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) 195–203
Piecewise rational approximation to continuous functions
with characteristic singularities
J. Illan 1
Departamento de Matematicas Aplicadas, Vigo University, Spain
Received 30 October 1997; received in revised form 23 April 1998
Abstract
Let Hp [a; b] be the class of continuous functions in the interval [a; b], which admit analytic continuation to H p in
the subintervals of a -subdivition of [a; b]. Let Sn; [a; b] be the set of piecewise rational functions which have pieces
of degree n and no more than breakpoints. The paper deals with the construction of fn; ∈ Sn; [a; b] close to a given
f ∈ Hp [a; b], to estimate the order of the least Lp (w) distance from f to Sn; [a; b]. It is concluded that the piecewise
rational approximation is better for Hp [a; b] than the rational one. This theory extends results of Gonchar (1967) on
the rational approximation to f ∈ H0∞ [−1; 1], and can be applied to study the asymptotic behavior of piecewise rational
methods to solve a singular identi cation problem associated with the equation w(2) + (f + ) w = 0, in terms of the error
c 1998 Elsevier Science B.V. All rights reserved.
equation criterion.
Keywords: Piecewise rational approximation; Best Lp approximation; Singular identi cation problem
1. Introduction and statement of results
Let w ∈ L∞ (dx; [a; b]), w ¿ 0, and k kp be the Lp (w) = Lp (w; [a; b]) norm, 1 6 p 6 ∞. For f
continuous in [a; b], we de ne R n (f; [a; b])p = inf {kf − rkp ; r ∈ Fn; n }, where Fn; m = {p=q; p ∈ n ;
q ∈ m }; and n is the set of all polynomials of degree 6 n. Let f be a function in the Hardy space
H ∞ such that f is continuous in [−1; 1]. Inspired by the major paper [6], Gonchar [3] obtained the
following result:
nc
R n (f; [−1; 1])∞ = O inf !(f; e ; [−1; 1]) + t exp −
t¿1
t
−t
;
(1)
where !(f; ; [a; b]) = sup{|f(x) − f(y)|; |x − y|¡; a 6 x; y 6 b} is the uniform modulus of
continuity of f, and c¿0 is an absolute constant.
1
This research was carried out while the author was a visiting professor at the Departmento de Matematica Aplicada,
Universidad de Vigo, Spain.
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 5 7 - 5
196
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
The estimate (1) has been extended to R n (f; [−1; 1])p for f ∈ H p , 16p6∞ (cf. [5]). In this
context, a way to connect a feature of the measure with the order of convergence of R n (f; [−1; 1])p
has been established
in [5] by the following result. Given a Carleson measure , such that () =
R1
sup{¿0;
d(x)=(1
− x2 ) ¡∞}¡∞, and f ∈ H p ∩ Lip
, then R n (f; [−1; 1])p =
p −1
O(exp(−d n())); d¿0.
As for the uniqueness of the element of the best rational approximation, see [1]. In what follows
the set {xk }k=
∈ N, will be a -subdivision of the
k=1 , such that a = x0 ¡x1 ¡ · · · ¡x ¡x+1 = b;
interval I with end points a; b; a¡b.
In order to introduce the problem of approximating piecewise analytic functions in the interval I ,
we consider rst the class of the continuous functions f which are analytic in every Ik () = [xk−1 +
; xk − ]; k = 1; : : : ; + 1, 0¡¡ min16k6+1 21 (xk − xk−1 ), and
q
n
max lim sup
16k6+1
n
maxx∈Ik () |f(n) (x)|
n
6
M
(cf : [7]):
This class is denoted by D( M ; ; I ). Szabados’ work [7] shows that for f ∈ D( M ; ; [−1; 1])
√
R n (f; [−1; 1])∞ = O(!(f; exp(−c1 3 n); [−1; 1])) (c1 ¿0):
(2)
Next, we de ne two subclasses of D(1=; ; I ), which are concerned with obtaining the approximation
results in the paper.
Let B( ; ) = {z; |z − (( + )=2)| ¡ |( − )=2|}, where and are complex numbers.
Let f be a continuous function in I such that for some -subdivition {x1 ; : : : ; x }, each fk (x)=f(x);
xk−1 ¡x¡xk , has analytic continuation to the Hardy space H p (B(xk−1 ; xk )), k = 1; : : : ; + 1, 1 6 p 6
∞. This class is denoted by Hp I . In particular, H0∞ [−1; 1] and H0p ] − 1; 1[ = H p , are considered in
[3] and [5], respectively. For ¿0 let H; I be the subclass of Hp I of those functions f, for which
fk has analytic continuation to H ∞ (B(xk−1 − ; xk + )), k = 1; : : : ; + 1.
Notice that the singularities of every f ∈ Hp I have the same characteristic of being all situated
in the boundary of some disk where f is analytic (cf. [3, 5, 6]). It is well known that the order of
rational approximation for the class H0; I is O(e−d1 n ); d1 ¿0 (cf. [3, 8]).
Estimates of R n (f; [−1; 1])∞ for some functions f ∈ H1p [−1; 1] are given in [3, 7]. The class of
approximants to be used in the paper, denoted by Sn; [a; b], ¿ 0, is formed by piecewise rational
functions. By de nition Sn; 0 [a; b] = Fn; n , and fn; ∈ Sn; [a; b], ¿ 1, when it is continuous in [a; b],
and there exist points 1 ; : : : ; m ; 1 6 m 6 , a = 0 ¡ 1 ¡ · · · ¡ m ¡ m+1 = b, and m + 1 rational
functions rk ∈ Fn; n , such that for x ∈ [ k−1 ; k ], fn; (x) = rk (x), where {x; rk (x) = ∞} ⊂ C\[ k−1 ; k ],
k = 1; : : : ; m + 1.
Let R n; (f)p = inf {kf−fn; kp ; fn; ∈ Sn; [a; b]}, 1 6 p 6 ∞. The purpose of this paper is to prove
estimates of R n; (f)p (Theorems 1–3), for f ∈ H , where H is one of the above de ned classes.
Theorem 1. Let f ∈ Hp [a; b]; 1 6 p 6 ∞. Then
R n; (f)p = O inf !(f; e−t ; [a; b]) + t exp −
t¿1
nc
t
+
t
p
;
(3)
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
197
where c is a positive absolute constant; and !(f; ; [a; b]) is the uniform modulus of continuity
of f.
Theorem 2. Let f be a function in H; [a; b]; with internal singularities {x1 ; : : : ; x }. Then there
exists d = d(; {x1 ; : : : ; x })¿0 such that
R n; (f)∞ = O(e−dn ):
(4)
The estimate (3), applied to f ∈ Hp [a; b] ∩ Lip , shows an order of convergence somewhat greater
than that given in (2) (see Corollary 1, Section 2). Likewise, Theorem 2 asserts that the piecewise
rational approximation for f ∈ H; [a; b] is generally better than the rational one. Thus, the corresponding class of internal characteristic singularities should be strongly associated with the expected
breakpoints attraction (see Proposition 1, Section 3).
The estimates (3)–(4) can be used to study the behavior of the rational method used in [4] when
the number of parameters is large. An outline of [4] is given in Section 3.
The uniform modulus of continuity !(f; ; ]a; b[) does not tend to zero as → 0 for every
f ∈ Hp ]a; b[. For this case we prove the following:
Theorem 3. Let f be a function in Hp ]a; b[; 1 6 p 6 ∞; with internal singularities {x1 ; : : : ; x }.
Let dk (x) = x lk + ck ; where ck = 21 (xk−1 + xk ); lk = 12 (xk − xk−1 ); k = 1; : : : ; + 1. Then
R n; (f)p = O inf
t¿1
max !(f; xk ; e−t ; [ck ; ck+1 ])
16k6
t
nc
+ max !p; k (f; e ) + t exp − +
16k6+1
t
p
−t
;
where !(f; x; ; I ) = supy∈I ; |x−y|¡ |f(x) − f(y)|; and
!p; k (f; ) = sup
0¡h¡
Z
1
−1
p
|f ◦ dk (x) − f ◦ dk ((1 − h)x)| w ◦ dk (x) dx
!1=p
:
(5)
The paper is organized as follows.
Some details of Gonchar’s technique are used in Section 2, to construct a rational piece on each
subinterval [xk−1 ; xk ], close to a given f ∈ Hp I . To prove Theorems 1–3, these pieces are transformed
according to a suitable equation system so the new ones turn into a continuous approximant of f.
This section also contains a corollary of Theorem 1.
Finally, we divide Section 3 into three short paragraphs in order to present some connected
problems in the theory of piecewise rational approximation.
2. Construction of fn; close to f
In this section dk , lk , and ck ; k = 1; : : : ; + 1; are the same as those in Theorem 3.
198
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Proof of Theorem 1. In order to simplify the notation, henceforth M (f; n; p; t; I ) = !(f; e−t ; I ) +
t exp(−(nc=t) + (t=p)), where c¿0 is the constant given by (1).
From [5] we can assure that there exists a rational function n; t; k ∈ Fn−1; n , such that
Z
1
−t
−1
p
|f ◦ dk (x) − n; t; k ((1 − e )x)| w ◦ dk (x) dx
6 M0 !p; k (f; e−t ) + t exp −
t
nc
+
t
p
!1=p
(6)
;
where t ¿ 1, M0 ¿0 only depends on kf ◦ dk kH p . Besides, !p; k (f; ) (see (5)) satis es
1=p
!p; k (f; ) 6 kwk1=p
([lk ] + 1)!(f; ; [xk−1 ; xk ]):
∞ 2
(7)
The formulation in (5) for !p; k (f; ), corresponds to 1 6 p¡∞. For p = ∞, the integral norm is
replaced by kgk∞ = sup ess |g(x) w(x)|.
Via calculations, we can obtain from [5] that n; t; k has n poles pn; j; t
e−t
1−
2
¡pn; j; t =
e−t
1−
2
!
Mn; j; t + 1
;
Mn; j; t − 1
and f ◦ dk (xn; j; t ) = n; t; k (xn; j; t ), where
0¡xn; j; t =
e−t
1−
2
!
Mn; j; t − 1
e−t
¡ 1−
Mn; j; t + 1
2
;
and Mn; j; t = (4 − 3e−t ) exp(t(1 − (j=n))); j = 1; : : : ; n; t ¿ 1:
Let
gn; t (z) =
n; t
n
Y
j=1
z − xn; j; t
z − pn; j; t
!
;
where
=
n; t
n
Y
j=1
1 + Mn; j; t
1 − Mn; j; t
!
:
The following property, which allows to obtain (6), is satis ed by gn; t (cf. [3, 5]).
|gn; t (x)| 6 M1
nc
exp −
t
(|x| 6 1 − e−t ):
Now let n; t; k (x) = n; t; k ((1 − e−t )d−1
k (x)), x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1. We have that n; t ; k ∈ Fn−1; n ,
it has poles at pn; j; t; k = dk (pn; j; t =(1 − e−t )), and satis es n; t ; k (xn; j; t; k ) = f(xn; j; t; k ), where xn; j; t; k =
dk (xn; j; t =(1 − e−t )), j = 1; : : : ; n.
The following upper bound is derived from (6) and (7).
Z
xk
xk−1
|f(x) − n; t ; k (x)|p w(x) dx
!1=p
= O(M (f; n; p; t; [xk−1 ; xk ])):
(8)
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
199
Next, we give an equation system to construct a continuous approximant fn; . Let
an; k xkn + · · · + a0; k an; k+1 xkn + · · · + a0; k+1
;
=
qt; k (xk )
qt; k+1 (xk )
k = 1; : : : ; ;
an; k xn;n j; t; k + · · · + a0; k = qt; k (xn; j; t; k )f(xn; j; t; k );
j = 1; : : : ; n; k = 1; : : : ; + 1;
(9)
(10)
where qt; k (x) = nj=1 (x − pn; j; t; k ); t ¿ 1. The system (9)–(10), which is linear with respect to the
unknowns aj; k , has system matrix with rank ( + 1)(n + 1) − 1, and condition number (based on the
maximum norm) cn; ; t ¿ M2 e t , M2 = M2 (n; ; a; x1 ; : : : ; x ; b)¿0, = (n; )¿0 (v.g. (2; 2) = 0:5,
M2 (2; 2; 0; 0:01; 1; 2) ≈ 103 ). Fixing an; 1 = n; t we obtain a solution {aj; k }, whose corresponding
fn; ; t ∈ Sn; [a; b], has + 1 pieces rn; t ; k ∈ Fn; n . That is
Q
fn; ; t (x) = rn; t ; k (x) =
an; k xn + · · · + a0; k
;
qt; k (x)
x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1:
For z = (1 − e−t ) d−1
k (x), x ∈ [xk−1 ; xk ] Eqs. (10) imply that
rn; t ; k (x) − n; t ; k (x) = an; k
n
Y
z − xn; j; t
z − pn; j; t
j=1
!
(11)
:
On the other hand, for k = 1; : : : ; ; t ¿ 1, the equalities given below can be obtained from (9) and
(11) (cf. [5]).
−t
−t
−t
f ◦ dk (1 − e ) − gn; t (1 − e )In; t ; k (1 − e ) + an; k
−t
j=1
1 − e−t − xn; j; t
1 − e−t − pn; j; t
!
= f ◦ dk+1 (−(1 − e )) − gn; t (−(1 − e ))In; t ; k+1 (−(1 − e−t ))
+an; k+1
n
Y
j=1
−t
n
Y
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
;
k = 1; : : : ; + 1; t ¿ 1;
(12)
where
1
In; t; ; k (x) =
2i
Z
Kt
f ◦ dk (s)
ds;
gn; t (s)(s − x)
with Kt = {s; |s| = (1 − 21 e−t )}.
The following inequality (cf. [2]) produces the term et=p in (6) and (14).
1
t
+1
|In; t ; k (x)| 6 2 p kf ◦ dk kH p e p ;
k = 1; : : : ; + 1; |x| 6 1 − e−t :
(13)
From (12) and (13) we obtain
n
Y
an; k
j=1
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
6 M3 M (f; n; p; t; [a; b]);
where k = 1; : : : ; + 1; t ¿ 1, and M3 ¿0 does not depend on n and t.
(14)
200
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Notice that the relation xn; j; t pn; j; t = (1 − 21 e−t 2)2 ; j = 1; : : : ; n; t ¿ 1, implies that
Y
n
j=1
z − xn; j; t
z − pn; j; t
!
n
Y
6
j=1
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
(15)
;
where z ∈ R; |z| 6 1 − e−t .
There exists M4 = M4 (f)¿0 such that (11), (14) and (15) prove the following:
xk
Z
xk−1
p
|rn; t ; k − n; t ; k | w dx
!1=p
6 M4 M (f; n; p; t; [a; b]);
(16)
where t ¿ 1; k = 1; : : : ; + 1:
Using well known properties of the Lebesgue integral, (8), (16) and the inequality ( + )p 6 2p−1
p
( + p ); ; ¿0; p ¿ 1; we can conclude that
R n; (f)p 6
Z
a
b
p
|fn; ; t − f| w dx
!1=p
= O(M (f; n; p; t; [a; b])):
(17)
The term R n; (f)p does not depend on t, therefore (17) yields (3).
Corollary 1. Let f be a function such that f ∈ Hp [a; b]; and !(f; ; [a; b]) = O( ), 0¡ 6 1.
There exists a constant c0 ¿0 such that
√
R n; (f)p = O(exp(−c0 n)):
(18)
√
Proof. Let t =
n. From (17) and a convenient selection of
¿0 we prove (18).
Proof of Theorem 2. Given a function f ∈ H; [a; b], which has internal singularities x1 ; : : : ; x , we
also apply here the method used in the proof of Theorem 1 to construct rational functions rn; k; ∈ Fn; n ,
each one close to f in the respective interval [xk−1 − ; xk + ].
For = min16k6+1 {lk }=(e − 1), let 1 be de ned as 1 = if ¿ , and 1 = otherwise. Thus
tk; = log(lk; =1 ) ¿ 1, where lk; = lk + 1 ; k = 1; : : : ; + 1.
Let dk; (z) = z lk; + ck , |z|¡1, k = 1; : : : ; + 1.
Observe that, for k = 1; : : : ; ,
dk; 1 − e−tk; = xk ;
dk+1; − 1 − e−tk+1;
For k = 1; : : : ; + 1, we obtain from [5] that
Z
= xk :
1
−1
|f ◦ dk; ((1 − e
′
6 M tk;
−tk;
)x) − n; t ; k ((1 − e
nc
:
exp −
tk;
−tk;
p
)x)| w ◦ dk; (x) dx
(19)
!1=p
(20)
Now we consider the system (9)–(10) with the parameters xn; j; t; k = dk; (xn; j; tk; =(1 − exp(−tk; ))),
pn; j; t; k = dk; (pn; j; tk; (1 − exp(−tk; ))), and the denominators qk; = qt; k , with t = tk; .
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
201
Let aj; k ; j = 1; : : : ; n; k = 1; : : : ; + 1, be the solution of the modi ed system (9)–(10) with
an; 1 = n; t , t = tk; ; and let fn; ; be the corresponding piecewise approximant. Equalities (19) produce
that both terms in (12) have the same summand f(xk ), so they are canceled. Hence, we have for
k = 1; : : : ; + 1
n
Y
an; k
j=1
1 − e−tk; + xn; j; tk;
1 − e−tk; + xn; j; tk;
!
= O exp − c n
:
tk;
(21)
Then, for x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1, (20) and (21) give
|fn; ; (x) − f(x)| w(x) 6 |fn; ; (x) − n; t ; k (x)| w(x)
+ |n; ; k (x) − f(x)| w(x) 6 M5 exp −
c
tk;
n :
From the above inequality we have R n; (f)∞ 6 M5 exp(−dn), where d = c=t and t = max{tk; ;
k = 1; : : : ; + 1}. Theorem 2 is proved.
Remark 1. From the proof of Theorem 2; it can be easily deduced that d 6 c, where c is the
constant given by Theorem 1: The equality d = c takes place if ¿(b − a)=[2( + 1)(e − 1)]; and
f has equidistant singularities. Small values of one of the parameters or = (e − 1) make d
small as well.
Proof of Theorem 3. We shall make a slight modi cation to the proof of Theorem 1. The inequality
(7) is not used here because of the unknown behavior of f ∈ Hp ]a; b[ at the ends of the interval.
Hence, the integral modulus of continuity (5) must remain in what follows. The rest of the proof
is the same except (14) which has to be improved according to the following inequalities for
k = 1; : : : ; .
|f ◦ dk+1 (−(1 − e−t )) − f ◦ dk ((1 − e−t ))| 6 M4 !(f; xk ; e−t ; [ck ; ck+1 ]):
3. Connected problems
3.1. The discrete approximation method.
Let kfkY = supy∈Y |f(y)w(y)|, where ∅ 6= Y ⊂ X = [a; b], and w ∈ C[a; b], w 6= 0; a:e:
The existence of a best approximant fY ∈ Sn; [a; b] of f ∈ C[a; b] with respect to the seminorm
k:kY , is guaranteed by [4], Proposition 2.2, by adding a parameter 1 to the de nition of Sn; [a; b].
k−1 + 1 6 k ;
k = 1; : : : ; + 1;
pj; k 6 k−1 − 1 or k + 1 6 pj; k ; j = 1; : : : ; n; k = 1; : : : ; + 1; ¿ 0
(22)
(23)
where k , k = 1; : : : ; ; are the breakpoints of fn; ; pj; k , j = 1; : : : ; n; are the poles of the k th rational
piece of fn; , k = 1; : : : ; + 1, and 1 ¿0 is suciently small.
Theorem 2.1 [4] assures that lim|Y |→0 kf − fY kX = kf − fX kX , where |Y | = supx ∈ X d(x; Y ) and
fY ∈ Sn; ; 1 [a; b] (the subindex 1 denotes that all the approximants fn; are subject to (22)–(23)).
202
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Theorem 1 is also valid considering the optimal approximation from Sn; ; n [a; b], for some sequences (n ), lim n = 0. For Theorem 2 it is sucient to x a small 1 ¿0. The selection of a
numerical procedure to obtain an approximant fn; of f depends on what information we have on f.
Therefore any method must not be based upon the linear system (9), (10) whose associated system
matrix is ill-conditioned. In fact, every solution of (9), (10), is not necessarily optimal at all. A best
approximation fY to a sample {f(y); y ∈ Y }, is directly calculated as a solution of the nonlinear
optimization problem minfn; kfn; − fkY , subject to fn; ∈ Sn; ; 1 [a; b]. In this respect some advantages
can be gotten using a modi cation of (9), (10) to nd feasible initial data, possibly close to a global
minimal point.
3.2. An identi cation problem
We applied a discrete version of the above piecewise rational method to solve the following
problem (cf. [4]). Let w(2) + f w + F = 0 be a linear di erential equation of second order, with
boundary values w(a) = wa ; w(b) = wb . The values w(k ), w(2) (k ), F(k ), k = 1; : : : ; m; are given as
data, and f is unknown.
The above problem is a singular identi cation problem when we assume that the real state of
nature f is analytic in [a; b]\{a; x1 ; : : : ; x ; b}. In principle, we say that a solution has been obtained
when especi c information on f has been found, especially that related to singularities location.
The expected behavior of poles when optimal approximants are considered, has been taken into
account in selecting a rational method. On the other hand, a piecewise scheme seems to be convenient
in dealing with real functions whose singularities under interest are all located in the interval [a; b].
The equality |w(f−fn; )| = |w(2) +wfn; +F|, being w the solution of the equation w(2) +wf+F = 0,
is the link which connects [4] with this paper.
3.3. Asymptotical behavior of the optimal breakpoints
Having an optimal solution of this approximation problem, a new question arises on whether
the breakpoints of fY = fn; ; 1 tend to the singularities of f as n → ∞. An interesting example is
given by the function f(x) = |x|; |x| 6 1, when f is to be approximated by a best approximant
fYn = fn; 1; 1 ∈ Sn; 1; 1 [−1; 1], and |Yn | tend to zero as n → ∞. Let n; 1 be the breakpoint of fYn , and
0¡1 ¡ 31 . The following proposition takes place
Proposition 1. The sequence ( n; 1 ) converges to zero provided that 0¡¡w(x); x ∈ [−; ]; for
some ; 0¡¡1.
Proof. The proof is √
based upon the fact that the nth best rational approximation to |x|, x ∈ [−; ],
has order O(exp(− n)), for every , 0¡ 6 1, and |x| ∈ H1; [−1; 1], for every ¿0.
Let ( n; 1 )n ∈ J be a convergent subsequence with limit x1 6= 0. Thus, for 0¡¡ 12 |x1 | and n large,
n; 1 ∈= [−; ], n ∈ J .
From Theorem 2 we have R n (|x|; [−; ])∞ 6 kfYn − |x|k∞ = O(exp(−cn)), n ∈ J , which is
impossible. Hence x1 = 0.
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
203
Many numerical evidences show that it is dicult to locate a singularity in a region where the
corresponding weight function value is zero or too small. Indeed, if w() ≈ 0, the point should
be considered in advance as a singularity of f.
References
[1] L. Baratchart, F. Wielonsky, Rational approximation in the real Hardy space H2 and Stieltjes integrals: a uniqueness
theorem, Constr. Approx. 9 (1993) 1–21.
[2] P.L. Duren, Theory of H p Spaces, Academic Press, New York, 1970.
[3] A.A. Gonchar, On the rate of rational approximation to continuous functions with characteristic singularities, Mat.
Sbornik T. 73, 115, (4) (1967) 630–638.
[4] F. Guerra, J. Illan, I. Romero, Rational methods to solve an identi cation problem of a non-homogeneous second
order linear equation with singularities, in: (Eds.), M. Florenzano et al., Proc. Internat. Conf. on Approximation and
Optimization in the Caribbean, Peter Lang Series in Approximation and Optimization, vol. 8, 1995, pp. 367–378.
[5] J. Illan, On the rational approximation of H p functions in the Lp () metric, Lectures Notes in Mathematics,
vol. 1354, Springer, Berlin, 1987, pp. 155–163.
[6] D.J. Newman, Rational approximation to |x|, Michigan Math. J. 11 MR 30 (1344) (1964) 11–14.
[7] J. Szabados, Rational approximation in certain classes of functions, Acta Math. Acad. Scientiarum Hungaricae Tomus
19 (1–2) (1968) 81–85.
[8] H. Widom, Rational approximation and n-dimensional diameter, J. Approx. Theory 5 (1972) 343–361.
Piecewise rational approximation to continuous functions
with characteristic singularities
J. Illan 1
Departamento de Matematicas Aplicadas, Vigo University, Spain
Received 30 October 1997; received in revised form 23 April 1998
Abstract
Let Hp [a; b] be the class of continuous functions in the interval [a; b], which admit analytic continuation to H p in
the subintervals of a -subdivition of [a; b]. Let Sn; [a; b] be the set of piecewise rational functions which have pieces
of degree n and no more than breakpoints. The paper deals with the construction of fn; ∈ Sn; [a; b] close to a given
f ∈ Hp [a; b], to estimate the order of the least Lp (w) distance from f to Sn; [a; b]. It is concluded that the piecewise
rational approximation is better for Hp [a; b] than the rational one. This theory extends results of Gonchar (1967) on
the rational approximation to f ∈ H0∞ [−1; 1], and can be applied to study the asymptotic behavior of piecewise rational
methods to solve a singular identi cation problem associated with the equation w(2) + (f + ) w = 0, in terms of the error
c 1998 Elsevier Science B.V. All rights reserved.
equation criterion.
Keywords: Piecewise rational approximation; Best Lp approximation; Singular identi cation problem
1. Introduction and statement of results
Let w ∈ L∞ (dx; [a; b]), w ¿ 0, and k kp be the Lp (w) = Lp (w; [a; b]) norm, 1 6 p 6 ∞. For f
continuous in [a; b], we de ne R n (f; [a; b])p = inf {kf − rkp ; r ∈ Fn; n }, where Fn; m = {p=q; p ∈ n ;
q ∈ m }; and n is the set of all polynomials of degree 6 n. Let f be a function in the Hardy space
H ∞ such that f is continuous in [−1; 1]. Inspired by the major paper [6], Gonchar [3] obtained the
following result:
nc
R n (f; [−1; 1])∞ = O inf !(f; e ; [−1; 1]) + t exp −
t¿1
t
−t
;
(1)
where !(f; ; [a; b]) = sup{|f(x) − f(y)|; |x − y|¡; a 6 x; y 6 b} is the uniform modulus of
continuity of f, and c¿0 is an absolute constant.
1
This research was carried out while the author was a visiting professor at the Departmento de Matematica Aplicada,
Universidad de Vigo, Spain.
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 5 7 - 5
196
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
The estimate (1) has been extended to R n (f; [−1; 1])p for f ∈ H p , 16p6∞ (cf. [5]). In this
context, a way to connect a feature of the measure with the order of convergence of R n (f; [−1; 1])p
has been established
in [5] by the following result. Given a Carleson measure , such that () =
R1
sup{¿0;
d(x)=(1
− x2 ) ¡∞}¡∞, and f ∈ H p ∩ Lip
, then R n (f; [−1; 1])p =
p −1
O(exp(−d n())); d¿0.
As for the uniqueness of the element of the best rational approximation, see [1]. In what follows
the set {xk }k=
∈ N, will be a -subdivision of the
k=1 , such that a = x0 ¡x1 ¡ · · · ¡x ¡x+1 = b;
interval I with end points a; b; a¡b.
In order to introduce the problem of approximating piecewise analytic functions in the interval I ,
we consider rst the class of the continuous functions f which are analytic in every Ik () = [xk−1 +
; xk − ]; k = 1; : : : ; + 1, 0¡¡ min16k6+1 21 (xk − xk−1 ), and
q
n
max lim sup
16k6+1
n
maxx∈Ik () |f(n) (x)|
n
6
M
(cf : [7]):
This class is denoted by D( M ; ; I ). Szabados’ work [7] shows that for f ∈ D( M ; ; [−1; 1])
√
R n (f; [−1; 1])∞ = O(!(f; exp(−c1 3 n); [−1; 1])) (c1 ¿0):
(2)
Next, we de ne two subclasses of D(1=; ; I ), which are concerned with obtaining the approximation
results in the paper.
Let B( ; ) = {z; |z − (( + )=2)| ¡ |( − )=2|}, where and are complex numbers.
Let f be a continuous function in I such that for some -subdivition {x1 ; : : : ; x }, each fk (x)=f(x);
xk−1 ¡x¡xk , has analytic continuation to the Hardy space H p (B(xk−1 ; xk )), k = 1; : : : ; + 1, 1 6 p 6
∞. This class is denoted by Hp I . In particular, H0∞ [−1; 1] and H0p ] − 1; 1[ = H p , are considered in
[3] and [5], respectively. For ¿0 let H; I be the subclass of Hp I of those functions f, for which
fk has analytic continuation to H ∞ (B(xk−1 − ; xk + )), k = 1; : : : ; + 1.
Notice that the singularities of every f ∈ Hp I have the same characteristic of being all situated
in the boundary of some disk where f is analytic (cf. [3, 5, 6]). It is well known that the order of
rational approximation for the class H0; I is O(e−d1 n ); d1 ¿0 (cf. [3, 8]).
Estimates of R n (f; [−1; 1])∞ for some functions f ∈ H1p [−1; 1] are given in [3, 7]. The class of
approximants to be used in the paper, denoted by Sn; [a; b], ¿ 0, is formed by piecewise rational
functions. By de nition Sn; 0 [a; b] = Fn; n , and fn; ∈ Sn; [a; b], ¿ 1, when it is continuous in [a; b],
and there exist points 1 ; : : : ; m ; 1 6 m 6 , a = 0 ¡ 1 ¡ · · · ¡ m ¡ m+1 = b, and m + 1 rational
functions rk ∈ Fn; n , such that for x ∈ [ k−1 ; k ], fn; (x) = rk (x), where {x; rk (x) = ∞} ⊂ C\[ k−1 ; k ],
k = 1; : : : ; m + 1.
Let R n; (f)p = inf {kf−fn; kp ; fn; ∈ Sn; [a; b]}, 1 6 p 6 ∞. The purpose of this paper is to prove
estimates of R n; (f)p (Theorems 1–3), for f ∈ H , where H is one of the above de ned classes.
Theorem 1. Let f ∈ Hp [a; b]; 1 6 p 6 ∞. Then
R n; (f)p = O inf !(f; e−t ; [a; b]) + t exp −
t¿1
nc
t
+
t
p
;
(3)
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
197
where c is a positive absolute constant; and !(f; ; [a; b]) is the uniform modulus of continuity
of f.
Theorem 2. Let f be a function in H; [a; b]; with internal singularities {x1 ; : : : ; x }. Then there
exists d = d(; {x1 ; : : : ; x })¿0 such that
R n; (f)∞ = O(e−dn ):
(4)
The estimate (3), applied to f ∈ Hp [a; b] ∩ Lip , shows an order of convergence somewhat greater
than that given in (2) (see Corollary 1, Section 2). Likewise, Theorem 2 asserts that the piecewise
rational approximation for f ∈ H; [a; b] is generally better than the rational one. Thus, the corresponding class of internal characteristic singularities should be strongly associated with the expected
breakpoints attraction (see Proposition 1, Section 3).
The estimates (3)–(4) can be used to study the behavior of the rational method used in [4] when
the number of parameters is large. An outline of [4] is given in Section 3.
The uniform modulus of continuity !(f; ; ]a; b[) does not tend to zero as → 0 for every
f ∈ Hp ]a; b[. For this case we prove the following:
Theorem 3. Let f be a function in Hp ]a; b[; 1 6 p 6 ∞; with internal singularities {x1 ; : : : ; x }.
Let dk (x) = x lk + ck ; where ck = 21 (xk−1 + xk ); lk = 12 (xk − xk−1 ); k = 1; : : : ; + 1. Then
R n; (f)p = O inf
t¿1
max !(f; xk ; e−t ; [ck ; ck+1 ])
16k6
t
nc
+ max !p; k (f; e ) + t exp − +
16k6+1
t
p
−t
;
where !(f; x; ; I ) = supy∈I ; |x−y|¡ |f(x) − f(y)|; and
!p; k (f; ) = sup
0¡h¡
Z
1
−1
p
|f ◦ dk (x) − f ◦ dk ((1 − h)x)| w ◦ dk (x) dx
!1=p
:
(5)
The paper is organized as follows.
Some details of Gonchar’s technique are used in Section 2, to construct a rational piece on each
subinterval [xk−1 ; xk ], close to a given f ∈ Hp I . To prove Theorems 1–3, these pieces are transformed
according to a suitable equation system so the new ones turn into a continuous approximant of f.
This section also contains a corollary of Theorem 1.
Finally, we divide Section 3 into three short paragraphs in order to present some connected
problems in the theory of piecewise rational approximation.
2. Construction of fn; close to f
In this section dk , lk , and ck ; k = 1; : : : ; + 1; are the same as those in Theorem 3.
198
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Proof of Theorem 1. In order to simplify the notation, henceforth M (f; n; p; t; I ) = !(f; e−t ; I ) +
t exp(−(nc=t) + (t=p)), where c¿0 is the constant given by (1).
From [5] we can assure that there exists a rational function n; t; k ∈ Fn−1; n , such that
Z
1
−t
−1
p
|f ◦ dk (x) − n; t; k ((1 − e )x)| w ◦ dk (x) dx
6 M0 !p; k (f; e−t ) + t exp −
t
nc
+
t
p
!1=p
(6)
;
where t ¿ 1, M0 ¿0 only depends on kf ◦ dk kH p . Besides, !p; k (f; ) (see (5)) satis es
1=p
!p; k (f; ) 6 kwk1=p
([lk ] + 1)!(f; ; [xk−1 ; xk ]):
∞ 2
(7)
The formulation in (5) for !p; k (f; ), corresponds to 1 6 p¡∞. For p = ∞, the integral norm is
replaced by kgk∞ = sup ess |g(x) w(x)|.
Via calculations, we can obtain from [5] that n; t; k has n poles pn; j; t
e−t
1−
2
¡pn; j; t =
e−t
1−
2
!
Mn; j; t + 1
;
Mn; j; t − 1
and f ◦ dk (xn; j; t ) = n; t; k (xn; j; t ), where
0¡xn; j; t =
e−t
1−
2
!
Mn; j; t − 1
e−t
¡ 1−
Mn; j; t + 1
2
;
and Mn; j; t = (4 − 3e−t ) exp(t(1 − (j=n))); j = 1; : : : ; n; t ¿ 1:
Let
gn; t (z) =
n; t
n
Y
j=1
z − xn; j; t
z − pn; j; t
!
;
where
=
n; t
n
Y
j=1
1 + Mn; j; t
1 − Mn; j; t
!
:
The following property, which allows to obtain (6), is satis ed by gn; t (cf. [3, 5]).
|gn; t (x)| 6 M1
nc
exp −
t
(|x| 6 1 − e−t ):
Now let n; t; k (x) = n; t; k ((1 − e−t )d−1
k (x)), x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1. We have that n; t ; k ∈ Fn−1; n ,
it has poles at pn; j; t; k = dk (pn; j; t =(1 − e−t )), and satis es n; t ; k (xn; j; t; k ) = f(xn; j; t; k ), where xn; j; t; k =
dk (xn; j; t =(1 − e−t )), j = 1; : : : ; n.
The following upper bound is derived from (6) and (7).
Z
xk
xk−1
|f(x) − n; t ; k (x)|p w(x) dx
!1=p
= O(M (f; n; p; t; [xk−1 ; xk ])):
(8)
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
199
Next, we give an equation system to construct a continuous approximant fn; . Let
an; k xkn + · · · + a0; k an; k+1 xkn + · · · + a0; k+1
;
=
qt; k (xk )
qt; k+1 (xk )
k = 1; : : : ; ;
an; k xn;n j; t; k + · · · + a0; k = qt; k (xn; j; t; k )f(xn; j; t; k );
j = 1; : : : ; n; k = 1; : : : ; + 1;
(9)
(10)
where qt; k (x) = nj=1 (x − pn; j; t; k ); t ¿ 1. The system (9)–(10), which is linear with respect to the
unknowns aj; k , has system matrix with rank ( + 1)(n + 1) − 1, and condition number (based on the
maximum norm) cn; ; t ¿ M2 e t , M2 = M2 (n; ; a; x1 ; : : : ; x ; b)¿0, = (n; )¿0 (v.g. (2; 2) = 0:5,
M2 (2; 2; 0; 0:01; 1; 2) ≈ 103 ). Fixing an; 1 = n; t we obtain a solution {aj; k }, whose corresponding
fn; ; t ∈ Sn; [a; b], has + 1 pieces rn; t ; k ∈ Fn; n . That is
Q
fn; ; t (x) = rn; t ; k (x) =
an; k xn + · · · + a0; k
;
qt; k (x)
x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1:
For z = (1 − e−t ) d−1
k (x), x ∈ [xk−1 ; xk ] Eqs. (10) imply that
rn; t ; k (x) − n; t ; k (x) = an; k
n
Y
z − xn; j; t
z − pn; j; t
j=1
!
(11)
:
On the other hand, for k = 1; : : : ; ; t ¿ 1, the equalities given below can be obtained from (9) and
(11) (cf. [5]).
−t
−t
−t
f ◦ dk (1 − e ) − gn; t (1 − e )In; t ; k (1 − e ) + an; k
−t
j=1
1 − e−t − xn; j; t
1 − e−t − pn; j; t
!
= f ◦ dk+1 (−(1 − e )) − gn; t (−(1 − e ))In; t ; k+1 (−(1 − e−t ))
+an; k+1
n
Y
j=1
−t
n
Y
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
;
k = 1; : : : ; + 1; t ¿ 1;
(12)
where
1
In; t; ; k (x) =
2i
Z
Kt
f ◦ dk (s)
ds;
gn; t (s)(s − x)
with Kt = {s; |s| = (1 − 21 e−t )}.
The following inequality (cf. [2]) produces the term et=p in (6) and (14).
1
t
+1
|In; t ; k (x)| 6 2 p kf ◦ dk kH p e p ;
k = 1; : : : ; + 1; |x| 6 1 − e−t :
(13)
From (12) and (13) we obtain
n
Y
an; k
j=1
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
6 M3 M (f; n; p; t; [a; b]);
where k = 1; : : : ; + 1; t ¿ 1, and M3 ¿0 does not depend on n and t.
(14)
200
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Notice that the relation xn; j; t pn; j; t = (1 − 21 e−t 2)2 ; j = 1; : : : ; n; t ¿ 1, implies that
Y
n
j=1
z − xn; j; t
z − pn; j; t
!
n
Y
6
j=1
1 − e−t + xn; j; t
1 − e−t + pn; j; t
!
(15)
;
where z ∈ R; |z| 6 1 − e−t .
There exists M4 = M4 (f)¿0 such that (11), (14) and (15) prove the following:
xk
Z
xk−1
p
|rn; t ; k − n; t ; k | w dx
!1=p
6 M4 M (f; n; p; t; [a; b]);
(16)
where t ¿ 1; k = 1; : : : ; + 1:
Using well known properties of the Lebesgue integral, (8), (16) and the inequality ( + )p 6 2p−1
p
( + p ); ; ¿0; p ¿ 1; we can conclude that
R n; (f)p 6
Z
a
b
p
|fn; ; t − f| w dx
!1=p
= O(M (f; n; p; t; [a; b])):
(17)
The term R n; (f)p does not depend on t, therefore (17) yields (3).
Corollary 1. Let f be a function such that f ∈ Hp [a; b]; and !(f; ; [a; b]) = O( ), 0¡ 6 1.
There exists a constant c0 ¿0 such that
√
R n; (f)p = O(exp(−c0 n)):
(18)
√
Proof. Let t =
n. From (17) and a convenient selection of
¿0 we prove (18).
Proof of Theorem 2. Given a function f ∈ H; [a; b], which has internal singularities x1 ; : : : ; x , we
also apply here the method used in the proof of Theorem 1 to construct rational functions rn; k; ∈ Fn; n ,
each one close to f in the respective interval [xk−1 − ; xk + ].
For = min16k6+1 {lk }=(e − 1), let 1 be de ned as 1 = if ¿ , and 1 = otherwise. Thus
tk; = log(lk; =1 ) ¿ 1, where lk; = lk + 1 ; k = 1; : : : ; + 1.
Let dk; (z) = z lk; + ck , |z|¡1, k = 1; : : : ; + 1.
Observe that, for k = 1; : : : ; ,
dk; 1 − e−tk; = xk ;
dk+1; − 1 − e−tk+1;
For k = 1; : : : ; + 1, we obtain from [5] that
Z
= xk :
1
−1
|f ◦ dk; ((1 − e
′
6 M tk;
−tk;
)x) − n; t ; k ((1 − e
nc
:
exp −
tk;
−tk;
p
)x)| w ◦ dk; (x) dx
(19)
!1=p
(20)
Now we consider the system (9)–(10) with the parameters xn; j; t; k = dk; (xn; j; tk; =(1 − exp(−tk; ))),
pn; j; t; k = dk; (pn; j; tk; (1 − exp(−tk; ))), and the denominators qk; = qt; k , with t = tk; .
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
201
Let aj; k ; j = 1; : : : ; n; k = 1; : : : ; + 1, be the solution of the modi ed system (9)–(10) with
an; 1 = n; t , t = tk; ; and let fn; ; be the corresponding piecewise approximant. Equalities (19) produce
that both terms in (12) have the same summand f(xk ), so they are canceled. Hence, we have for
k = 1; : : : ; + 1
n
Y
an; k
j=1
1 − e−tk; + xn; j; tk;
1 − e−tk; + xn; j; tk;
!
= O exp − c n
:
tk;
(21)
Then, for x ∈ [xk−1 ; xk ]; k = 1; : : : ; + 1, (20) and (21) give
|fn; ; (x) − f(x)| w(x) 6 |fn; ; (x) − n; t ; k (x)| w(x)
+ |n; ; k (x) − f(x)| w(x) 6 M5 exp −
c
tk;
n :
From the above inequality we have R n; (f)∞ 6 M5 exp(−dn), where d = c=t and t = max{tk; ;
k = 1; : : : ; + 1}. Theorem 2 is proved.
Remark 1. From the proof of Theorem 2; it can be easily deduced that d 6 c, where c is the
constant given by Theorem 1: The equality d = c takes place if ¿(b − a)=[2( + 1)(e − 1)]; and
f has equidistant singularities. Small values of one of the parameters or = (e − 1) make d
small as well.
Proof of Theorem 3. We shall make a slight modi cation to the proof of Theorem 1. The inequality
(7) is not used here because of the unknown behavior of f ∈ Hp ]a; b[ at the ends of the interval.
Hence, the integral modulus of continuity (5) must remain in what follows. The rest of the proof
is the same except (14) which has to be improved according to the following inequalities for
k = 1; : : : ; .
|f ◦ dk+1 (−(1 − e−t )) − f ◦ dk ((1 − e−t ))| 6 M4 !(f; xk ; e−t ; [ck ; ck+1 ]):
3. Connected problems
3.1. The discrete approximation method.
Let kfkY = supy∈Y |f(y)w(y)|, where ∅ 6= Y ⊂ X = [a; b], and w ∈ C[a; b], w 6= 0; a:e:
The existence of a best approximant fY ∈ Sn; [a; b] of f ∈ C[a; b] with respect to the seminorm
k:kY , is guaranteed by [4], Proposition 2.2, by adding a parameter 1 to the de nition of Sn; [a; b].
k−1 + 1 6 k ;
k = 1; : : : ; + 1;
pj; k 6 k−1 − 1 or k + 1 6 pj; k ; j = 1; : : : ; n; k = 1; : : : ; + 1; ¿ 0
(22)
(23)
where k , k = 1; : : : ; ; are the breakpoints of fn; ; pj; k , j = 1; : : : ; n; are the poles of the k th rational
piece of fn; , k = 1; : : : ; + 1, and 1 ¿0 is suciently small.
Theorem 2.1 [4] assures that lim|Y |→0 kf − fY kX = kf − fX kX , where |Y | = supx ∈ X d(x; Y ) and
fY ∈ Sn; ; 1 [a; b] (the subindex 1 denotes that all the approximants fn; are subject to (22)–(23)).
202
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
Theorem 1 is also valid considering the optimal approximation from Sn; ; n [a; b], for some sequences (n ), lim n = 0. For Theorem 2 it is sucient to x a small 1 ¿0. The selection of a
numerical procedure to obtain an approximant fn; of f depends on what information we have on f.
Therefore any method must not be based upon the linear system (9), (10) whose associated system
matrix is ill-conditioned. In fact, every solution of (9), (10), is not necessarily optimal at all. A best
approximation fY to a sample {f(y); y ∈ Y }, is directly calculated as a solution of the nonlinear
optimization problem minfn; kfn; − fkY , subject to fn; ∈ Sn; ; 1 [a; b]. In this respect some advantages
can be gotten using a modi cation of (9), (10) to nd feasible initial data, possibly close to a global
minimal point.
3.2. An identi cation problem
We applied a discrete version of the above piecewise rational method to solve the following
problem (cf. [4]). Let w(2) + f w + F = 0 be a linear di erential equation of second order, with
boundary values w(a) = wa ; w(b) = wb . The values w(k ), w(2) (k ), F(k ), k = 1; : : : ; m; are given as
data, and f is unknown.
The above problem is a singular identi cation problem when we assume that the real state of
nature f is analytic in [a; b]\{a; x1 ; : : : ; x ; b}. In principle, we say that a solution has been obtained
when especi c information on f has been found, especially that related to singularities location.
The expected behavior of poles when optimal approximants are considered, has been taken into
account in selecting a rational method. On the other hand, a piecewise scheme seems to be convenient
in dealing with real functions whose singularities under interest are all located in the interval [a; b].
The equality |w(f−fn; )| = |w(2) +wfn; +F|, being w the solution of the equation w(2) +wf+F = 0,
is the link which connects [4] with this paper.
3.3. Asymptotical behavior of the optimal breakpoints
Having an optimal solution of this approximation problem, a new question arises on whether
the breakpoints of fY = fn; ; 1 tend to the singularities of f as n → ∞. An interesting example is
given by the function f(x) = |x|; |x| 6 1, when f is to be approximated by a best approximant
fYn = fn; 1; 1 ∈ Sn; 1; 1 [−1; 1], and |Yn | tend to zero as n → ∞. Let n; 1 be the breakpoint of fYn , and
0¡1 ¡ 31 . The following proposition takes place
Proposition 1. The sequence ( n; 1 ) converges to zero provided that 0¡¡w(x); x ∈ [−; ]; for
some ; 0¡¡1.
Proof. The proof is √
based upon the fact that the nth best rational approximation to |x|, x ∈ [−; ],
has order O(exp(− n)), for every , 0¡ 6 1, and |x| ∈ H1; [−1; 1], for every ¿0.
Let ( n; 1 )n ∈ J be a convergent subsequence with limit x1 6= 0. Thus, for 0¡¡ 12 |x1 | and n large,
n; 1 ∈= [−; ], n ∈ J .
From Theorem 2 we have R n (|x|; [−; ])∞ 6 kfYn − |x|k∞ = O(exp(−cn)), n ∈ J , which is
impossible. Hence x1 = 0.
J. Illan / Journal of Computational and Applied Mathematics 99 (1998) 195–203
203
Many numerical evidences show that it is dicult to locate a singularity in a region where the
corresponding weight function value is zero or too small. Indeed, if w() ≈ 0, the point should
be considered in advance as a singularity of f.
References
[1] L. Baratchart, F. Wielonsky, Rational approximation in the real Hardy space H2 and Stieltjes integrals: a uniqueness
theorem, Constr. Approx. 9 (1993) 1–21.
[2] P.L. Duren, Theory of H p Spaces, Academic Press, New York, 1970.
[3] A.A. Gonchar, On the rate of rational approximation to continuous functions with characteristic singularities, Mat.
Sbornik T. 73, 115, (4) (1967) 630–638.
[4] F. Guerra, J. Illan, I. Romero, Rational methods to solve an identi cation problem of a non-homogeneous second
order linear equation with singularities, in: (Eds.), M. Florenzano et al., Proc. Internat. Conf. on Approximation and
Optimization in the Caribbean, Peter Lang Series in Approximation and Optimization, vol. 8, 1995, pp. 367–378.
[5] J. Illan, On the rational approximation of H p functions in the Lp () metric, Lectures Notes in Mathematics,
vol. 1354, Springer, Berlin, 1987, pp. 155–163.
[6] D.J. Newman, Rational approximation to |x|, Michigan Math. J. 11 MR 30 (1344) (1964) 11–14.
[7] J. Szabados, Rational approximation in certain classes of functions, Acta Math. Acad. Scientiarum Hungaricae Tomus
19 (1–2) (1968) 81–85.
[8] H. Widom, Rational approximation and n-dimensional diameter, J. Approx. Theory 5 (1972) 343–361.