡1, (17) can be reduced to
1
Qn−1
− a
=0
(ak a − 1)
a0 · · · an
a
k
=
An; k =
:
· Q=0
n
Qn
1
1
=0 (a
k − a )
ak
6=k
=0
−
6=k
ak
a
Qn−1
(18)
It is well known that system of functions (14) is orthonormal on the unit circle |s| = 1 with respect
to the inner product
1
(u; v) =
2i
1
ds
=
u(s)v(s)
s
2
|s| = 1
I
Z
u(ei )v(ei ) d:
(19)
−
Namely, ( n ; m ) = nm (n; m = 0; 1; : : :). This generalises the Szeg˝o class of polynomials orthogonal
on the unit circle (see [19]).
We note also that
1
(u; v) =
2i
I
u(s)v(s∗ )
|s| = 1
ds
;
s
where s∗ = 1= s on the unit circle |s| = 1.
For the sake of completeness we mention the following result (cf. [21]):
Theorem 8. If the system of rational functions {Wn }+∞
n = 0 de ned by (15) and the inner product
(· ; ·) by (19), then
(Wn ; Wm ) = kWn k2 nm ;
where
kWn k2 =
|a0 a1 · · · an |2
:
1 − |an |2
In order to prove Theorem 6 we need the following auxiliary result:
Lemma 9. Let −1 6 t 6 1 and let F be de ned by
1
F(t) =
2i
I
Wn (s)Wm (ts)
|s| = 1
ds
;
s
(20)
where the system functions {Wn (s)} is de ned by (15) with mutually di erent numbers a
( = 0; 1; : : :) in the unit circle |s| = 1. Then
F(t) =
n X
m
X
An; i Am; j
i=0 j=0
a∗i a∗j − t
;
where the numbers An; k are given in (17).
(21)
306
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. For −1 6 t 6 1 we conclude that the integrand in
1
F(t) =
2i
I
|s| = 1
Qm−1 t
Qn−1
−
=0
(s
−
a
)
s
Q=0
· Qm
n
∗
t
(s − a )
=0
(−1)a0 · · · am 1
= ∗
·
a0 · · · a∗m−1
2i
=0
I
|s| = 1
s
−
1
a∗
1
a
·
ds
s
Qn−1
Qm−1
∗
=0 (s − a )
=0 (s − a t)
Qn
Q
ds
·
m
∗
=0
(s − a )
=0
(s − a t)
has (m + 1) poles inside the circle |s| = 1: aj t (j = 0; 1; : : : ; m). By Cauchy’s residue theorem we
nd that
Qm−1
m Qn−1
∗
(−1)a0 · · · am X
=0 (aj t − a t)
=0 (aj t − a )
Q
Q
F(t) = ∗
:
·
n
m
∗
=0 (aj t − a t)
a0 · · · am−1 j=0 =0 (aj t − a∗ )
6=j
De ne Gm; j by
Qm−1
(aj t − a∗ t)
:
=0 (aj t − a t)
Gm; j = Q=0
m
6=j
Since
Qm−1
Gm; j = Q=0
m
=0
6=j
1
(aj a − 1)
·
;
(aj − a ) a0 · · · am−1
because of (18), we have
Gm; j =
aj
1
aj am Am; j
· Am; j ·
=
:
a0 · · · am
a0 · · · am−1 |a0 · · · am |2
Thus,
a0 · · · am
Gm; j = aj Am; j ;
a∗0 · · · a∗m−1
and we obtain
F(t) = −
m
X
j=0
Qn−1
=0
Am; j Qn
=0
t−
a
aj
t−
a∗
aj
:
On the other hand, expanding Qj (t) =
Qj (t) =
n
X
i=0
Qn−1 a∗i
=0
Qn
=0
6=i
aj
a∗
i
aj
−
a
aj
−
a
aj
·
∗
1
t−
a∗
i
aj
Qn−1
=0 (t
=
n
X
i=0
− a =aj )=
Qn
=0 (t
Qn−1 ∗
(ai − a )
Q=0
·
n
∗
∗
=0
6=i
− a∗ =aj ) in partial fractions, we get
1
:
(ai − a ) t − a∗i a∗j
Because of (17), the right hand side of the last equality becomes
n
X
i=0
An; i
;
t − a∗i a∗j
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
307
so that we obtain
F(t) =
m
X
A m; j
n
X
An; i
i=0
j=0
1
;
a∗i a∗j − t
i.e., (21).
For t = 1, from (20) we see that F(1) = (Wn ; Wm ). Thus,
(Wn ; Wm ) =
n X
m
X
An; i Am; j
a∗i a∗j − 1
i=0 j=0
:
(22)
Now, we give a connection between the Malmquist system of rational functions (15) and a Muntz
system, which is orthogonal with respect to the inner product (5).
At rst, we assume that = {0 ; 1 ; : : :} is a complex sequence satisfying (4) and de ne a = 1= .
Then we use the Malmquist system (15) with these a . The corresponding numbers a∗ ( = 1= a = )
∗
∗
∗
which appear in (15) are outside the unit circle and they form the Muntz system {xa0 ; xa1 ; : : : ; xan }.
In order to shorten our notation and to be consistent with the previous section, we write a∗ =
( = 0; 1; : : :)
Finally, we can prove Theorem 6:
Proof of Theorem 6. According to (6) and (10), we have
[Qn ; Qm ] =
n X
m
X
An; i Am; j
i=0 j=0
i j − 1
;
where An; k is given by (11).
Using Lemma 9 with t = 1, i.e., equality (22), we conclude that
[Qn ; Qm ] = (Wn ; Wm );
where Wn (s) is determined by (8).
Since a = 1= ( = 0; 1; : : :), (12) follows from Theorem 8.
For the norm of the polynomial Qn (x) we obtain
kQn k =
p
[Qn ; Qn ] =
1
1
·p 2
;
|0 1 · · · n−1 |
|n | − 1
where the complex numbers satisfy the condition (4).
4. Recurrence formulae
Some recurrence formulae for polynomials Qn (x) are given in the following theorem. Similar
relations were obtained in [3] for Muntz–Legendre polynomials.
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G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Theorem 10. Suppose that is a complex sequence satisfying (4). Then the polynomials Qn (x);
de ned by (9), satisfy the following relations:
′
(x) + n Qn (x) − (1= n−1 )Qn−1 (x);
xQn′ (x) = xQn−1
xQn′ (x) = n Qn (x) +
n−1
X
(k − 1= k )Qk (x);
(23)
(24)
k =0
xQn′′ (x) = (n − 1)Qn′ (x) +
n−1
X
(k − 1= k )Qk′ (x);
(25)
k =0
Qn (1) = 1;
Qn′ (1) = n +
n−1
X
(k − 1= k );
(26)
k =0
Qn (x) = Qn−1 (x) − (n − 1= n−1 )xn
Z
1
t −n −1 Qn−1 (t) dt:
(27)
x
In proving this theorem we use the same method as in [3]. Because of that, we omit the proof.
5. Real zeros of Muntz polynomials
Now we consider the orthogonal real Muntz polynomials when the sequence is real and such
that
1¡0 ¡1 ¡ · · ·
(28)
Theorem 11. Let be a real sequence satisfying (28). Then the polynomial Qn (x); de ned by (9),
has exactly n simple zeros in (0; 1) and no other zeros in [1; ∞).
Proof. Because of orthogonality we have
Z
1
x ⊙ Qn (x)
0
dx
= 0 ( = 0; 1; : : : ; n − 1):
x2
Since x ⊙ Qn (x) = Qn (x ), these orthogonality conditions reduce to
Z
1
Qn (x )
0
dx
= 0 ( = 0; 1; : : : ; n − 1);
x2
or, after changing variable x = t,
Z
1
Qn (t)t − dt = 0 ( = 0; 1; : : : ; n − 1);
(29)
0
where = 1 + 1= . Note that 1¡ ¡2 and are mutually di erent under condition (28).
According to (29) the polynomial Qn (t) has at least one zero in (0; 1) in which Qn (t) changes
the sign. On the other side, it is easy to prove by induction (cf. [11, Theorem. 4.1]) that every real
Muntz polynomial from Mn () has at most n zeros in (0; ∞).
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
309
In order to prove our theorem, we suppose that the number of sign changes of Qn (t) in (0; 1) is
k¡n. Let
; : : : ; k be the points where Qn (t) changes sign. Now, we can construct a polynomial
P 1
Pk (t) = ki=0 bi t −i , such that Pk ( ) = 0 for each = 1; : : : ; k, and, for example, b0 = 1. We note also
that Pk (t) has no other zeros in (0; ∞). For existence and uniqueness of such a polynomial see [2,
Chap. 3].
As we can see now, Pk (t)Qn (t) has a constant sign in (0; 1), and therefore
Z
1
Pk (t)Qn (t) dt 6= 0:
0
On the other side, using (29), we have
Z
1
Pk (t)Qn (t) dt =
0
Z
0
1
k
X
i=0
bi t
−i
!
Qn (t) dt =
k
X
i=0
bi
Z
1
t −i Qn (t) dt = 0
0
for k¡n, which gives a contradiction. Thus, k = n. The points 1 ; : : : ; n are simple zeros of Qn (t)
in (0; 1) and this polynomial has no other positive zeros.
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[2] P. Borwein, T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995.
[3] P. Borwein, T. Erdelyi, J. Zhang, Muntz systems and orthogonal Muntz-Legendre polynomials, Trans. Amer. Math.
Soc. 342 (1994) 523–542.
[4] A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njastad, The computation of orthogonal rational functions and
their interpolating properties, Numer. Algorithms 2 (1992) 85–114.
[5] A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njastad, A Favard theorem for orthogonal rational functions on
the unit circle, Numer. Algorithms 3 (1992) 81– 89.
[6] A. Bultheel, P. Gonzalez-Vera, E. Hendriksen, O. Njastad, Moment problems and orthogonal functions, J. Comput.
Appl. Math. 48 (1993) 49 – 68.
[7] M.M. Djrbashian, Orthogonal systems of rational functions on the circle with given set of poles, Dokl. Akad. Nauk
SSSR 147 (1962) 1278 –1281.
[8] M.M. Djrbashian, Orthogonal systems of rational functions on the circle, Akad. Nauk. Armyan. SSR Izv. Ser. Mat.
1 (1966) 3–24, 106–125 (in Russian).
[9] M.M. Djrbashian, A survey on the theory of orthogonal systems and some open problems, in: P. Nevai (Ed.),
Orthogonal Polynomials – Theory and Practice, NATO ASI Series, Series C; Mathematical and Physical Sciences,
vol. 294, Kluwer, Dordrecht, 1990, pp. 135–146.
[10] S. Kus, On the completeness of certain systems in L2 (|z| = 1), Demonstratio Math. 18 (1985) 395– 408.
[11] P.C. McCarthy, J.E. Sayre, B.L.R. Shawyer, Generalized Legendre polynomials, J. Math. Anal. Appl. 177 (1993)
530 –537.
[12] G.V. Milovanovic, Numerical Analysis, I Naucna Knjiga, Belgrade, 1988.
[13] T. Muir, The Theory of Determinants, I, Macmillan London, 1906.
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circle” by M.M. Dzrbasian, Report TW 253, Katholieke Univ. Leuven, Leuven, 1997.
[15] K. Pan, Strong and weak convergence of orthonormal systems of rational functions on the unit circle, J. Comput.
Appl. Math. 46 (1993) 427– 436.
[16] K. Pan, Extensions of Szeg˝o’s theory of rational functions orthogonal on the unit circle, J. Comput. Appl. Math. 62
(1995) 321–331.
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[17] K. Pan, Convergence of orthogonal rational functions outside of the unit circle, J. Comput. Appl. Math. 69 (1996)
345–351.
[18] K. Pan, On the convergence of rational functions orthogonal on the unit circle, J. Comput. Appl. Math. 76 (1996)
315–324.
[19] G. Szeg˝o, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence,
RI, 1975.
[20] A.K. Taslakyan, Some properties of Legendre quasi-polynomials with respect to a Muntz system, Mathematics,
Erevan
University, Erevan
2 (1984) 179 –189 (in Russian, Armenian summary).
[21] J.L. Walsh, Interpolation and Approximation in the Complex Plane, Amer. Math. Soc., Providence, RI, 1969.
Some Muntz orthogonal systems1
Gradimir V. Milovanovic a; ∗ , Bratislav Dankovic b , Snezana Lj. Rancic b
a
Department Mathematics, University of Nis, Faculty of Electronic Engineering, P.O. Box 73, 18000 Nis,
Serbia, Yugoslavia
b
Department Automatics, University of Nis, Faculty of Electronic Engineering, P.O. Box 73, 18000 Nis,
Serbia, Yugoslavia
Received 30 October 1997; received in revised form 20 March 1998
Abstract
Let = {0 ; 1 ; : : :} be a given
of complex P
numbers such that |i |¿1 and Re(i j − 1)¿0 for each i and j.
Psequence
n
m
i
For Muntz polynomials P(x) =
p
x
and
Q(x)
=
q xj we de ne an inner product [P; Q] by
i
i=0
j=0 j
[P; Q] =
Z
1
(P ⊙ Q)(x)
0
dx
;
x2
Pn Pm
where (P ⊙ Q)(x) =
p q xi j . We obtain the Muntz polynomials (Qn (x)) orthogonal with respect to this inner
i=0
j=0 i j
product and connect them with the Malmquist rational functions
Qn−1
=0
Wn (z) = Q
n
(z − 1= )
=0
(z − )
orthogonal on the unit circle with respect to the inner product
(u; v) =
1
2i
I
|z|=1
u(z)v(z)
dz
:
z
Several interesting properties of polynomials Qn (x) are given.
Keywords: Muntz systems; Muntz polynomials; Malmquist systems; Rational functions; Orthogonality; Inner product;
Recurrence formulas; Zeros
∗
1
Corresponding author. E-mail: grade@elfak.ni.ac.yu.
This work was supported in part by the Serbian Ministry of Science and Technology, grant number 04M03.
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 6 4 - 2
300
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
1. Introduction
This paper is devoted to some classes of orthogonal Muntz polynomials, as well as to the associated
Malmquist systems of orthogonal rational functions. The rst papers for orthogonal rational functions
on the unit circle whose poles are xed were given by Djrbashian [7, 8]. These rational functions
generalise the orthogonal polynomials of Szeg˝o [19, pp. 287–295]. Recently, the paper of Djrbashian
[8], which originally appeared in two parts, has been translated to English by Muller and Bultheel
[14]. A survey on the theory of such orthogonal systems and some open problems was written also
by Djrbashian [9]. Several papers in this direction have been appeared in the last period (cf. [4–6,
10, 15–18]).
The orthogonal Muntz systems were considered rst by Armenian mathematicians Badalyan [1]
and Taslakyan [20]. Recently, it was investigated by McCarthy et al. [11] and more completely by
Borwein et al. [3] (see also the recent book [2]).
In this paper we consider a class of Muntz polynomials orthogonal on (0; 1) with respect to an
inner product introduced in an unusual way. The paper is organised as follows. Section 2 is devoted
to one class of orthogonal Muntz systems on (0; 1). The Malmquist systems of orthogonal rational
functions and a connection with the orthogonal Muntz systems are considered in Section 3. Some
recurrence formulae of orthogonal Muntz polynomials are given in Section 4. Finally, real zeros of
the real Muntz polynomials are analysed in Section 5.
2. Orthogonal Muntz systems
Let = {0 ; 1 ; : : :} be a given sequence of complex numbers. Taking the following de nition
for x :
x = e log x ;
x ∈ (0; ∞); ∈ C;
and the value at x = 0 is de ned to be the limit of x as x → 0 from (0; ∞) whenever the limits
exists, we will consider orthogonal Muntz polynomials as linear combinations of the Muntz system
{x0 ; x1 ; : : : ; xn } (see [2, 3]). The set of all such polynomials we will denote by Mn (), i.e.,
Mn () = span{x0 ; x1 ; : : : ; xn };
where the linear span is over the complex numbers C in general. The union of all Mn () is denoted
by M ().
For realPnumbers 0 6 0 ¡1 ¡ · · · → ∞, it is well P
known that the real Muntz polynomials of
n
−1
k
2
the form k=0 ak x are dense in L [0; 1] if and only if ∞
k=1 k = + ∞. In addition, if 0 = 0 this
condition also characterises the denseness of the Muntz polynomials in C[0; 1] in the uniform norm.
The rst considerations of orthogonal Muntz systems were made by Badalyan [1] and Taslakyan
[20]. Recently, it was investigated by McCarthy et al. [11] and more completely by Borwein et al. [3].
Supposing that Re(k )¿ − 21 (k = 0; 1; : : :) they introduced the Muntz-Legendre polynomials on
(0; 1] as (see [3, 20]):
1
Pn (0 ; : : : ; n ; x) =
2i
where the simple contour
I n−1
Y s + + 1 xs ds
=0
s −
s − n
(n = 0; 1; : : :);
surrounds all the zeros of the denominator in the integrand.
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
301
The polynomials Pn (x) ≡ Pn (0 ; : : : ; n ; x) satisfy an orthogonality relation on (0; 1)
Z
1
Pn (x)Pm (x) dx = n; m =(1 + n + n );
0
for every n; m = 0; 1; : : : .
De nition 1. For ; ∈ C we have
x ⊙ x = x
(x ∈ (0; ∞)):
(1)
Using (1) we can introduce an external operation for the Muntz polynomials from M ().
De nition 2. For polynomials P ∈ Mn () and Q ∈ Mm (), i.e.,
P(x) =
n
X
pi xi
and
Q(x) =
i=0
m
X
q j x j ;
(2)
j=0
we have
(P ⊙ Q)(x) =
n X
m
X
p i q j x i j :
(3)
i=0 j=0
Under restrictions that for each i and j we have
|i |¿1;
Re(i j − 1)¿0;
(4)
then we can introduce a new inner product for Muntz polynomials.
De nition 3. Let the conditions (4) be satis ed for the Muntz polynomials P(x) and Q(x) given by
(2). Their inner product [P; Q] is de ned by
[P; Q] =
Z
0
1
(P ⊙ Q)(x)
dx
;
x2
(5)
where (P ⊙ Q)(x) is determined by (3).
It is not clear immediately that (5) represents an inner product. Therefore, we prove the following
result:
Theorem 4. Let = {0 ; 1 ; : : :} be a sequence of the complex numbers such that the conditions
(4) hold. Then
(i) [P; P]¿0;
(ii) [P; P] = 0 ⇔ P(x) ≡ 0;
(iii) [P + Q; R] = [P; R] + [Q; R];
(iv) [cP; Q] = c[P; Q];
(v) [P; Q] = [Q; P]
for each P; Q; R ∈ M () and each c ∈ C.
302
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. Let P(x) and Q(x) be given by (2). Using (5) and (3) we have
[P; Q] =
Z
1
(P ⊙ Q)(x)
0
Z 1
m
n X
dx X
xi j −2 dx:
q
−
p
i
j
2
x
0
i=0 j=0
Because of (4), we nd that
[P; Q] =
m
n X
X
i=0 j=0
R1
0
xi j −2 dx = 1=(i j − 1) for each i and j, so that we get
pi q j
:
i j − 1
(6)
In order to prove (i) and (ii) it is enough to conclude that the quadratic form
[P; P] =
n X
n
X
i=0
1
pi p j ;
j = 0 i j − 1
i.e., its matrix Hn = [1=(i j −1)]ni; j=0 , is positive de nite. Therefore, we use the Sylvester’s necessary
and sucient conditions (cf. [12, p. 214])
Dk = det Hk = det[1=(i j − 1)]ki; j=0 ¿0
(k = 0; 1; : : : ; n):
In order to evaluate the determinants Dk , we use Cauchy’s formula (see [13, p. 345])
det
"
1
ai + bj
#k
=
i; j = 0
Qk
i¿j=0
(ai − aj )(bi − bj )
Qk
i; j=0
(ai + bj )
with ai = i and bj = − 1= j . Thus, we obtain
Dk = Qk
1
j=0
= Qk
1
j=0
i.e.,
j
j
det
·
"
Qk
1
i − 1= j
i¿j=0
#k
i; j=0
(i − j )((−1= i ) + (1= j ))
;
Qk
(i − (1= j ))
i; j=0
2
i¿j=0 |i − j | = i j
· Qk
:
j=0 j
i; j=0 (i j − 1)= j
Dk = Qk
1
Qk
Since D0 = 1=(|0 |2 − 1)¿0 (because of (4) and
Dk =
Y |k − i |2
Dk−1 k−1
;
|k |2 − 1 i=0 |i k − 1|2
by induction we conclude that Dk ¿0 for all k.
The properties (iii)–(v) follow directly from (5) or (6).
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
303
Now we are ready to de ne the Muntz polynomials
Qn (x) ≡ Qn (0 ; 1 ; : : : ; n ; x)
(n = 0; 1; : : :)
(7)
orthogonal with respect to the inner product (5).
De nition 5. Let = {0 ; 1 ; : : :} be a sequence of the complex numbers such that the conditions
(4) hold and let
Qn−1
(s − 1= )
Wn (s) = Q=0
:
n
=0
(8)
(s − )
The nth Muntz polynomial Qn (x), associated to the rational function (8), is de ned by
Qn (x) =
1
2i
I
Wn (s)xs ds;
where the simple contour
(9)
surrounds all the points ( = 0; 1; : : : ; n).
Using (9), (16), (17) and Cauchy’s residue theorem we get a representation of (7) in the form
Qn (x) =
n
X
An; k xk ;
(10)
k=0
where
Qn−1
=0 (k − 1= )
An; k = Q
n
=0
6=k
(k − )
(k = 0; 1; : : : ; n);
(11)
and where we assumed that i 6= j (i 6= j).
The following theorem gives an orthogonality relation for polynomials Qn (x).
Theorem 6. Under conditions (4) on the sequence ; the Muntz polynomials Qn (x); n = 0; 1; : : : ;
de ned by (9), are orthogonal with respect to the inner product (5), i.e.,
[Qn ; Qm ] =
1
n; m :
(|n |2 − 1)|0 1 · · · n−1 |2
(12)
The proof of this theorem will be given in the next section using the orthogonality of a Malmquist
system of rational functions.
One particular result of Theorem 6, when → for each , can be interesting:
Corollary 7. Let Qn (x) be de ned by (9) and let 0 = 1 = · · · = n = . Then
log x);
Qn (x) = x Ln (−( − 1= )
(13)
where Ln (x) is the Laguerre polynomial orthogonal with respect to e−x on [0; ∞) and such that
Ln (0) = 1.
304
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. For 0 = 1 = · · · = n = , (9) reduces to
Qn (x) =
1
2i
n
(s − 1= )
xs ds;
(s − )n+1
I
where ∈ int . Since
#
"
n
dn
1
(s − 1= )
s
n xs ];
lim
=
x
[(s − 1= )
Res
z=
(s − )n+1
n! z→ dsn
we obtain by Cauchy’s residue theorem
Qn (x) =
n
n
1 X
k x (log x)k
n(n − 1) · · · (k + 1)( − 1= )
n! k = 0 k
=x
n
X
1 n
k!
k =0
k
k (log x)k ;
( − 1= )
which gives (13).
3. Orthogonal Malmquist systems and a connection with an orthogonal Muntz system
Let A = {a0 ; a1 ; : : :} be an arbitrary sequence of complex numbers in the unit circle (|a |¡1). The
Malmquist system of rational functions (see [7–9],[21]) is de ned in the following way:
n (s) =
Y a − s |a |
(1 − |an |2 )1=2 n−1
·
1 − an s =0 1 − a s a
(n = 0; 1; : : :);
(14)
where for a = 0 we put |a |=a = a =|a | = − 1. In the case n = 0, an empty product should be taken
to be equal 1. Such system of functions was intensively investigated in several papers by Djrbashian
[7–9], Bultheel et al. [4–6], Pan [15–18], etc.
In this section we want to prove some auxiliary results in order to connect this system of orthogonal
functions with some Muntz orthogonal system of polynomials.
Excluding the normalisation constants, the system (14) can be represented in the form
Qn−1
(s − a )
Wn (s) = Q=0
;
n
∗
=0 (s
(15)
− a )
where a∗ = 1= a . For a = 0 we put only s instead of (s − a )=(s − a∗ ).
Suppose now that a 6= a for 6= . Then (15) can be written in the form
Wn (s) =
n
X
An; k
k=0
s − a∗k
(16)
;
where
Qn−1
(a∗k − a )
∗
∗
=0 (a − a )
k
An; k = Q=0
n
(k = 0; 1; : : : ; n):
6=k
The case when a = a can be considered as a limiting process a → a .
(17)
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
305
Alternatively, for |a |¡1, (17) can be reduced to
1
Qn−1
− a
=0
(ak a − 1)
a0 · · · an
a
k
=
An; k =
:
· Q=0
n
Qn
1
1
=0 (a
k − a )
ak
6=k
=0
−
6=k
ak
a
Qn−1
(18)
It is well known that system of functions (14) is orthonormal on the unit circle |s| = 1 with respect
to the inner product
1
(u; v) =
2i
1
ds
=
u(s)v(s)
s
2
|s| = 1
I
Z
u(ei )v(ei ) d:
(19)
−
Namely, ( n ; m ) = nm (n; m = 0; 1; : : :). This generalises the Szeg˝o class of polynomials orthogonal
on the unit circle (see [19]).
We note also that
1
(u; v) =
2i
I
u(s)v(s∗ )
|s| = 1
ds
;
s
where s∗ = 1= s on the unit circle |s| = 1.
For the sake of completeness we mention the following result (cf. [21]):
Theorem 8. If the system of rational functions {Wn }+∞
n = 0 de ned by (15) and the inner product
(· ; ·) by (19), then
(Wn ; Wm ) = kWn k2 nm ;
where
kWn k2 =
|a0 a1 · · · an |2
:
1 − |an |2
In order to prove Theorem 6 we need the following auxiliary result:
Lemma 9. Let −1 6 t 6 1 and let F be de ned by
1
F(t) =
2i
I
Wn (s)Wm (ts)
|s| = 1
ds
;
s
(20)
where the system functions {Wn (s)} is de ned by (15) with mutually di erent numbers a
( = 0; 1; : : :) in the unit circle |s| = 1. Then
F(t) =
n X
m
X
An; i Am; j
i=0 j=0
a∗i a∗j − t
;
where the numbers An; k are given in (17).
(21)
306
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Proof. For −1 6 t 6 1 we conclude that the integrand in
1
F(t) =
2i
I
|s| = 1
Qm−1 t
Qn−1
−
=0
(s
−
a
)
s
Q=0
· Qm
n
∗
t
(s − a )
=0
(−1)a0 · · · am 1
= ∗
·
a0 · · · a∗m−1
2i
=0
I
|s| = 1
s
−
1
a∗
1
a
·
ds
s
Qn−1
Qm−1
∗
=0 (s − a )
=0 (s − a t)
Qn
Q
ds
·
m
∗
=0
(s − a )
=0
(s − a t)
has (m + 1) poles inside the circle |s| = 1: aj t (j = 0; 1; : : : ; m). By Cauchy’s residue theorem we
nd that
Qm−1
m Qn−1
∗
(−1)a0 · · · am X
=0 (aj t − a t)
=0 (aj t − a )
Q
Q
F(t) = ∗
:
·
n
m
∗
=0 (aj t − a t)
a0 · · · am−1 j=0 =0 (aj t − a∗ )
6=j
De ne Gm; j by
Qm−1
(aj t − a∗ t)
:
=0 (aj t − a t)
Gm; j = Q=0
m
6=j
Since
Qm−1
Gm; j = Q=0
m
=0
6=j
1
(aj a − 1)
·
;
(aj − a ) a0 · · · am−1
because of (18), we have
Gm; j =
aj
1
aj am Am; j
· Am; j ·
=
:
a0 · · · am
a0 · · · am−1 |a0 · · · am |2
Thus,
a0 · · · am
Gm; j = aj Am; j ;
a∗0 · · · a∗m−1
and we obtain
F(t) = −
m
X
j=0
Qn−1
=0
Am; j Qn
=0
t−
a
aj
t−
a∗
aj
:
On the other hand, expanding Qj (t) =
Qj (t) =
n
X
i=0
Qn−1 a∗i
=0
Qn
=0
6=i
aj
a∗
i
aj
−
a
aj
−
a
aj
·
∗
1
t−
a∗
i
aj
Qn−1
=0 (t
=
n
X
i=0
− a =aj )=
Qn
=0 (t
Qn−1 ∗
(ai − a )
Q=0
·
n
∗
∗
=0
6=i
− a∗ =aj ) in partial fractions, we get
1
:
(ai − a ) t − a∗i a∗j
Because of (17), the right hand side of the last equality becomes
n
X
i=0
An; i
;
t − a∗i a∗j
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
307
so that we obtain
F(t) =
m
X
A m; j
n
X
An; i
i=0
j=0
1
;
a∗i a∗j − t
i.e., (21).
For t = 1, from (20) we see that F(1) = (Wn ; Wm ). Thus,
(Wn ; Wm ) =
n X
m
X
An; i Am; j
a∗i a∗j − 1
i=0 j=0
:
(22)
Now, we give a connection between the Malmquist system of rational functions (15) and a Muntz
system, which is orthogonal with respect to the inner product (5).
At rst, we assume that = {0 ; 1 ; : : :} is a complex sequence satisfying (4) and de ne a = 1= .
Then we use the Malmquist system (15) with these a . The corresponding numbers a∗ ( = 1= a = )
∗
∗
∗
which appear in (15) are outside the unit circle and they form the Muntz system {xa0 ; xa1 ; : : : ; xan }.
In order to shorten our notation and to be consistent with the previous section, we write a∗ =
( = 0; 1; : : :)
Finally, we can prove Theorem 6:
Proof of Theorem 6. According to (6) and (10), we have
[Qn ; Qm ] =
n X
m
X
An; i Am; j
i=0 j=0
i j − 1
;
where An; k is given by (11).
Using Lemma 9 with t = 1, i.e., equality (22), we conclude that
[Qn ; Qm ] = (Wn ; Wm );
where Wn (s) is determined by (8).
Since a = 1= ( = 0; 1; : : :), (12) follows from Theorem 8.
For the norm of the polynomial Qn (x) we obtain
kQn k =
p
[Qn ; Qn ] =
1
1
·p 2
;
|0 1 · · · n−1 |
|n | − 1
where the complex numbers satisfy the condition (4).
4. Recurrence formulae
Some recurrence formulae for polynomials Qn (x) are given in the following theorem. Similar
relations were obtained in [3] for Muntz–Legendre polynomials.
308
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
Theorem 10. Suppose that is a complex sequence satisfying (4). Then the polynomials Qn (x);
de ned by (9), satisfy the following relations:
′
(x) + n Qn (x) − (1= n−1 )Qn−1 (x);
xQn′ (x) = xQn−1
xQn′ (x) = n Qn (x) +
n−1
X
(k − 1= k )Qk (x);
(23)
(24)
k =0
xQn′′ (x) = (n − 1)Qn′ (x) +
n−1
X
(k − 1= k )Qk′ (x);
(25)
k =0
Qn (1) = 1;
Qn′ (1) = n +
n−1
X
(k − 1= k );
(26)
k =0
Qn (x) = Qn−1 (x) − (n − 1= n−1 )xn
Z
1
t −n −1 Qn−1 (t) dt:
(27)
x
In proving this theorem we use the same method as in [3]. Because of that, we omit the proof.
5. Real zeros of Muntz polynomials
Now we consider the orthogonal real Muntz polynomials when the sequence is real and such
that
1¡0 ¡1 ¡ · · ·
(28)
Theorem 11. Let be a real sequence satisfying (28). Then the polynomial Qn (x); de ned by (9),
has exactly n simple zeros in (0; 1) and no other zeros in [1; ∞).
Proof. Because of orthogonality we have
Z
1
x ⊙ Qn (x)
0
dx
= 0 ( = 0; 1; : : : ; n − 1):
x2
Since x ⊙ Qn (x) = Qn (x ), these orthogonality conditions reduce to
Z
1
Qn (x )
0
dx
= 0 ( = 0; 1; : : : ; n − 1);
x2
or, after changing variable x = t,
Z
1
Qn (t)t − dt = 0 ( = 0; 1; : : : ; n − 1);
(29)
0
where = 1 + 1= . Note that 1¡ ¡2 and are mutually di erent under condition (28).
According to (29) the polynomial Qn (t) has at least one zero in (0; 1) in which Qn (t) changes
the sign. On the other side, it is easy to prove by induction (cf. [11, Theorem. 4.1]) that every real
Muntz polynomial from Mn () has at most n zeros in (0; ∞).
G.V. Milovanovic et al. / Journal of Computational and Applied Mathematics 99 (1998) 299–310
309
In order to prove our theorem, we suppose that the number of sign changes of Qn (t) in (0; 1) is
k¡n. Let
; : : : ; k be the points where Qn (t) changes sign. Now, we can construct a polynomial
P 1
Pk (t) = ki=0 bi t −i , such that Pk ( ) = 0 for each = 1; : : : ; k, and, for example, b0 = 1. We note also
that Pk (t) has no other zeros in (0; ∞). For existence and uniqueness of such a polynomial see [2,
Chap. 3].
As we can see now, Pk (t)Qn (t) has a constant sign in (0; 1), and therefore
Z
1
Pk (t)Qn (t) dt 6= 0:
0
On the other side, using (29), we have
Z
1
Pk (t)Qn (t) dt =
0
Z
0
1
k
X
i=0
bi t
−i
!
Qn (t) dt =
k
X
i=0
bi
Z
1
t −i Qn (t) dt = 0
0
for k¡n, which gives a contradiction. Thus, k = n. The points 1 ; : : : ; n are simple zeros of Qn (t)
in (0; 1) and this polynomial has no other positive zeros.
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