Piecewise rational approximation to continuous functions
with characteristic singularities
J. Illan1
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 Hp 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∈H∞
0 [−1;1], and can be applied to study the asymptotic behavior of piecewise rational
methods to solve a singular identication problem associated with the equationw(2)+ (f+)w= 0, in terms of the error
equation criterion. c1998 Elsevier Science B.V. All rights reserved.
Keywords:Piecewise rational approximation; BestLp approximation; Singular identication problem
1. Introduction and statement of results
Let w∈L∞(dx;[a; b]), w¿0, and k k
p be the Lp(w) =Lp(w;[a; b]) norm, 16p6∞. For f
continuous in [a; b], we dene Rn(f;[a; b])p= inf{kf−rkp; r∈Fn; n}, where Fn; m={p=q; p∈n;
q∈m};andn is the set of all polynomials of degree 6n. Let f be a function in the Hardy space
H∞ such thatf is continuous in [−1;1]. Inspired by the major paper [6], Gonchar [3] obtained the
following result:
Rn(f;[−1;1])∞= O
inf
t¿1
!(f;e−t;[−1;1]) +texp
−nct
; (1)
where !(f; ;[a; b]) = sup{|f(x)−f(y)|; |x −y|¡; a6x; y6b} is the uniform modulus of continuity of f, and c¿0 is an absolute constant.
1This research was carried out while the author was a visiting professor at the Departmento de Matematica Aplicada,
Universidad de Vigo, Spain.
The estimate (1) has been extended to Rn(f;[−1;1])p for f∈Hp, 16p6∞ (cf. [5]). In this
context, a way to connect a feature of the measurewith the order of convergence ofRn(f;[−1;1])p
has been established in [5] by the following result. Given a Carleson measure , such that () = sup{¿0; R1
As for the uniqueness of the element of the best rational approximation, see [1]. In what follows the set {xk}kk==1, such that a=x0¡x1¡· · ·¡x¡x+1=b; ∈N, will be a -subdivision of the 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+
Next, we dene two subclasses ofD(1=; ; I), which are concerned with obtaining the approximation results in the paper.
Notice that the singularities of every f∈Hp
I have the same characteristic of being all situated
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
Rn; (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
corre-sponding 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[; 16p6∞; with internal singularities {x1; : : : ; x}. Let dk(x) =x lk+ck; where ck=12(xk−1+xk); lk=12(xk−xk−1); k= 1; : : : ; + 1. Then
Rn; (f)p= O
inf
t¿1
max
16k6!(f; xk;e −t;[c
k; ck+1])
+ max
16k6+1!p; k(f;e
−t) +t exp
−nct + t
p
;
where !(f; x; ; I) = supy∈I;|x−y|¡|f(x)−f(y)|; and
!p; k(f; ) = sup
0¡h¡
Z 1
−1|
f◦dk(x)−f◦dk((1−h)x)|pw◦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 givenf∈HpI. 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
Proof of Theorem 1. In order to simplify the notation, henceforth M(f; n; p; t; I) =!(f;e−t; I) +
The following property, which allows to obtain (6), is satised by gn; t (cf. [3, 5]).
|gn; t(x)|6M1
The following upper bound is derived from (6) and (7).
Z xk
xk−1
|f(x)−n; t ; k(x)|pw(x) dx
!1=p
Next, we give an equation system to construct a continuous approximant fn; . Let
The following inequality (cf. [2]) produces the term et=p in (6) and (14).
Notice that the relation xn; j; t pn; j; t= (1−12e−t2)2; j= 1; : : : ; n; t¿1, implies that
There exists a constant c0¿0 such that
Rn; (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 functionf∈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 functionsrn; k; ∈Fn; n,
Let aj; k; j= 1; : : : ; n; k= 1; : : : ; + 1, be the solution of the modied system (9)–(10) with
an;1= n; t, t=tk; ; and letfn; ; 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
an; k n
Y
j=1
1−e−tk; +x
n; j; tk;
1−e−tk; +x
n; j; tk; !
= O
exp
−tc
k;
n
: (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)6M5exp
−tc
k;
n
:
From the above inequality we have Rn;(f)∞6M5exp(−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 d6c, 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 modication 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))|6M4!(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], w6= 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 denition of Sn;[a; b].
k−1+16k; k= 1; : : : ; + 1; (22)
pj; k6k−1−1 or k+16pj; k; j= 1; : : : ; n; k= 1; : : : ; + 1; ¿0 (23)
where k, k= 1; : : : ; ; are the breakpoints of fn; ; pj; k, j= 1; : : : ; n; are the poles of the kth rational
piece of fn; , k= 1; : : : ; + 1, and 1¿0 is suciently small.
Theorem 2.1 [4] assures that lim|Y|→0kf−fYkX=kf−fXkX, where |Y|= supx∈Xd(x; Y) and
Theorem 1 is also valid considering the optimal approximation from Sn; ; n[a; b], for some
se-quences (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 approximantfn; off 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 modication of (9), (10) to nd feasible initial data, possibly close to a global minimal point.
3.2. An identication 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 dierential 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 identication 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 especic 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|, beingw the solution of the equationw(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|61, 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¡13. 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¡61, and |x| ∈H1; [−1;1], for every ¿0.
Let (n;1)n∈J be a convergent subsequence with limit x16= 0. Thus, for 0¡¡12|x1| and n large, n;1∈= [−; ], n∈J.
From Theorem 2 we have Rn(|x|;[−; ])∞6kfYn − |x|k∞= O(exp(−cn)), n∈J, which is
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.
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