On conformal invariants in problems of constructive function theory on sets of the real line
This is a survey of some recent results by the author and his collaborators in the constructive theory of functions of a real variable. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory in the complex plane.
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2004
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| citation_txt | On conformal invariants in problems of constructive function theory on sets of the real line / V.V. Andrievskii // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 21-46. — Бібліогр.: 50 назв. — англ. |
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| description | This is a survey of some recent results by the author and his collaborators in the constructive theory of functions of a real variable. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory in the complex plane.
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Український математичний вiсник
Том 1 (2004), № 1, 21 – 46
On conformal invariants in problems of
constructive function theory on sets of the real
line
V. V. Andrievskii
(Presented by V.Ya. Gutlyanskǐı 15.02.2003 )
Abstract. This is a survey of some recent results by the author and his
collaborators in the constructive theory of functions of a real variable.
The results are achieved by the application of methods and techniques of
modern geometric function theory and potential theory in the complex
plane.
2000 MSC. 30C10, 30E10, 41A20, 31A15, 41A10, 41A25.
1. Introduction
Let E ⊂ C be a compact set of positive logarithmic capacity cap(E)
with connected complement Ω := C \ E with respect to the extended
complex plane C = C ∪ {∞}, gΩ(z) = gΩ(z,∞) be the Green function
of Ω with pole at infinity, and µE be the equilibrium measure for the
set E (see [48], [41] for further details on logarithmic potential theory).
The properties of gΩ and µE play an important role in many problems
concerning polynomial approximation of continuous functions on E and
the behavior of polynomials with a known uniform norm along E.
In this survey we discuss some of these problems for the case when
E is a subset of the real line R. The main idea of our approach is to use
conformal invariants such as the extremal length and module of a family
of curves. The basic conformal mapping can be described as follows.
Let E ⊂ [0, 1] be a regular set such that 0 ∈ E, 1 ∈ E. Then [0, 1] \
E =
∑N
j=1(aj , bj), where N is finite or infinite.
Received 03.02.2003
Key words and phrases. polynomial approximation, constructive description, continu-
ous functions, compact sets, Remez inequality, exponential of a potential, logarithmic
capacity, Green’s function, equilibrium measure.
ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України
22 On conformal invariants...
Denote by H := {z : ℑ(z) > 0} the upper half-plane and consider the
function
F (z) = FE(z) := exp
(∫
E
log(z − ζ) dµE(ζ) − log cap(E)
)
, z ∈ H.
(1.1)
It is analytic in H.
Since
gΩ(z) = log
1
cap(E)
−
∫
log
1
|z − t|dµE(t), z ∈ Ω, (1.2)
the function F has the following obvious properties:
|F (z)| = egΩ(z) > 1, z ∈ H,
ℑ(F (z)) = egΩ(z) sin
(∫
E
arg(z − ζ) dµE(ζ)
)
> 0, z ∈ H.
Moreover, F can be extended from H continuously to H such that
|F (z)| = 1, z ∈ E,
F (x) = egΩ(x) > 1, x ∈ R, x > 1,
F (x) = −egΩ(x) < −1, x ∈ R, x < −1.
Next, for any 1 ≤ j ≤ N and aj ≤ x1 < x2 ≤ bj , we have
arg
(
F (x2)
F (x1)
)
= arg exp
(∫
E
log
x2 − ζ
x1 − ζ
dµE(ζ)
)
= 0,
that is,
argF (x1) = argF (x2), aj ≤ x1 < x2 ≤ bj .
Our next objective is to show that F is univalent in H. We shall use
the following simple result. Let
√
z2 − 1, z ∈ C \ [−1, 1], be the analytic
function defined in a neighborhood of infinity as
√
z2 − 1 = z
(
1 − 1
2z2
+ . . .
)
.
Then for any −1 ≤ x ≤ 1 and z ∈ H,
ux(z) := ℜ
(√
z2 − 1
z − x
)
≥ 0. (1.3)
V. V. Andrievskii 23
Using the reflection principle we can extend F to a function analytic in
C \ [0, 1] by the formula
F (z) := F (z), z ∈ C \ H,
and consider the function
h(w) :=
1
F (J(w))
, w ∈ D := {w : |w| < 1},
where J is a linear transformation of the Joukowski mapping, namely
J(w) :=
1
2
(
1
2
(
w +
1
w
)
+ 1
)
,
which maps the unit disk D onto C\ [0, 1]. Note that the inverse mapping
is defined as follows
w = J−1(z) = (2z − 1) −
√
(2z − 1)2 − 1, z ∈ C \ [0, 1].
Therefore, for z ∈ H and w = J−1(z) ∈ D, we obtain
wh′(w)
h(w)
= w(log h(w))′ =
= −w
(∫
E
log(J(w) − ζ) dµE(ζ)
)′
= −w J ′(w)
∫
E
dµE(ζ)
z − ζ
=
= −1
4
(
w − 1
w
)∫
E
dµE(ζ)
z − ζ
=
1
2
∫
E
√
(2z − 1)2 − 1
z − ζ
dµE(ζ) =
=
∫
E
√
(2z − 1)2 − 1
(2z − 1) − (2ζ − 1)
dµE(ζ).
According to (1.3) for w under consideration we obtain
ℜ
(
wh′(w)
h(w)
)
≥ 0.
Because of the symmetry and the maximum principle for harmonic func-
tions we have
ℜ
(
wh′(w)
h(w)
)
> 0, w ∈ D.
It means that h is a conformal mapping of D onto a starlike domain (cf.
[35, p. 42]).
Hence, F is univalent and maps C \ [0, 1] onto a (with respect to ∞)
starlike domain C\KE with the following properties: C\KE is symmetric
24 On conformal invariants...
with respect to the real line R and coincides with the exterior of the unit
disk without 2N slits.
Note that
gΩ(z) = log |F (z)|, z ∈ C \ E. (1.4)
There is a close connection between the capacities of the compact sets
KE and E, namely
cap(E) =
1
4 cap(KE)
(cf. [5]).
The main idea of the results below is the investigation of the local
properties of the Green function gΩ, i.e., local properties of conformal
mapping F .
The paper is organized as follows. In section 2 we give a new inter-
pretation (and a generalization) of recent remarkable result by Totik [47]
concerning the smoothness properties of gΩ and µE . We also demonstrate
that if for E ⊂ [0, 1] the Green function satisfies the 1/2-Hölder condition
locally at the origin, then the density of E at 0, in terms of logarithmic
capacity, is the same as that of the whole interval [0, 1].
In section 3 we describe the connection between C-dense compact sets
and John domains. It allows us to extend classical Bernshtein’s theorem
about constructive description of infinitely differentiable functions to the
case of C-dense compact subset of R.
In section 4 the Nikol’skii-Timan-Dzjadyk theorem concerning poly-
nomial approximation of functions on the interval [−1, 1] is generalized
to the case of approximation of functions given on a compact set on the
real line.
A new necessary condition and a new sufficient condition for the ap-
proximation of the reciprocal of an entire function by reciprocals of poly-
nomials on [0,∞) with geometric speed of convergence are provided in
section 5.
In section 6 we give sharp uniform bounds for exponentials of loga-
rithmic potentials if the logarithmic capacity of the subset, where they
are at most 1, is known.
We shall use c, c1, c2, . . . to denote positive constants. These constants
may be either absolute or they may depend on E depending on the con-
text. We may use the same symbol for different constants if this does not
lead to confusion.
V. V. Andrievskii 25
2. On the Green function for a complement of a finite
number of real intervals
The main purpose of this section is to discuss the following recent
remarkable result by Totik [47].
Let E ⊂ [0, 1] be a compact set of positive logarithmic capacity. The
smoothness of gΩ and µE at 0 depends on the density of E at 0. This
smoothness can be measured by the function
θE(t) := |[0, t] \ E|, t > 0,
where | · | denotes linear Lebesgue measure.
Theorem 1. (Totik [47, (2.8) and (2.12)]) There are absolute positive
constants C1, C2, D1 and D2 such that for 0 < r < 1,
gΩ(−r) ≤ C1
√
r exp
(
D1
∫ 1
r
θ2
E(t)
t3
dt
)
log
2
cap(E)
, (2.1)
µE([0, r]) ≤ C2
√
r exp
(
D2
∫ 1
r
θ2
E(t)
t3
dt
)
. (2.2)
The results in [47] are formulated and proven for general compact sets
of the unit disk. The theorem above is one of the main steps in their ver-
ification. Even though the statement of this theorem is rather particular,
the theorem has several notable applications, such as Phragmén-Lindelöf
type theorems, Markov and Bernstein type, Remez and Schur type poly-
nomial inequalities, etc.
Observe that we can simplify geometrical nature of a compact set E
under consideration. Indeed, it is well-known that there exists a sequence
of compact sets En ⊂ [0, 1], n ∈ N := {1, 2, . . .}, such that
(i) E ⊂ En and each En consists of a finite number of closed intervals,
(ii) for 0 < r < 1, we have
gΩ(−r) = lim
n→∞
gΩn(−r), Ωn := C \ En,
µE([0, r]) = lim
n→∞
µEn([0, r]).
The set [0, 1] \ En is smaller and simpler then [0, 1] \ E. For example,
θEn(t) ≤ θE(t), t > 0.
However, gΩn and µEn can be arbitrary close to gΩ and µE . Thus, in
order to establish Totik type results it is natural to concentrate only on
compact sets consisting of a finite number of real intervals.
26 On conformal invariants...
Let
E = ∪k
j=1[aj , bj ], 0 ≤ a1 < b1 < a2 < . . . < ak < bk ≤ 1,
and let
E∗ := (0, 1)\E = ∪m
j=1(αj , βj), 0 ≤ α1 < β1 < α2 < . . . < αm < βm ≤ 1.
For 0 < r < 1, we set E∗
r := E∗ \ (0, r]. We are interested in the case
when E∗
r 6= ∅, i.e.,
E∗
r = ∪mr
j=1(αj,r, βj,r), r ≤ α1,r < β1,r < α2,r < . . . < αmr,r < βmr,r ≤ 1.
Theorem 2. ([4]) For 0 < r < 1
gΩ(−r) ≥ c1
√
r exp
d1
mr
∑
j=1
βj,r − αj,r
βj,r
log
βj,r
αj,r
, (2.3)
where c1 = 1/16, d1 = 10−13.
Theorem 2 provides the lower bound of the Green function (cf. [47,
(3.5)]). Since in (2.3) only the size of components of E∗
r influences this
bound, one cannot expect to find an upper bound of the same form. We
believe that in a Totik type theorem not only the size of the components
(αj,r, βj,r) but also their mutual disposition must be important.
We fix q > 1. The set of a finite number of closed intervals
{[δj , νj ]}n
j=1 = {[δj(r, q), νj(r, q)]}n
j=1,
where 0 ≤ δ1 < ν1 ≤ δ2 < . . . ≤ δn < νn ≤ 1, is called a q-covering of E∗
r
if
(i) E∗
r ⊂ ∪n
j=1[δj , νj ],
(ii) either 2δj ≤ νj , or q|E∗
r ∩ [δj , νj ]| ≤ νj − δj .
Theorem 3. For 0 < r < 1, q > 1 and any finite q-covering of E∗
r the
inequalities
gΩ(−r) ≤ c2
√
r exp
d2
n
∑
j=1
νj − δj
νj
log
νj
δj
log
2
cap(E)
, (2.4)
µE([0, r]) ≤ c3
√
r exp
d2
n
∑
j=1
νj − δj
νj
log
νj
δj
(2.5)
hold with c2 = 24, c3 = 5 and
d2 = max
(
1,
2q2
π(q − 1)2
)
.
V. V. Andrievskii 27
Notice that the factor log(2/cap(E)) on the right of (2.1) and (2.4)
appears only to cover pathological cases. It is useful to keep in mind that
|E| ≤ 4 cap(E) ≤ 1.
Corollary 1. The estimates (2.1) and (2.2) hold with C1 = 384, C2 = 80
and D1 = D2 = 120.
Corollary 2. For the compact set
Ẽ := {0} ∪
∞
⋃
n=1
n2
⋃
j=1
[
n2 + j − 1
2n+1n2
,
2n2 + 2j − 1
2n+2n2
]
.
we have
g
C\Ẽ(−r) ≤ c
√
r, 0 < r < 1,
with some absolute constant c > 0, which is better then (2.1).
Indeed, let
Ẽr := Ẽ ∩ [r, 1], 0 < r < 1.
For Ẽ∗
r = (r, 1) \ Ẽr with 2−k−2 < r ≤ 2−k−1 we construct a 2-covering
[r, 2−k],
{
{[
n2 + j − 1
2n+1n2
,
n2 + j
2n+1n2
]}n2
j=1
}k−1
n=1
,
[
1
2
, 1
]
.
By the monotonicity of the Green function and Theorem 3, for any 0 <
r < 1 and some absolute constant c > 0 we obtain
g
C\Ẽ(−r) ≤ g
C\Ẽr
(−r) ≤ c
√
r.
In what follows in this section we assume that 0 is a regular point of
E, i.e., gΩ(z) extends continuously to 0 and gΩ(0) = 0.
The monotonicity of the Green function yields
gΩ(z) ≥ g
C\[0,1](z), z ∈ C \ [0, 1],
that is, if E has the "highest density" at 0, then gΩ has the "highest
smoothness" at the origin. In particular
gΩ(−r) ≥ g
C\[0,1](−r) >
√
r
2
, 0 < r < 1.
In this regard, we would like to explore properties of E whose Green’s
function has the “highest smoothness" at 0, that is, of E conforming to
the following condition
gΩ(z) ≤ c|z|1/2, c = const > 0, z ∈ C,
28 On conformal invariants...
which is known to be the same as
lim sup
r→0
gΩ(−r)
r1/2
<∞ (2.6)
(cf. [41, Corollary III.1.10]). Various sufficient conditions for (2.6) in
terms of metric properties of E are stated in [47], where the reader can
also find further references.
There are compact sets E ⊂ [0, 1] of linear Lebesgue measure 0 with
property (2.6) (see e.g. [47, Corollary 5.2]), hence (2.6) may hold, though
the set E is not dense at 0 in terms of linear measure. On the contrary,
our first result states that if E satisfies (2.6) then its density in a small
neighborhood of 0, measured in terms of logarithmic capacity, is arbitrary
close to the density of [0, 1] in that neighborhood.
Theorem 4. ([5]) The condition (2.6) implies
lim
r→0
cap(E ∩ [0, r])
r
=
1
4
. (2.7)
Recall that cap([0, r]) = r/4 for any r > 0.
The converse of Theorem 4 is slightly weaker.
Theorem 5. ([5]) If E satisfies (2.7), then
lim
r→0
gΩ(−r)
r1/2−ε
= 0, 0 < ε <
1
2
. (2.8)
The connection between properties (2.6), (2.7) and (2.8) is quite del-
icate. For example, even a slight alteration of (2.6) can lead to the vio-
lation of (2.7). As an illustration of this phenomenon we formulate
Theorem 6. ([5]) There exists a regular compact set E ⊂ [0, 1] such
that (2.8) holds and
lim inf
r→0
cap(E ∩ [0, r])
r
= 0. (2.9)
3. Constructive description of infinitely differentiable
functions on a compact set of the real line
Let E ⊂ R be a compact set. Denote C(E) the set of all real functions
f continuous on E. Let Πn, n ∈ N, be the set of all polynomials with
real coefficients of degree at most n. For f ∈ C(E) and n ∈ N let
En(f,E) := inf
p∈Πn
||f − p||E ,
where || · ||E is the uniform norm over E.
We start with the following classical theorem.
V. V. Andrievskii 29
Theorem 7. (Bernstein [10]) The function f ∈ C([a, b]) is infinitely
differentiable on [a, b] iff
lim
n→∞
En(f, [a, b])nc = 0, c > 0. (3.1)
Our purpose is to describe the result analogous to Bernstein’s theorem
for functions given on a general compact set E instead of the interval.
The condition
lim
n→∞
En(f,E)nc = 0, c > 0. (3.2)
is obvious analogue of (3.1).
We have to pay more attention to the generalization of the notion
of differentiability on a subset of R. We follow the approach due to
H. Whitney [49] involving the Taylor formula (in finite form). That is,
we say that f =: f0 ∈ C(E) is of class C∞(E) if there exist functions
{fj(x)}∞j=1, {Rsm(x′, x)}∞s≤m=0, where x, x′ ∈ E, such that
fs(x
′) =
m
∑
j=s
fj(x)
(j − s)!
(x′ − x)j−s +Rsm(x′, x)
and for each s,m and every ε > 0 there is δ = δ(ε, s,m) > 0 such that
the inequality
∣
∣
∣
∣
Rsm(x′, x)
(x′ − x)m−s
∣
∣
∣
∣
< ε
holds for any x′, x ∈ E satisfying |x′ − x| < δ.
A description of f ∈ C∞(E) in terms of the divided differences of f
with respect to points of E can be found in [50]. A classical Whitney
theorem [49, Theorem 1] asserts that f ∈ C∞(E) iff f is a restriction on
E of some function infinitely differentiable on R. Hence, we have
Corollary 3. Let E ⊂ R be an arbitrary compact set. For any f ∈
C∞(E) the condition (3.2) holds.
This statement cannot be converted for an arbitrary set E (see [33],
[46]). However, the following result is valid.
Theorem 8. (Pleśniak [34]) Let E be such that the Green function gΩ
satisfies the Hölder condition, i.e., there are constants c > 0 and 0 <
α ≤ 1/2 such that
gΩ(z) ≤ cdist(z, E)α, z ∈ Ω. (3.3)
If f ∈ C(E) satisfies (3.2), then f ∈ C∞(E).
30 On conformal invariants...
Thus, Theorem 8 and Corollary 3 extend Bernstein’s theorem to com-
pact sets of R satisfying (3.3). It is natural to study metric properties
of such sets. It turns out that C-dense compact subsets of R known in
approximation theory of a complex variable satisfy (3.3). This can be
derived from the description of the connection between C-dense sets and
John domains well-known in geometrical function theory and the theory
of quasiconformal mappings.
Following Tamrazov [44, p. 61] we call E C-dense if
lim inf
t→0+
inf
x∈E
cap(E ∩ [x− t, x+ t])
t
> 0.
C-dense sets arise in many areas of complex analysis under the name
uniformly perfect sets. This class of sets was originally considered by
Beardon and Pommerenke [9]. Almost at the same time Pommerenke
[36] proved the equivalence of these notions (for general unbounded closed
sets in C).
Let E ⊂ [0, 1], 0 ∈ E, 1 ∈ E, and let F = FE be a conformal mapping
defined in section 1, i.e., F map C\[0, 1] onto C\KE =: ∆E . The domain
∆E is called a John domain (see [35, p. 96]) if, for every rectilinear
crosscut [z, ζ] of ∆E ,
diam(H) ≤ c |z − ζ|
holds with some constant c > 0 for the bounded component H of ∆E \
[z, ζ].
Theorem 9. ([6]) E is a C-dense compact set iff ∆E is a John domain.
This theorem builds a bridge between two quite different concepts of
analysis and can be used to study properties of C-dense sets. In partic-
ular, from general properties of John domains and (1.4) we obtain the
following.
Corollary 4. For any C-dense compact set E the Green function gΩ
satisfies (3.3).
Hence, infinitely differentiable functions given on a C-dense compact
sets can be characterized by the condition (3.2).
4. The Nikol’skii-Timan-Dzjadyk-type theorem
Let E ⊂ R be a compact set, and let ω(δ), δ > 0, be a function of
modulus of continuity type, i.e., a positive nondecreasing function with
ω(0+) = 0 such that for some constant c ≥ 1,
ω(tδ) ≤ c t ω(δ), δ > 0, t > 1.
V. V. Andrievskii 31
Let Cω(E) consist of all f ∈ C(E) such that
|f(x1) − f(x2)| ≤ c1 ω(|x2 − x1|), x1, x2 ∈ E,
with some c1 = c1(f) > 0.
For ω(δ) = δα, 0 < α ≤ 1, we set Cω(E) =: Cα(E).
One of the central problems in approximation theory is to describe the
relation between the smoothness of functions and the rate of decrease of
their approximation by polynomials when the degree of these polynomials
tends to infinity. The following well-known statement is the starting point
of our consideration.
Theorem 10 (Nikol’skii [32], Timan [45], Dzjadyk [17]). Let f ∈
C([−1, 1]) and let ω be a function of modulus of continuity type satisfying
the inequality
δ
1
∫
δ
ω(t)
t2
dt ≤ c2 ω(δ), 0 < δ < 1, (4.1)
with some constant c2 > 0. Then the following assertions are equivalent:
(i) f ∈ Cω([−1, 1]);
(ii) for any n ∈ N there exists pn ∈ Πn such that the inequality
|f(x) − pn(x)| ≤ c3 ω
(
1
n2
+
√
1 − x2
n
)
, −1 ≤ x ≤ 1, (4.2)
holds with some constant c3 > 0.
In the late 50s - early 60s Dzyadyk [18], [19] laid the foundation of a
new constructive theory of functions on continua in the complex plane (a
survey of the results and a bibliography can be found in the monographs
[20], [44], [24], [42], [7]). He used the following simple but fundamental
idea.
Denote by I1/n, n ∈ N, the ellipse with foci at ±1 and sum of semiaxes
equal to 1 + 1/n. Such an ellipse is the image of the circle {w : |w| = 1 +
1/n} under the conformal mapping z = 1
2(w+1/w) of ∆ := {w : |w| > 1}
onto C \ [−1, 1], i.e., I1/n is the level line of the conformal mapping
Φ(z) = z +
√
z2 − 1
of C \ [−1, 1] onto ∆, where the square root is chosen so that Φ(z) =
2z +O( 1
|z|) in a neighborhood of ∞.
Then for −1 ≤ x ≤ 1 and n ∈ N,
1
n2
+
√
1 − x2
n
≍ ρ1/n(x),
32 On conformal invariants...
where a ≍ b means a double inequality
a
c
≤ b ≤ c a
with some constant c ≥ 1, and
ρ1/n(x) := dist(x, I1/n),
where
dist(A,B) := inf
z∈A,ζ∈B
|z − ζ|, A,B ⊂ C.
The concepts of Cω,Φ, I1/n and ρ1/n(x) are also meaningful for an
arbitrary bounded continuum in the complex plane. This is the key to
a generalization of the Nikol’skii-Timan-Dzjadyk theorem to classes of
functions on continua in C.
If E ⊂ C is a compact set, then the interpretation of the Nikol’skii-
Timan-Dzjadyk theorem above can be corrected by consideration of the
Green function gΩ and its level lines. If E consists of a finite number of
components, Nguen Tu Than’ [31] has found a simple way to reduce the
problem to the case of a continuum. The case of infinitely connected Ω
is extremely difficult to handle. This can be seen from the recent paper
of Shirokov [43] (the first work devoted to this quite new situation).
We are going to discuss the case E ⊂ R, where the number of compo-
nents of E can be infinite. It turns out that the appropriate analogue of
the Nikol’skii-Timan-Dzjadyk theorem is valid for some E that are not
"too scarce" (see Theorem 12) and that in general a result of such kind
is not true (see Theorem 11).
More precisely, let E ⊂ R be a regular compact set. For δ > 0 and
z ∈ C set
Eδ := {z ∈ Ω : gΩ(z) = δ},
ρδ(z) := dist(z, Eδ).
It turns out that even for f ∈ Cα(E), polynomials satisfying an analogue
of (4.2) cannot be constructed for any E under consideration.
Theorem 11. ([1]) There exist a regular compact set E0 ⊂ R and for
any 0 < α ≤ 1 a function fα ∈ Cα(E0) such that the following assertion
is false: for any n ∈ N there is a polynomial pn ∈ Πn with the property:
|fα(x) − pn(x)| ≤ c ρα
1/n(x), x ∈ E0, (4.3)
where the constant c > 0 is independent of n and x.
V. V. Andrievskii 33
The analysis of the proof of Theorem 11 shows that E0 is “too scarce"
in a neighbourhood of 0 ∈ E0. Hence, to admit estimates like (4.2) or
(4.3), E has to be “thick enough" in a neighbourhood of each of its points.
In order to formulate the appropriate restrictions we need some notations.
The set R \ E consists of a finite or infinite number of components,
i.e., disjoint open intervals.
We say that E ∈ E(α, c), α > 0, c > 0, if for any bounded component
J of R \ E the inequality
dist(J, (R \ E) \ J) ≥ c |J |1/(1+α)
holds.
By definition, we relate a single closed interval to E(α, c).
We can now state the analogue of the Nikol’skii-Timan-Dzjadyk the-
orem for functions continuous on a compact subset of the real line.
Theorem 12. ([1]) Let the regular set E ⊂ R consist of a finite number
of disjoint compact sets, each of which belongs to the class E(α, c) with
some α, c > 0. Suppose that f ∈ C(E) and that the function ω of the
modulus of continuity type satisfies (4.1).
Then the following conditions are equivalent:
(i) f ∈ Cω(E);
(ii) for any n ∈ N there exists a polynomial pn ∈ Πn such that
|f(x) − pn(x)| ≤ c1 ω(ρ1/n(x)), x ∈ E,
where the constant c1 > 0 does not depend on x and n.
The simplest example of E satisfying the assumptions of Theorem 12
is the union of a finite number of disjoint closed intervals. The compact
set
Eα := {0} ∪
∞
⋃
n=nα
[
1
n+ 1
,
1
n
− 1
n2+α
]
, α > 0, nα > 21/α,
which obviously satisfies the conditions of Theorem 12, illustrates a non-
trivial extension of (4.2) to compact subsets of the real line.
If E consists of an infinite number of components, then taking ±1 as
values of f on different subintervals of E (that is, f ′ ≡ 0 on E) we can
construct a function f which is not even continuous on E. It shows that
unlike to the case of an interval, Theorem 12 does not in general admit
a generalization to the classes of functions continuously differentiable on
E with given majorant for the modulus of continuity of their derivatives.
34 On conformal invariants...
5. Approximation on an unbounded interval
Let R
+ denote the non–negative real axis. We will consider functions
f , continuous in the complex plane C, real valued on R
+ and possessing
also the basic properties
f > 0 on R
+, lim
x→∞
f(x) = ∞ . (5.1)
For every positive integer n ∈ N, we define ρn(f) as follows:
ρn(f) := inf
p∈Πn
||1/f − 1/p||R+ . (5.2)
In the present section, we discuss necessary and sufficient conditions
for the geometric convergence of reciprocals of polynomials to the recip-
rocal of the function f on the half–axis R
+, i.e., we shall discuss the
inequality
lim sup
n→∞
ρn(f)1/n =
1
q
< 1 . (5.3)
The first results in this area were due to Cody, Meinardus and Varga
[15] concerning the function exp(x). Later, Meinardus and Varga [28]
extended these results to the class of entire functions of completely regular
growth. The paper [30] gave rise to investigations devoted to enlarging
the class of functions that admit geometric approximation by reciprocals
of polynomials on R
+.
We introduce some notations. Given numbers r > 0 and s > 1, denote
by Er(s) the closed ellipse with foci at the points x = 0 and x = r such
that the ratio between the semimajor axis and semiminor axis equals
(s2 + 1)/(s2 − 1). Further, let µ(r) := minx≥r{f(x)}.
The remarkable result concerning necessary conditions for geometric
convergence is the following
Theorem 13. (Meinardus [29], Meinardus, Reddy, Taylor, Varga [30])
Let f satisfy (5.3). Then
(i) the function f can be extended from R
+ to an entire function of
finite order,
(ii) for every number s > 1, there exist positive constants c1 =
c1(s, q), θ = θ(s, q) and r0 = r0(s, q) such that the inequality
||f ||Er(s) ≤ c1||f ||θ[0,r] (5.4)
holds for all r ≥ r0.
After the appearance of [30], a lot of work was done to find sufficient
conditions for (5.3) (cf. [11]–[13], [27], [38], [40]). The most general
known result in this direction is the following
V. V. Andrievskii 35
Theorem 14. (Blatt, Kovacheva [13]) Assume that f is an entire func-
tion with (5.1) and, in addition to condition (5.4), the inequality
||f ||[0,r] ≤ µ(r)λ (5.5)
holds for some number λ > 1 and for every r > r0. Then (5.3) is true.
On the other hand, Henry and Roulier [27] have shown that the con-
ditions (i) and (ii) of Theorem 13 are not sufficient for geometric conver-
gence. For example, in [27] it was proved that
f(x) = 1 + x+ ex sin2 x (5.6)
cannot be approximated with geometric speed. Their proof was based on
the fact that f satisfying (5.3) cannot oscillate too often.
The main aim of this section is to discuss a new necessary and a
new sufficient condition for geometric convergence. In this context it is
important to note that up to now all proofs for geometric convergence
are based on the classical Bernstein theorem [10], i.e.,
En(f, [0, 1]) ≤ c ||f ||Er(s) s
−n , (5.7)
where the function f is analytic in Er(s) and the constant c is independent
of the interval [0, r]. In our reasoning, we use a new generalization of (5.7)
for a finite number of intervals, new in the sense that the constant c is
independent of the geometry (Theorem 19).
We begin with a necessary condition for geometric convergence of best
approximants (5.2). Let f be as above, i.e., f is an entire function with
(5.1). For r > 0, we define the set
Zr := { 0 ≤ x <∞ : f(x) < r } .
Then Zr is the union of a finite number of disjoint open intervals. This
follows from (5.1) and the uniqueness theorem for analytic functions.
Now, we consider the closure Z̄r of Zr, which is regular and possesses
a Green’s function gr(z) with respect to the region C \ Z̄r with pole at
infinity, where gr := 0 on Z̄r. For s > 1 we denote by Er(f, s) the set
which consists of the interior of the level set of gr(z) and the level set
itself for a fixed parameter s, i.e.,
Er(f, s) := { z ∈ C : 0 ≤ gr(z) ≤ log s } .
Then the new necessary condition for geometric convergence can be for-
mulated as follows.
36 On conformal invariants...
Theorem 15. ([8]) Let f satisfy (5.3). Then for every 1 < s < q there
exist positive constants c = c(s, q), θ = θ(s, q) and r0 = r0(s, q) such that
||f ||Er(f,s) ≤ c rθ, r ≥ r0 . (5.8)
Next, we are going to discuss the geometrical meaning of condition
(5.8). For ∞ > H > h > minx∈R+ f(x) > 0, we introduce the strip
domain
S(h,H) := { (x, y) : −∞ < x <∞, h < y < H }
as well as the intersection of this strip with the graph of f , i.e.,
Y (f, h,H) := S(h,H) ∩ { (x, y) : x ≥ 0, y = f(x) } ,
and define N(f, h,H) to be the number of connected components of
Y (f, h,H) joining the line {ℑz = h} with the line {ℑz = H}. Since
f satisfies (5.1), the number N(f, h,H) is finite and, moreover, it is odd.
Theorem 16. ([8]) Let f be entire and satisfy (5.1). If, in addition,
for some s > 1 and θ > 1, the function f satisfies (5.8), then, for each
M > θ,
lim sup
h→∞
N(f, h, hM )
log h
<∞ . (5.9)
Note that the result of Theorem 16 is sharp in the following sense:
For each M > 1 there exists an entire function f = fM which satisfies
(5.8) with some s > 1 and 1 < θ < M and
lim sup
h→∞
N(f, h, hM )
log h
> 0 . (5.10)
Indeed, consider the function
fM (x) := ex + e2Mx sin2 πx .
Obviously, it satisfies the conditions of Theorem 14. Therefore, fM guar-
antees the geometrical convergence of best approximants in the sense
of (5.3), and, by Theorem 15, f satisfies (5.8) in which we can take s
so close to 1 that θ < M . The relation (5.10) easily follows, if we set
h = ek, k ∈ N, and let k → ∞.
The sufficient condition for geometrical convergence of best approxi-
mants can be stated in the following form.
V. V. Andrievskii 37
Theorem 17. ([8]) Let f be entire and satisfy (5.1) and (5.8) with
some s > 1 and θ > 1. In addition, assume that there exists a constant
M = M(f) > 1 such that
lim sup
h→∞
N(f, h, hM ) <∞ . (5.11)
Then f satisfies (5.3).
It is easy to see that Theorem 14 follows from Theorem 17, because
under the assumptions of Theorem 14, for M > λ and h sufficiently
large, we have N(f, h, hM ) = 1. At the same time, condition (5.4) is
weaker than (5.8). The simple example of the function (5.6) shows that
conditions (5.4) and (5.8) are not equivalent. Indeed, f given by (5.6)
obviously satisfies (5.4). On the other hand, some simple calculations
show that relation (5.9) does not hold for this function. Thus, f does not
possess (5.8).
The fact that Theorem 17 is essentially stronger than Theorem 14 is
not so obvious.
Theorem 18. ([8]) There exists an entire function f satisfying the as-
sumptions of Theorem 17, but not possessing property (5.5).
The proof of Theorem 17 is based on an analogue of the classical re-
sult due to Bernstein, concerning polynomial approximation of functions
analytic in the neighborhood of a subinterval of the real axis, for the case
of several intervals.
Let E =
⋃k
j=1 Ij be the union of k disjoint intervals Ij = [αj , βj ] of
the real axis R and let Ω := C\E. The set
Es := {z ∈ Ω : gΩ(z) = log s }, s > 1,
consists of at most k (mutually exterior) curves. Denote by ext(Es) the
unbounded component of C\Es and set int(Es) := C\ext(Es). Moreover,
let the function f ∈ C(E) satisfy the following two conditions:
For some s > 1, f can be extended analytically into int(Es) , (5.12)
f has at least one zero on each Ij . (5.13)
Theorem 19. ([8]) For each function f satisfying (5.12) and (5.13),
there exist constants q > 1 and c > 0 depending only on s and k such
that
En(f,E) ≤ c ||f ||Es q−n, n ∈ N . (5.14)
Note that (5.14) can be interpreted as a result concerning geometric
convergence of the polynomials of best approximation to the function f ,
independent of the geometry of E.
38 On conformal invariants...
6. Remez-type inequalities in terms of capacity
The Remez inequality [39] (see also [22, 14, 25]) asserts that
||pn||I ≤ Tn
(
2 + s
2 − s
)
(6.1)
for every polynomial pn ∈ Πn such that
|{x ∈ I : |pn(x)| ≤ 1}| ≥ 2 − s, 0 < s < 2, (6.2)
where I := [−1, 1] and Tn is the Chebyshev polynomial of degree n.
Since
Tn(x) ≤ (x+
√
x2 − 1)n, x > 1,
we have by (6.1) that a polynomial pn with (6.2) satisfies
||pn||I ≤
(√
2 +
√
s√
2 −√
s
)n
. (6.3)
It is easy to see that the last inequality (more precisely its n-th root) is
asymptotically sharp.
Our aim is to discuss an analogue of (6.2)-(6.3) in which we use log-
arithmic capacity instead of the length. Our main result deals not only
with polynomials, but also with exponentials of potentials (see [22, 23]).
We refer to the basic notions of potential theory (such as capacity, po-
tential, Green’s function, equilibrium measure, Fekete polynomials, etc.)
without special citations. All these notions and their properties can be
found in [48, 35, 41].
Given a nonnegative Borel measure ν with compact support (in the
complex plane C) and finite total mass ν(C) > 0 as well as a constant
c ∈ R, we say that
Qν,c(z) := exp(c− Uν(z)), z ∈ C,
where
Uν(z) :=
∫
log
1
|ζ − z| dν(ζ), z ∈ C,
is the logarithmic potential of ν, is an exponential of a potential of degree
ν(C).
Let
Eν,c := {x ∈ I : Qν,c(x) ≤ 1}.
Theorem 2.1 and Corollary 2.11 in [23] assert that for 0 < s < 2 the
condition
|Eν,c| ≥ 2 − s (6.4)
V. V. Andrievskii 39
implies
||Qν,c||I ≤
(√
2 +
√
s√
2 −√
s
)ν(C)
. (6.5)
Our result can be formulated as follows.
Theorem 20. ([2] Let 0 < δ < 1/2. Then the condition
cap(Eν,c) ≥
1
2
− δ (6.6)
yields that
||Qν,c||I ≤
(
1 +
√
2δ
1 −
√
2δ
)ν(C)
. (6.7)
Remark 1 Since |Eν,c| ≤ 4 cap(Eν,c) [35, p. 337], the assertion (6.4)-
(6.5) follows from (6.6)-(6.7).
Remark 2 For 0 < δ < 1/2, set
ν = νδ := µ[−1,1−4δ], c = cδ := log
2
1 − 2δ
,
where µE denotes the equilibrium measure of a compact set E ⊂ C.
Then ν(C) = 1,
Eν,c = [−1, 1 − 4δ],
Qν,c(x) =
1
1 − 2δ
(
x+ 2δ + ((x+ 2δ)2 − (1 − 2δ)2)1/2
)
, x ≥ 1 − 4δ.
Therefore, in this case
cap(Eν,c) =
1
2
− δ,
||Qν,c||I = Qν,c(1) =
1 +
√
2δ
1 −
√
2δ
,
which shows the sharpness of Theorem 20.
Remark 3 Let pn(z) = c
∏n
j=1(z − zj), 0 6= c ∈ C, be a complex
polynomial of degree n, and let
νn :=
n
∑
j=1
δzj ,
where δz is the Dirac unit measure in a point z ∈ C. For z ∈ C, we have
Qνn,log |c| = exp (log |c| + log
n
∏
j=1
|z − zj |) = |pn(z)|.
40 On conformal invariants...
Therefore, applying the theorem we obtain for 0 < δ < 1/2: the condition
cap({x ∈ I : |pn(x)| ≤ 1}) ≥ 1
2
− δ
implies
||pn||I ≤
(
1 +
√
2δ
1 −
√
2δ
)n
(cf. (6.2)-(6.3)).
Remark 4 The previous remark can be rewritten in a form as in [16,
Theorem 1.1]. Namely, let r > 0 and pn be a complex polynomial of
degree n such that ||pn||[−r,r] = 1. Then for 0 < ε < 1,
cap({x ∈ [−r, r] : |pn(x)| ≤ εn}) ≤ 2rε
(1 + ε)2
.
This inequality is asymptotically sharp for any fixed ε and r.
Note that Theorem 20 is a straightforward consequence of its following
particular case.
Lemma 1. ([2]) Let E ⊂ I be a compact set with 0 < cap(E) < 1/2.
Then
sup
x∈I\E
g
C\E(x) ≤ log
(
1 + (1 − 2 cap(E))1/2
1 − (1 − 2 cap(E))1/2
)
.
Next we present an analogue of results above for complex polynomials.
Let |A| be the linear measure (length) of a set A in the complex plane
C. By Pn we denote the set of all complex polynomials of degree at most
n ∈ N. Let
Π(p) := {z ∈ C : |p(z)| > 1}, p ∈ Pn.
From the numerous generalizations of the Remez inequality, we cite
one result which is a direct consequence of the trigonometric version of
the Remez inequality (and is equivalent to this trigonometric version, up
to constants).
Assume that p ∈ Pn, T := {z : |z| = 1} and
|T ∩ Π(p)| ≤ s, 0 < s ≤ π
2
. (6.8)
Then, q(t) := |p(eit)|2 is a trigonometric polynomial of degree at most n
and, by the Remez-type inequality on the size of trigonometric polyno-
mials (cf. [21, Theorem 2], [14, p. 230]), we have
||p||T ≤ e2sn, 0 < s ≤ π
2
. (6.9)
V. V. Andrievskii 41
Here, || · ||A means the uniform norm along A ⊂ C.
Our next aim is to discuss an analogue of (6.8)-(6.9) in which we
use logarithmic capacity instead of the length. As before our main result
deals not only with polynomials, but also with exponentials of potentials.
Theorem 21. ([3]) Let 0 < δ < 1. Then, the condition
cap(Eν,c) ≥ δ
implies that
||Qν,c||T ≤
(
1 +
√
1 − δ2
δ
)ν(C)
.
Remark 5 In order to examine the sharpness of Theorem 21 we
consider the following example.
Let 0 < α < π/2, and let L = Lα := {eiθ : 2α ≤ θ ≤ 2π − 2α}. Since
the function
z = Ψ(w) = −w w − a
1 − aw
,
where a = 1/ cosα, maps ∆ onto Ω := C\L (cf. [26]) and since the Green
function of Ω with pole at ∞ can be defined via the inverse function
Φ := Ψ−1 by the formula
gΩ(z) = log |Φ(z)|, z ∈ Ω,
we have
cap(L) = lim
w→∞
Ψ(w)
w
=
1
a
= cosα, (6.10)
as well as
max
z∈T\L
gΩ(z) = gΩ(1) = log |Φ(1)|
= log(a+
√
a2 − 1) = log
1 +
√
1 − cap(L)2
cap(L)
. (6.11)
Let c = cα := − log cap(L) and let ν = να be the equilibrium measure
for L; that is, ν(C) = 1. Since for z ∈ C,
Uν(z) = −g
C\L(z) − log cap(L),
and therefore
Qν,c(z) = exp (g
C\L(z)),
we have Eν,c = L as well as
||Qν,c||T =
1 +
√
1 − cap(L)2
cap(L)
.
42 On conformal invariants...
This shows the exactness of Theorem 21.
Remark 6 Let p(z) = c
∏n
j=1(z − zj), 0 6= c ∈ C, be a complex
polynomial of degree n, and let
νn :=
n
∑
j=1
δzj .
For z ∈ C, we have
Qνn,log |c| = exp (log |c| + log
n
∏
j=1
|z − zj |) = |p(z)|.
Therefore, applying the above theorem, we obtain the following, for 0 <
δ < 1: For p ∈ Pn the condition
cap(T \ Π(p)) ≥ δ (6.12)
implies
||p||T ≤
(
1 +
√
1 − δ2
δ
)n
. (6.13)
Remark 7 Since for any E ⊂ T we have cap(E) ≥ sin |E|
4 (see [37]),
(6.12)-(6.13) imply the following refinement of (6.8)-(6.9): For p ∈ Pn the
condition
|T ∩ Π(p)| ≤ s, 0 < s < 2π, (6.14)
implies
||p||T ≤
(
1 + sin s
4
cos s
4
)n
. (6.15)
This result is also sharp in the following sense. Let 0 < s < 2π, α = s/4,
and let L = Lα be defined as in Remark 5. By (6.10) and (6.11),
gΩ(1) = log
1 + sin s
4
cos s
4
.
We denote by fn(z) the n-th Fekete polynomial for a compact set L (see
[41]). Hence, condition (6.14) holds for the polynomial p(z) = pn(z) :=
fn(z)/||fn||L. At the same time, since
lim
n→∞
( |fn(z)|
||fn||L
)1/n
= exp(gΩ(z)), z ∈ Ω \ {∞}
(see [41, p. 151]), we have
lim
n→∞
|p(1)|1/n =
1 + sin s
4
cos s
4
.
V. V. Andrievskii 43
Theorem 21 is a straightforward consequence of its following particu-
lar case.
Lemma 2. ([3]) Let E ⊂ T be a compact set with 0 < cap(E) < 1.
Then,
sup
z∈T\E
g
C\E(z) ≤ log
(
1 +
√
1 − cap(E)2
cap(E)
)
.
In our approach, we exploit the following simple connection between
estimates which express the possible growth of a polynomial with a known
norm on a given compact set E ⊂ C and the behavior of the Green’s
function for C \ E.
Let the logarithmic capacity of E be positive and let Ω := C \ E
be connected. For z ∈ Ω and u > 0, the following two conditions are
equivalent:
(i) gΩ(z) ≤ u;
(ii) for any p ∈ Pn and n ∈ N,
|p(z)| ≤ eun||p||E .
Indeed, (i) ⇒ (ii) follows from the Bernstein–Walsh lemma and (ii)
⇒ (i) is a simple consequence of a result by Myrberg and Leja (see [35,
p. 333]).
We study the properties of the Green function by methods of geomet-
ric function theory (involving symmetrization) which allows, according
to the implication (i) ⇒ (ii), to obtain results similar to (6.12) - (6.13).
Note that the sharpness of results for the Green function, by virtue of
the equivalence of (i) and (ii), implies the (asymptotic) sharpness of the
corresponding Remez-type inequalities for polynomials (6.12) - (6.13).
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Contact information
V. V. Andrievskii Department of Mathematical Sciences, Kent
State University, Kent, OH, 44242
E-Mail: andriyev@mcs.kent.edu
|
| id | nasplib_isofts_kiev_ua-123456789-124608 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | English |
| last_indexed | 2025-11-26T06:26:33Z |
| publishDate | 2004 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Andrievskii, V.V. 2017-09-30T08:04:40Z 2017-09-30T08:04:40Z 2004 On conformal invariants in problems of constructive function theory on sets of the real line / V.V. Andrievskii // Український математичний вісник. — 2004. — Т. 1, № 1. — С. 21-46. — Бібліогр.: 50 назв. — англ. 1810-3200 2000 MSC. 30C10, 30E10, 41A20, 31A15, 41A10, 41A25. https://nasplib.isofts.kiev.ua/handle/123456789/124608 This is a survey of some recent results by the author and his collaborators in the constructive theory of functions of a real variable. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory in the complex plane. en Інститут прикладної математики і механіки НАН України Український математичний вісник On conformal invariants in problems of constructive function theory on sets of the real line Article published earlier |
| spellingShingle | On conformal invariants in problems of constructive function theory on sets of the real line Andrievskii, V.V. |
| title | On conformal invariants in problems of constructive function theory on sets of the real line |
| title_full | On conformal invariants in problems of constructive function theory on sets of the real line |
| title_fullStr | On conformal invariants in problems of constructive function theory on sets of the real line |
| title_full_unstemmed | On conformal invariants in problems of constructive function theory on sets of the real line |
| title_short | On conformal invariants in problems of constructive function theory on sets of the real line |
| title_sort | on conformal invariants in problems of constructive function theory on sets of the real line |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/124608 |
| work_keys_str_mv | AT andrievskiivv onconformalinvariantsinproblemsofconstructivefunctiontheoryonsetsoftherealline |