Sign changes in rational Lw1-approximation
Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. The...
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| author | Blatt, H. P. Grothmann, R. Kovacheva, R. K. Блат, Х.П. Гросман, Р. Ковачева, Р. К. |
| author_facet | Blatt, H. P. Grothmann, R. Kovacheva, R. K. Блат, Х.П. Гросман, Р. Ковачева, Р. К. |
| author_sort | Blatt, H. P. |
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| description | Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real
rational functions of degree at most n in the numerator and of degree at most m in the denominator,
let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. Then we show that the counting measures of certain subsets of sign
changes of $f - r_{n,m}(f)$ converge weakly to the equilibrium measure on $[-1, 1]$ as $n\rightarrow \infty$.
Moreover, we prove discrepancy estimates between these counting measures and the equilibrium measure. |
| first_indexed | 2026-03-24T02:42:49Z |
| format | Article |
| fulltext |
K O R O T K I P O V I D O M L E N N Q
UDC 517.5
H.-P. Blatt, R. Grothmann (Kathol. Univ. Eichstätt-Ingolstadt, Germany),
R. K. Kovacheva (Inst. Math. and Inform. Bulg. Acad. Sci., Sofia, Bulgaria)
SIGN CHANGES IN RATIONAL L1
w-APPROXIMATION
ZNAKOZMINY U RACIONAL\NOMU L1
w-NABLYÛENNI
Let f ∈ L1
w[−1, 1], let rn,m(f) be a best rational L1
w-approximation for f with respect to real rational
functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and
let limn→∞(n − m(n)) = ∞. Then we show that the counting measures of certain subsets of sign changes
of f − rn,m(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove
discrepancy estimates between these counting measures and the equilibrium measure.
Nexaj f ∈ L1
w[−1, 1] i rn,m(f) — najkrawe L1
w-nablyΩennq dlq f vidnosno dijsnyx racional\nyx
funkcij stepenq ne bil\ße niΩ n u çysel\nyku ta stepenq ne bil\ße niΩ m u znamennyku, m = m(n)
i limn→∞(n−m(n)) = ∞. U c\omu vypadku prodemonstrovano, wo liçyl\ni miry pevnyx pidmnoΩyn
znakozmin f − rn,m(f) slabko zbihagt\sq do rivnovaΩno] miry na [−1, 1] pry n → ∞. TakoΩ dovedeno
ocinky vidxylennq cyx liçyl\nyx mir vid rivnovaΩno] miry.
1. Introduction. Letw be a weight function on I = [−1, 1] positive a.e. on [−1, 1] in the
sense of Lebesgue. Let L1
w[−1, 1] denote the class of all real-valued measurable functions
f on [−1, 1] such that f(t)w(t) is Lebesgue-integrable on [−1, 1] and let ‖f‖1,w be the
weighted L1-norm in L1
w[−1, 1], i.e.,
‖f‖1,w :=
1∫
−1
|f(t)|w(t)dt =
∫
|f |w.
Let
Rn,m =
{
r =
p
q
: deg p ≤ n, deg q ≤ m
}
be the class of all real-valued rational functions with numerator in Pn and denominator in
Pm. Here, Pk denotes the class of all algebraic polynomials of degree not exceeding k.
For given f ∈ L1
w[−1, 1], denote by rn,m(f) a best L1
w-approximation of f with respect
to Rn,m.
We say that a function g ∈ L1
w[−1, 1] does not change its sign at x0 ∈ [−1, 1] if f > 0
(or f < 0) a.e. in some neighborhood U of x0 in [−1, 1]. All other points of [−1, 1] are
called sign changes of f .
We denote by µ the equilibrium measure of [−1, 1].
In [1], Króo and Peherstorfer have proved the following result. Let n,m ∈ N0, 0 ≤
≤ m < n+ 1, and f ∈ L1
w[−1, 1] with w(x) ≡ 1. If f − rn,m(f) has no sign change on
(α, β) ⊂ [−1, 1], then
µ[α, β] =
arccosα− arccosβ
π
≤ 1
n−m+ 2
c© H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVA, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2 283
284 H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVA
and this upper bound is, in general, best possible. Our aim is to make more precise
statements about the distribution of sign changes of the error function, and to generalize
to general weights.
To formulate our results, we need the notation of counting measures. Let A be a finite
point set of k points. Then we define the discrete measure νA that associates the mass
1
k
to every point of A. For specified subsets A of the set of sign changes of f − rn,m(f),
the normalized counting measure νA will be compared with the equilibrium measure µ of
[−1, 1]. In other words, we are interested in upper bounds for the discrepancy
D[νA − µ] := sup
−1≤α≤β≤1
∣∣(νA − µ)([α, β])
∣∣
between νA and µ. We obtain distribution estimates for the sign changes of f − rn,m(f)
on [−1, 1] that generalize our results for polynomial approximation in [2].
2. Distribution of sign changes. In the following, c, ci, i = 1, 2, . . . , will denote
positive constants independent of n. Let w be positive a.e. and measurable on [−1, 1],∫
w = 1,
and, for 0 < ε ≤ 1, let us define
ϕ(w, ε) := inf
∫
A
w : A ⊂ [−1, 1], µ(A) ≥ ε
.
By εn(w), we denote the unique solution of the equation
ϕ(w, ε) = e−nε.
Then εn(w) → 0 as n→ ∞ and
εn(w) ≥ c1
log n
n
, n = 1, 2, . . .
(cf. [3], Lemma 2.3). Concerning the rate of εn(w) as n → ∞, we have εn(w) =
= O
(
log
n
n
)
for Jacobi weights and εn(w) = O(n−1/(1+α)) for weights of typew(x) =
= e−|x|α with α > 0.
The best approximation r = rn,m(f) =
p0
q0
is well-known to be characterized by the
fact that, for every p ∈ Pn and q ∈ Pm, we have∫
pq0 − qp0
q20
sgn(f − r)w ≤
∫
Z(f−r)
|pq0 − qp0|
q20
w (1)
(cf. [1], Lemma 1), where we denote
Z(g) := {x : g(x) = 0}
for functions g ∈ L1
w[−1, 1] and
sgn g(x) =
1 if g(x) > 0,
0 if g(x) = 0,
−1 if g(x) < 0.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
SIGN CHANGES IN RATIONAL L1
w-APPROXIMATION 285
It can easily be seen from (1) that the error function f−r changes its sign at least n+m+1
times on [−1, 1] if µ(Z(f − r)) = 0 and r /∈ Rn−1,m−1.
The function ϕ ∈ L1
w[−1, 1] is called sign function if ϕ2 = 1 on [−1, 1]. ϕ is said to
be orthogonal to Pn (written ϕ ⊥ Pn) if∫
ϕPw = 0 for all P ∈ Pn.
Two sign functions ϕ and ψ are called equivalent if ϕ = ψ or ϕ = −ψ a.e. on [−1, 1].
Let r = rn(f) =
p0
q0
with no common factors, then the defect of r is defined as
dn,m = dn,m(r) = min(n− grad p0, m− grad q0).
Then (1) is equivalent to
1∫
−1
p
q20
sgn(f − r)w ≤
∫
Z(f−r)
|p|
q20
w for all p ∈ Pn+m−dn,m , (2)
i.e., 0 is the best approximation of f − r in the subspace q−2
0 Pn+m−dn,m
.
According to Proposition 20 of Cheney and Wulbert [4], there exists a sign function
ψ such that ψ(x) = sgn(f − r)(x) for all x ∈ [−1, 1] \ Z(f − r) and∫
p
q20
ψw = 0 for all p ∈ Pn+m−dn,m .
Then the sign changes of f − r are identical with the sign changes of ψ. Therefore, ψ
and, consequently, f − r has at least n+m− dm,n sign changes in (−1, 1).
Concerning the distribution of these sign changes, we are able to prove estimates for
a subset of n−m+ dn,m + 1 sign changes. To this end, we use the weaker condition∫
p sgn(f − r)w ≤
∫
Z(f−r)
|p|w for all p ∈ Pn−m+dn,m
, (3)
which holds because of (2).
Theorem 1. Let f ∈ L1
w[−1, 1] and let r = rn,m(f) be a best L1
w-approximation to
f on Rn,m, m < n + 1. Then there exist n −m + dn,m + 1 sign changes of f − r at
points
−1 < ξ(n,m)
0 < . . . < ξ
(n,m)
n−m+dn,m
< 1 (4)
such that the normalized counting measure νn,m of the these points satisfies
D[νn,m − µ] ≤ c
√
εn−m+dn,m(w),
where c is an absolute constant, not depending on n,m or f .
Corollary 1. Letm = m(n) and let
n−m(n) → ∞ as n→ ∞,
then the normalized counting measures νn,m of the point sets in (4) converge weakly to µ
for n→ ∞.
For special weights w, the numbers εn(w) are well-known and one gets explicit dis-
crepancy estimates between νn,m and µ.
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
286 H.-P. BLATT, R. GROTHMANN, R. K. KOVACHEVA
As a special example, we consider generalized Jacobi weights where, as in the poly-
nomial case, the estimate of Theorem 1 can be sharpened.
Theorem 2. Let −1 = t1 < t2 < . . . < tk = 1 be fixed points, a1, . . . , ak > −1 be
fixed numbers and w a weight function that satisfies
w(x) ≥ c1
k∏
i=1
|x− ti|ai
with c1 > 0. Then the normalized counting measure νn,m of (4) satisfy
D[νn,m − µ] ≤ c (log(n−m+ dn,m))2
n−m+ dn,m
for all n ≥ m+ 2, where c is an absolute constant.
3. Proofs. In the following, we use a lemma of Króo and Peherstorfer [5].
Lemma 1 [5]. Let ϕ be a sign function, ϕ ⊥ Pn, and assume that ϕ has exactly n+1
sign changes at the point y1, . . . , yn+1 with
y0 = −1 < y1 < y2 < . . . < yn+1 < 1 = yn+2.
If the sign function ψ is not equivalent to ϕ and ψ ⊥ Pn, then ψ has a sign change in
each interval (yi, yi+1), 0 ≤ i ≤ n+ 1.
Denote by Un,w(x) the Chebychev polynomial of second kind with respect to w; i.e.,
Un,w(x) is a monic polynomial in Pn and
‖Un,w‖1,w = min{‖P‖1,w : P monic in Pn},
Un,w is unique and characterized by exactly n sign changes (or zeros) at points
−1 < y(n)
1 < y
(n)
2 < . . . < y(n)
n < 1.
We denote by νn,w the normalized zero-counting measure of Un,w. Then for all n =
= 1, 2, . . . ,
D[νn,w − µ] ≤ c
√
εn(w) (5)
with some absolute constant c > 0 (Theorem 2 in [2]).
Proof of Theorem 1. Let
ϕ(x) :=
sgn Un−m+dn,m+1,w(x) for x with Un−m+1,w(x) �= 0,
1 elsewhere.
Then ϕ satisfies the conditions of Lemma 1.
Because of (3), 0 is a best L1
w-approximation to f − r from the space Pn−m+dn,m .
Thus, there exists a sign function ψ ⊥ Pn−m+dn,m
such that
ψ(x) = sgn (f − r)(x) for all x ∈ [−1, 1] \ Z(f − r)
(cf. [4], Proposition 20).
Then either ψ = ±ϕ a.e. on [−1, 1] or, due to Lemma 1, ψ has a sign change in each
interval (
y
(n−m+dn,m+1)
i , y
(n−m+dn,m+1)
i+1
)
, 0 ≤ i ≤ n−m+ dn,m + 1,
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
SIGN CHANGES IN RATIONAL L1
w-APPROXIMATION 287
where
−1 < y(n−m+dn,m+1)
1 < . . . < y
(n−m+dn,m+1)
n−m+dn,m+1 < 1
are the zeros of Un−m+dn,m+1,w(x) and
y
(n−m+dn,m+1)
0 = −1, y
(n−m+dn,m+1)
n−m+dn,m+2 = 1.
Finally, we get from (5) the desired discrepancy result.
Corollary 1 is an immediate consequence of Theorem 1. Theorem 2 follows from a
result in [2], stating that, for generalized Jacobi weights,
D[νn,w − µ] ≤ c (logn)2
n
for all n = 2, 3, . . . , where c is again an absolute constant.
1. Króo A., Peherstorfer Fr. Interpolatory properties of best rational L1-approximations // Constr. Approxim.
– 1988. – 4. – P. 97 – 106.
2. Blatt H.-P., Grothmann R., Kovacheva R. K. On sign changes in weighted polynomial L1-approximation
// Acta math. hung. – 2002.
3. Blatt H.-P. A discrepancy lemma for oscillating polynomials and sign changes of the error function of best
approximations // Ann. Numer. Math. – 1997. – 4. – P. 55 – 66.
4. Cheney E. W., Wulbert D. E. Existence and unicity of best approximation // Math. scand. – 1969. – 24. –
P. 113 – 140.
5. Króo A., Peherstorfer Fr. Interpolatory properties of best L1-approximation // Math. Z. – 1987. – 96. –
P. 249 – 255.
Received 16.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 2
|
| id | umjimathkievua-article-3452 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:42:49Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bd/2db350a884d9d38d1a5f03b156959cbd.pdf |
| spelling | umjimathkievua-article-34522020-03-18T19:54:47Z Sign changes in rational Lw1-approximation Sign changes in rational Lw1-approximation Blatt, H. P. Grothmann, R. Kovacheva, R. K. Блат, Х.П. Гросман, Р. Ковачева, Р. К. Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. Then we show that the counting measures of certain subsets of sign changes of $f - r_{n,m}(f)$ converge weakly to the equilibrium measure on $[-1, 1]$ as $n\rightarrow \infty$. Moreover, we prove discrepancy estimates between these counting measures and the equilibrium measure. Нехай $f \in L_{1}^{w}[-1, 1]$ і $r_{n, m}(f)$ — найкраще $L_{1}^{w}$-наближення для $f$ відносно дійсних раціональних функцій степеня не більше ніж n у чисельнику та степеня не більше ніж m у знаменнику, $m = m(n)$ і $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. У цьому випадку продемонстровано, що лічильні міри певних підмножин знакозмін $f - r_{n,m}(f)$ слабко збігаються до рівноважної міри на $[-1, 1]$ при $n\rightarrow \infty$. Також доведено оцінки відхилення цих лічильних мір від рівноважної міри. Institute of Mathematics, NAS of Ukraine 2006-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3452 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 2 (2006); 283–287 Український математичний журнал; Том 58 № 2 (2006); 283–287 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3452/3642 https://umj.imath.kiev.ua/index.php/umj/article/view/3452/3643 Copyright (c) 2006 Blatt H. P.; Grothmann R.; Kovacheva R. K. |
| spellingShingle | Blatt, H. P. Grothmann, R. Kovacheva, R. K. Блат, Х.П. Гросман, Р. Ковачева, Р. К. Sign changes in rational Lw1-approximation |
| title | Sign changes in rational Lw1-approximation |
| title_alt | Sign changes in rational Lw1-approximation |
| title_full | Sign changes in rational Lw1-approximation |
| title_fullStr | Sign changes in rational Lw1-approximation |
| title_full_unstemmed | Sign changes in rational Lw1-approximation |
| title_short | Sign changes in rational Lw1-approximation |
| title_sort | sign changes in rational lw1-approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3452 |
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