Influence of poles on equioscillation in rational approximation
The error curve for the rational best approximation of ƒ ? C[?1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in...
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| author | Blatt, H. P. Блат, Х.П. |
| author_facet | Blatt, H. P. Блат, Х.П. |
| author_sort | Blatt, H. P. |
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| datestamp_date | 2020-03-18T19:54:30Z |
| description | The error curve for the rational best approximation of ƒ ? C[?1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in [?1, 1]. The reason is the influence of the distribution of the poles of rational approximants. In this paper, we generalize the results known so far to situations where the requirements for the degrees of numerators and denominators are less restrictive. |
| first_indexed | 2026-03-24T02:42:23Z |
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UDC 517.5
H.-P. Blatt (Kathol. Univ. Eichstätt-Ingolstadt, Germany)
THE INFLUENCE OF POLES ON EQUIOSCILLATION
IN RATIONAL APPROXIMATION
VPLYV POLGSIV NA EKVIOSCYLQCI}
U RACIONAL\NOMU NABLYÛENNI
The error curve for rational best approximation of f ∈ C[−1, 1] is characterized by the well-known equioscil-
lation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the
equilibrium distribution. It is known that these points need not to be dense in [−1, 1]. The reason is the influence
of the distribution of the poles of the rational approximants. In this paper, we generalize the results known so
far to situations where the requirements for the degrees of numerators and denominators are less restrictive.
Kryva poxybok dlq racional\noho najkrawoho nablyΩennq f ∈ C[−1, 1] xarakteryzu[t\sq vidomog
vlastyvistg ekvioscylqcij. Na vidminu vid polinomial\noho vypadku rozpodil cyx zmin znaku ne vy-
znaça[t\sq rivnovaΩnym rozpodilom. Vidomo, wo ci toçky ne obov’qzkovo magt\ buty wil\nymy v
[−1, 1], wo zumovleno vplyvom rozpodilu polgsiv racional\nyx nablyΩen\. U danij roboti uza-
hal\neno vidomi rezul\taty na vypadky, de na stepeni çysel\nykiv ta znamennykiv nakladagt\sq menß
Ωorstki umovy.
1. Introduction. Let f ∈ C[−1, 1] be a real-valued function and let Rn,m denote the
family of real rational functions with numerator in Pn and denominator in Pm, where
Pk is the set of algebraic polynomials of degree at most k, k ∈ N0. For each pair of
nonnegative integers (n,m), there exists a unique function r∗n,m ∈ Rn,m that is the best
uniform approximation to f on I = [−1, 1] in the sense that
‖f − r∗n,m‖ < ‖f − r‖ for all r ∈ Rn,m, r �= r∗n,m,
where ‖· ‖ denotes the sup norm on I . Writing r = pn/qm,where pn ∈ Pn and qm ∈ Pm
have no common factor and qm is monic, the defect of r is defined by
dn,m(r) := min(n− deg pn,m− deg qm). (1)
Let us define
l(r) = n+m+ 1 − dn,m(r); (2)
then l(r) is the dimension of the tangential space with respect to the coefficients of the nu-
merator and denominator as parameter space. We write r∗n,m := p∗n/q
∗
m with no common
factors and define for abbreviation
ln,m := l(r∗n,m).
Then it is well known that the best approximation of f is characterized by the following
equioscillation property:
There exist ln,m + 1 points x(n,m)
k ,
−1 ≤ x
(n,m)
0 < . . . < x
(n,m)
ln,m
≤ 1,
c© H.-P. BLATT, 2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1 3
4 H.-P. BLATT
such that
λn,m(−1)k(f − r∗n,m)(x(n,m)
k ) = ‖f − r∗n,m‖, 0 ≤ k ≤ ln,m, (3)
where λn,m = +1 or λn,m = −1 is fixed. Such a point set {x(n,m)
k } is called alternation
set. In general, it is not unique. Therefore, in the following, we denote by
An,m = An,m(f) = {x(n,m)
k }ln,m
k=0
an arbitrary but fixed alternation set for the best approximation r∗n,m of f out of Rn,m.
Let νn,m denote the normalized counting measure of An,m, i.e.,
νn,m([α, β]) :=
#{x(n,m)
k : α ≤ x
(n,m)
k ≤ β}
ln,m + 1
. (4)
Kadec [1] has shown that there exists a subsequence Λ of N such that
νn,0
∗−→ µ as n ∈ Λ, n −→ ∞, (5)
where µ is the equilibrium measure of [−1, 1], i.e., the density of µ on I is
dµ(x) =
dx√
1 − x2
.
For rational approximation, Borwein et al. [2] have proved that denseness on [−1, 1] holds
for a subsequence of alternation sets An,m whenever m = m(n) and
n
m(n)
−→ κ > 1
as n −→ ∞. Moreover, they have shown in the case lim
n→∞
m(n)
n
= 0 that there exists
Λ ⊂ N such that
νn,m(n)
∗−→ as n ∈ Λ, n −→ ∞.
More quantitative results were obtained by Kroó and Peherstorfer in [3]. Namely, let
us denote by Nn,m(α, β) the number of points of An,m in [α, β]. Then the main result
can be stated as follows: let m(n) < n, then
Nn,m(n)(α, β)
n−m(n)
≥ µ([α, β]) − c
√
log n
n−m(n)
, (6)
where c is an absolute constant independent of f and n.
Braess et al. [4] considered the case m(n) = n + κ, κ ∈ Z, fixed. Their re-
sults were based on the number γn(ε) of poles of best approximants lying outside an
ε-neighbourhood of [−1, 1]. Roughly speaking, if γn(ε) is sufficiently big, then there is a
connection of the distribution of An,m with the equilibrium distribution µ.
The intimate relation between An,m(f) and the poles of r∗n,m was investigated in [5].
To be precise, let f be not a rational function and let n and m(n) satisfy
m(n) ≤ n; m(n) ≤ m(n+ 1) ≤ m(n) + 1. (7)
Moreover, let
Qn(x) = q∗m(n)(x)q
∗
m(n+1)(x) =
κn∏
i=1
(x− yi) (8)
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THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 5
be the product of the denominators of r∗n,m(n) and r∗n+1,m(n+1), then
τn(∆) :=
#{yi : yi ∈ A}
κn
(A ⊂ C)
denotes the normalized counting measure of all finite poles of r∗n,m(n) and r∗n+1,m(n+1)
counted with their multiplicities. Then it was proved in [5] that there exists a subsequence
Λ ⊂ N such that
νn,m(n) − αnτ̂n − (1 − αn)µ ∗−→ 0 as n→ ∞, n ∈ Λ, (9)
in the weak∗-topology, where
αn =
κn
ln,m(n) + 1
and τ̂n denotes the balayage measure of τn onto [−1, 1]. The purpose of the present paper
is to obtain a convergence result of type (9), where the restrictionm(n) ≤ n is weakened
to m(n) ≤ n+ 1. We point out that this weaker condition implies that the original proof
in [5] has to be substantially modified. Moreover, the weaker condition m(n) ≤ n + 1
allows to apply and to understand examples of [2].
Borwein et al. [2] have proved in the case m(n) = n + 1 that there exists a function
with no alternation points in a certain interval.
It is a challenge to generalize results of type (8) to m(n) > n+ 1.
2. Main results. We assume that m(n) depends on the parameter n ∈ N. Let
En,m(n) := inf
r∈Rn,m(n)
‖f − r‖ = ‖f − r∗n,m(n)‖
and define for abbreviation
r∗n := r∗n,m(n), p∗n = p∗n,m(n), q∗n = q∗n,m(n),
En = En,m(n), ln = ln,m(n), dn = dn,m(n),
x
(n)
k := x
(n,m(n))
k , k = 0, 1, . . . , ln.
Again, we use the normalized counting measure νn of the alternation set {x(n)
k }ln
k=0
and the normalized counting measure τn of the union of the (finite) poles of r∗n and r∗n+1.
All poles are counted with their multiplicities. For any finite Borel measure ν, the loga-
rithmic potential of ν is defined by
Uν(z) :=
∫
log
1
|z − t| dν(t).
A crucial role is played by the balayage measure τ̂n of τn onto [−1, 1]. τ̂n is the unique
measure supported on [−1, 1], for which ‖τ̂n‖ = ‖τn‖ and
U τ̂n(z) = Uτn(z) + c, z ∈ [−1, 1],
where
c =
∫
G(t,∞)dτn(t)
and G(z, a) denotes Green’s function of Ω = C \ [−1, 1] with pole at a ∈ Ω (cf. [6]).
Furthermore, τ̂n has the following properties:
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
6 H.-P. BLATT
a) U τ̂n(z) ≤ Uτn(z) + c, z ∈ C;
b) if h is continuous on C and harmonic in Ω, then
∫
hdτn =
∫
hdτ̂n.
Our main result can be formulated in the following theorem.
Theorem. Let f be not a rational function and let the parameters m(n), n ∈ N,
satisfy
m(n) ≤ n+ 1, m(n) ≤ m(n+ 1) ≤ m(n) + 1. (10)
Then there exists a subsequence Λ ⊂ N such that
νn − αnτ̂n − (1 − αn)µ ∗−→ 0 as n→ ∞, n ∈ Λ,
where
αn =
deg q∗n + deg q∗n+1
ln + 1
.
We note that condition (10) is less restrictive than (7).
It is possible to formulate the above result in a more concise manner such that only the
alternation counting measure νn and pole counting measures of r∗n and rn+1 are involved.
Let
Rn = r∗n+1 − r∗n =
p
q
,
where p and q have no common divisor. Then the degree of
p
q
is defined by
deg
p
q
:= max(deg p, deg q).
Then the number of zeros, resp. poles, of Rn in the closed complex plane C is degRn,
where all zeros and poles are counted with their multiplicity.
We define the normalized pole counting measure σpole,n of Rn in C by
σpole,n(A) =
#{poles of Rn inA}
degRn
(A ⊂ C)
and the normalized zero counting measure σzero,n of Rn in C by
σzero,n(A) =
#{zeros of Rn inA}
degRn
(A ⊂ C).
Corollary. Under the conditions of Theorem 1, there exists Λ ⊂ N such that
σ̂zero,n − σ̂pole,n
∗−→ 0 as n→ ∞, n ∈ Λ.
Especially,
νn − σ̂pole,n
∗−→ 0 as n→ ∞, n ∈ Λ,
if lim
n→∞
m(n)
n
≤ 1.
Let us discuss the second part of the corollary in Kadec’s case, i.e., (n,m(n)) =
(n, 0). Then Rn = p∗n+1 − p∗n and pn, p
∗
n+1 are the best approximating polynomials to f
with respect to Pn, resp. Pn+1 andRn has a pole of multiplicity n+1 at ∞ if p∗n �= p∗n+1.
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THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 7
Now, for the Dirac measure δ∞ at the point at ∞ we know that the balayage mesure δ̂∞ is
just the equilibrium measure µ (cf. [6]). Moreover, all zeros of p∗n+1 − p∗n are separating
the alternation points. Hence, σ̂zero,n = σzero,n and
lim
n→∞
n∈Λ
σzero,n = lim
n→∞
n∈Λ
νn,0 = µ
from the corollary. That is Kadec’s result (5).
3. Proofs. Since lim
n→∞
En = 0, by a well-known argument, there exists a subsequence
Λ ⊂ N such that
En + En+1
En − En+1
≤ n2 for n ∈ Λ (11)
(cf. [7, p. 243], Lemma 7.3.3). In particular, for n ∈ Λ we have r∗n �= r∗n+1 and, by (3),
(−1)k(r∗n+1 − r∗n)(x(n)
k ) ≥ En − En+1 (12)
for 0 ≤ k ≤ ln, where we have assumed without loss of generality that the number
λn,m(n) = 1 in (3). Writing
Rn = r∗n+1 − r∗n =
p∗n+1q
∗
n − p∗nq∗n+1
q∗nq
∗
n+1
=
Pn
q∗nq
∗
n+1
=
Pn
Qn
,
we obtain
(−1)kRn(x(n)
k ) ≥ En − En+1, 0 ≤ k ≤ ln. (13)
In the following, we assume that an is the highest coefficient of Pn(x), i.e.,
Pn(x) = anx
ln + . . . .
By (13), Pn orRn =
Pn
Qn
has at least ln zeros in (−1, 1). Since r∗n �= r∗n+1, condition (10)
implies that all zeros of Pn are in (−1, 1). As in [5], our next intention is to reconstruct
the polynomial Qn by interpolation at the points x(n)
k , 0 ≤ k ≤ ln. Since
κn = degQn = deg q∗n + deg q∗n+1 ≤ m(n) − dn + n+ 2 = ln + 1,
the degree of Qn is, in general, too big to be reconstructed by interpolation at x(n)
k , 0 ≤
≤ k ≤ ln.
In the case κn ≤ ln, we can use the method of proof in [5]. Therefore, we can restrict
ourselves in the following to the case κn = ln + 1.
First, we have to modify the polynomial Pn(x): Let ξn be such that
ξn ≥ n max(1, |y1|, |y2|, . . . , |yκn |), (14)
where y1, . . . , yκn are all zeros of Qn(x) in C. Then we define
P̃n(x) := (x− ξn)Pn(x) and R̃n :=
P̃n
Qn
. (15)
Then
deg P̃n = degQn = ln + 1
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
8 H.-P. BLATT
and we can reconstruct Qn by interpolation at the points x(n)
k , 0 ≤ k ≤ ln, and at the
point ξn. We obtain
Qn(z) =
ln∑
k=0
Qn(x(n)
k )w(z)
(z − x(n)
k )w′(x(n)
k )
+
Qn(ξn)w(z)
(z − ξn)w′(ξn)
, (16)
where
w(z) = (z − ξn)
ln∏
k=0
(z − x(n)
k ). (17)
For z �= ξn, x(n)
k , 0 ≤ k ≤ ln, relation (16) can be written as
Qn(z)
w(z)
=
ln∑
k=0
Qn(x(n)
k )
(z − x(n)
k )w′(x(n)
k )
+
Qn(ξn)
(z − ξn)w′(ξn)
. (18)
By definition, we have
‖Rn‖ ≤ ‖f − r∗n‖ + ‖f − r∗n+1‖ ≤ En + En+1
and, therefore, ∥∥∥R̃n
∥∥∥ ≤ (ξn + 1)(En + En+1). (19)
Moreover, by (13) we get
(−1)k+1R̃n(x(n)
k ) ≥ (ξn − 1)(En − En+1). (20)
Next, we consider the function
h(z) := log
∣∣∣R̃n(z)
∣∣∣ − κn∑
i=1
G(z, yi) +G(z, ξn).
The function h(z) is subharmonic in C; hence, the maximum principle applies and
h(∞) ≤ max
z∈I
h(z) = max
z∈I
log
∣∣∣R̃n(z)
∣∣∣ = log ‖R̃n‖,
and we obtain
h(∞) = log |an| −
κn∑
i=1
G(∞, yi) +G(∞, ξn).
Therefore, with (19)
log |an| ≤ log
(
(ξn + 1)(En + En+1)
)
+
κn∑
i=1
G(∞, yi) −G(∞, ξn). (21)
Next, let us consider the approximation of the function P̃n(x) at the points
x
(n)
k , 0 ≤ k ≤ ln,
with interpolation at the zero ξn with respect to Pln and the weight function
1
Qn(x)
. It
turns out that de la Vallée Poussin’s theorem implies together with (20) that the minimal
error ρ satisfies
ρ ≥ (ξn − 1)(En − En+1). (22)
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THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 9
On the other hand, for any P ∈ Pln with P (ξn) = 0, we have
ρ =
∣∣∣∣∑ln
k=0
βk
(
P̃n − P )(x(n)
k
)∣∣∣∣∑ln
k=0
∣∣βkQn(x(n)
k )
∣∣ , (23)
where
1
β k
= w′(x(n)
k
)
=
(
x
(n)
k − ξn
) ∏
i�=k
(
x
(n)
k − x(n)
i
)
. (24)
Now, fix the polynomial P ∈ Pln by P (ξn) = 0 and P
(
x
(n)
k
)
= P̃n
(
x
(n)
k
)
, 1 ≤ k ≤ ln.
Hence,
(P̃n − P )(x) = an(x− ξn)
ln∏
k=1
(x− x(n)
k )
and, therefore,
(P̃n − P )(x(n)
0 ) = an(x(n)
0 − ξn)
ln∏
k=1
(x(n)
0 − x(n)
k ).
By (22) – (24) we obtain
ρ =
|an|∑ln
k=0
∣∣βkQn(x(n)
k )
∣∣ ≥ (ξn − 1)(En − En+1).
Using representation (18), for z /∈ I we get∣∣∣∣Qn(z)
w(z)
∣∣∣∣ ≤ D(z)
ln∑
k=0
|βkQn(x(n)
k )| + 1
|z − ξn|
∣∣∣∣Qn(ξn)
w′(ξn)
∣∣∣∣ ≤
≤ D(z)
|an|
(ξn − 1)(En − En+1)
+
1
|z − ξn|
∣∣∣∣Qn(ξn)
w′(ξn)
∣∣∣∣ , (25)
where
D(z) = max
0≤k≤ln
∣∣∣z − x(n)
k
∣∣∣−1
.
Since κn = ln + 1 ≤ 2n+ 3, for n ≥ 2 we obtain
∣∣∣∣Qn(ξn)
w′(ξn)
∣∣∣∣ =
∣∣∣∣∣∣∣
∏κn
i=1
(ξn − yi)∏ln
k=0
(ξn − x(n)
k )
∣∣∣∣∣∣∣ ≤
[
ξn(1 + 1/n)
ξn − 1
]κn
≤
≤
(
1 + 1/n
1 − 1/ξn
)κn
≤
(
1 + 1/n
1 − 1/n
)κn
≤ c1, (26)
where c1 is independent of n.
In the following, we consider the level line
Γ1/n :=
{
z ∈ C : G(z,∞) = log
(
1 +
1
n
)}
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
10 H.-P. BLATT
of Green’s function G(z,∞). Then for z ∈ Γ1/n, n ≥ 2, we have
log
1
|z − ξn|
≤ log
1
|ξn|
+ c2,
where c2 is independent of n. Since
lim
z→∞
(
G(z,∞) + log
1
2
− log |z|
)
= 0
and lim
n→∞
ξn = ∞, there exists c3 > 0 such that
log
1
|z − ξn|
≤ −G(ξn,∞) + c3
for all n ≥ 2. Then from (26) we obtain for z ∈ Γ1/n and n ≥ 2 that
log
(
1
|z − ξn|
∣∣∣∣Qn(ξn)
w′(ξn)
∣∣∣∣) ≤ c4 −G(ξn,∞), (27)
where c4 > 0 is independent of n.
Define for abbreviation
An :=
|an|
(ξn − 1)(En − En+1)
. (28)
Then inequality (21) together with (11) implies
logAn ≤ log
ξn + 1
ξn − 1
+ log
En + En+1
En − En+1
+
κn∑
i=1
G(∞, yi) −G(∞, ξn) ≤
≤ log
n+ 1
n− 1
+ log(n2) +
κn∑
i=1
G(∞, yi) −G(∞, ξn)
and for z ∈ Γ1/n
log(D(z)An) ≤ c5 logn+
κn∑
i=1
G(∞, yi) −G(∞, ξn) (29)
for n ∈ Λ, n ≥ 2, with some constant c5 independent of n.
Comparing the right-hand sides of (27) and (29), we conclude from (25) that, for
n ∈ Λ, n ≥ 2, and z ∈ Γ1/n,
log
∣∣∣∣Qn(z)
w(z)
∣∣∣∣ ≤ c6 logn+
κn∑
i=1
G(∞, yi) −G(∞, ξn) (30)
with an absolute constant c6 independent of n.
The last inequality can be written with the logarithmic potentials Uνn(z), Uτn(z) and
the Dirac measure δξn
at the point ξn as
Uνn(z) − αnU
τn(z) +
1
ln + 1
Uδξn (z) ≤
≤ 1
ln + 1
(
c6 logn+
κn∑
i=1
G(∞, yi) −G(∞, ξn)
)
.
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THE INFLUENCE OF POLES ON EQUIOSCILLATION IN RATIONAL APPROXIMATION 11
Next, we use the balayage measure δ̂ξn of δξn onto the interval [−1, 1]. Since
U δ̂ξn (z) ≤ Uδξn (z) +G(∞, ξn), z ∈ C,
we obtain for z �= ξn
Uνn(z) − Uτn(z) +
1
ln + 1
U δ̂ξn (z) ≤ 1
ln + 1
(
c6 log n+
κn∑
i=1
G(∞, yi)
)
. (31)
Taking into account that we can choose the point ξn arbitrarily large on the positive real
axis and
lim
ξn→∞
δ̂ξn
= µ
in the weak∗-sense (cf. [6], Chapter II, formula 4.46), we can choose ξn such that
|U δ̂ξn − Uµ(z)| < 1
n
, z ∈ Γ1/n.
Then we obtain for z ∈ Γ1/n that
Uνn(z) − Uτn(z) ≤ c
log n
n
+
1
κn
κn∑
i=1
G(∞, yi).
The last inequality is of the same structure as inequality (30) in [5]. Hence, the remaining
proof follows the same lines as in this paper and is therefore omitted.
The proof of the corollary is left to the reader.
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Received 16.05.2005
ISSN 1027-3190. Ukr. mat. Ωurn., 2006, t. 58, # 1
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| id | umjimathkievua-article-3429 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:42:23Z |
| publishDate | 2006 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/0f431afc4c7cc003301f8667f408e494.pdf |
| spelling | umjimathkievua-article-34292020-03-18T19:54:30Z Influence of poles on equioscillation in rational approximation Вплив полюсів нa еквіосциляції у раціональному наближенні Blatt, H. P. Блат, Х.П. The error curve for the rational best approximation of ƒ ? C[?1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in [?1, 1]. The reason is the influence of the distribution of the poles of rational approximants. In this paper, we generalize the results known so far to situations where the requirements for the degrees of numerators and denominators are less restrictive. Крива похибок для раціонального найкращого наближення $f \in C[-1, 1]$ характеризується відомою властивістю еквіосциляцій. На відміну від поліноміального випадку розподіл цих змін знаку не визначається рівноважним розподілом. Відомо, що ці точки не обов'язково мають бути щільними в $[-1, 1]$, що зумовлено впливом розподілу полюсів раціональних наближень. У даній роботі узагальнено відомі результати на випадки, де на степені чисельників та знаменників накладаються менш жорсткі умови. Institute of Mathematics, NAS of Ukraine 2006-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3429 Ukrains’kyi Matematychnyi Zhurnal; Vol. 58 No. 1 (2006); 3–11 Український математичний журнал; Том 58 № 1 (2006); 3–11 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3429/3598 https://umj.imath.kiev.ua/index.php/umj/article/view/3429/3599 Copyright (c) 2006 Blatt H. P. |
| spellingShingle | Blatt, H. P. Блат, Х.П. Influence of poles on equioscillation in rational approximation |
| title | Influence of poles on equioscillation in rational approximation |
| title_alt | Вплив полюсів нa еквіосциляції у раціональному наближенні |
| title_full | Influence of poles on equioscillation in rational approximation |
| title_fullStr | Influence of poles on equioscillation in rational approximation |
| title_full_unstemmed | Influence of poles on equioscillation in rational approximation |
| title_short | Influence of poles on equioscillation in rational approximation |
| title_sort | influence of poles on equioscillation in rational approximation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3429 |
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