Application of Dzyadyk’s polynomial kernels in the constructive function theory
This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzjadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507175801061376 |
|---|---|
| author | Andrievskii, V. V. Андрієвський, В. В. |
| author_facet | Andrievskii, V. V. Андрієвський, В. В. |
| author_sort | Andrievskii, V. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:43Z |
| description | This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through
the application of the theory of Dzjadyk’s kernels combined with the methods and results from modern geometric function
theory and the theory of quasiconformal mappings. |
| first_indexed | 2026-03-24T02:05:08Z |
| format | Article |
| fulltext |
UDC 517.5
V. Andrievskii (Kent. State Univ., USA)
APPLICATION OF DZYADYK’S POLYNOMIAL KERNELS
IN THE CONSTRUCTIVE FUNCTION THEORY
ЗАСТОСУВАННЯ ПОЛIНОМIАЛЬНИХ ЯДЕР ДЗЯДИКА
В КОНСТРУКТИВНIЙ ТЕОРIЇ ФУНКЦIЙ
This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through
the application of the theory of Dzjadyk’s kernels combined with the methods and results from modern geometric function
theory and the theory of quasiconformal mappings.
Наведено огляд нових результатiв у конструктивнiй теорiї функцiй, що отриманi автором iз застосуванням теорiї
ядер Дзядика в поєднаннi з методами та результатами сучасної геометричної теорiї функцiй i теорiї квазiконформних
вiдображень.
1. Introduction. In [9] (see also [10], Chapter IX, \S 7) Dzjadyk introduced and thoroughly re-
searched his polynomial kernels Kr,m,k,n(\zeta , z). We believe that these kernels possess the strongest
approximation properties out of all the kernels approximating the Cauchy kernel 1/(\zeta - z). This is a
survey of some recent results in the constructive theory of functions of complex variable obtained by
the author through the application of the theory of Dzjadyk’s kernels Kr,m,k,n(\zeta , z) combined with
the methods and results from modern geometric function theory and the theory of quasiconformal
mappings.
The paper is organized as follows. In Section 2, we discuss a conjecture on the rate of polyno-
mial approximation on the compact set of the plane to a complex extension of the absolute value
function. The conjecture was stated by Grothmann and Saff in 1988 (see [12]). We also mention
Gaier’s conjecture on the polynomial approximation of piecewise analytic functions on a compact set
consisting of two touching discs.
Section 3 is devoted to a Jackson – Mergelyan type theorem on approximation of a function by
reciprocals of complex polynomials. The function is continuous on a quasismooth (in the sense of
Lavrentiev) arc in \BbbC .
2. Polynomial approximation on touching domains. In connection with the distribution of
the zeros of some “near best” approximating polynomials, Grothmann and Saff stated the following
conjecture.
Let \BbbP n be the set of complex algebraic polynomials of degree at most n \in \BbbN \cup \{ 0\} , where
\BbbN := \{ 1, 2, . . .\} . For a compact set K \subset \BbbC , n \in \BbbN , and a function f : K \rightarrow \BbbC , set
En(f,K) := \mathrm{i}\mathrm{n}\mathrm{f}
p\in \BbbP n
\| f - p\| K ,
where we use the notation \| f\| K := \mathrm{s}\mathrm{u}\mathrm{p}z\in K | f(z)| for the uniform norm.
For \alpha > 0, consider the piecewise analytic function
f\alpha (z) =
\left\{
z\alpha if \Re (z) > 0,
( - z)\alpha if \Re (z) < 0,
0 if z = 0,
(2.1)
c\bigcirc V. ANDRIEVSKII, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2 151
152 V. ANDRIEVSKII
which is the “analytic continuation” of | x| \alpha , x \in \BbbR , so that f1(x) = | x| for the real x \in \BbbR .
Grothmann and Saff conjectured that if
K =
\bigl\{
z \in \BbbC : | \Re (z)| \leq 2, | \Im (z)| \leq | \Re (z)| 2
\bigr\}
is a closed “parabolic region”, then
En(f1,K) = O(n - 1) as n\rightarrow \infty . (2.2)
Since En(f1,K) \geq En
\bigl(
| x| , [ - 1, 1]
\bigr)
, the rate of approximation in (2.2) cannot be improved.
Saff and Totik [23] confirmed the above conjecture by constructing polynomials that “overcon-
verge” to f on certain compact subsets of \BbbC \setminus i\BbbR , where i\BbbR :=
\bigl\{
z \in \BbbC : \Re (z) = 0
\bigr\}
. By the way,
another proof of (2.2) can be derived from the results of Dzjadyk [10, p. 440] (Theorem 1).
Motivated by the Grothmann and Saff conjecture, Anderson and Fuchs [1], in the same 1988,
inquired about the structure of a compact set E with [0, 1] \subset E \subset
\bigl\{
z \in \BbbC : \Re (z) > 0
\bigr\}
\cup \{ 0\}
such that f1 satisfies (2.2) with K = E \cup ( - E). Here - E :=
\bigl\{
z \in \BbbC : - z \in E
\bigr\}
. In Theorem 3
below, we use purely geometric terms to give full description to the type of “touching domains” with
the above property. Following [7, p. 322], we call such sets continua with the de la Vallée Poussin
property, or, for short, VP-property.
Also motivated by the Grothmann and Saff conjecture, Gaier [11] considered a more general
problem where he suggested to approximate the function f\alpha for arbitrary \alpha > 0 by polynomials. One
of his major results can be stated as follows. Let E be a closed domain in
\bigl\{
z \in \BbbC : \Re (z) > 0
\bigr\}
\cup \{ 0\}
which is symmetric with respect to \BbbR and bounded by a Jordan curve passing through z = 0 which
is smooth except at the origin. Let the upper half of \partial E be a Jordan arc J represented by
J : y = g(x), 0 \leq x \leq A, (2.3)
with g(0) = g(A) = 0. The behavior of J is only of consequence near z = 0.
Theorem 1 [11]. Let 0 < \alpha \leq 1 and assume that g in (2.3) satisfies the following conditions:
for some 0 < a < A, the function g(x)/x is continuous and increasing on (0, a) and
a\int
0
g(x)
x2
dx <\infty .
Then
En
\bigl(
f\alpha , E \cup ( - E)
\bigr)
= O(n - \alpha ) as n\rightarrow \infty . (2.4)
Hence, E\cup ( - E) from Theorem 1 possesses the VP-property. In Theorem 4 below, we generalize
(2.4) to the case of an arbitrary continuum with the VP-property and \alpha > 0.
In the same paper [11], Gaier also studied other types of continua. The case of two touching
discs seems to be the most difficult for his analysis. Let
D(z, r) :=
\bigl\{
\zeta \in \BbbC : | \zeta - z| < r
\bigr\}
, z \in \BbbC , r > 0.
Theorem 2 [11]. Let \alpha > 1 and assume that K satisfies
D(0, a) \cap K = D(0, a) \cap
\bigl(
D( - A,A) \cup D(A,A)
\bigr)
for some 0 < a < A. Then
En(f\alpha ,K) = O
\bigl(
(\mathrm{l}\mathrm{o}\mathrm{g} n) - \alpha +1
\bigr)
as n\rightarrow \infty . (2.5)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
APPLICATION OF DZYADYK’S POLYNOMIAL KERNELS IN THE CONSTRUCTIVE FUNCTION THEORY 153
Gaier conjectured that (2.5) holds with the exponent - \alpha instead of - \alpha + 1 and for all \alpha > 0.
This conjecture is surprising since the best known estimate of En(f\alpha ,K), 0 < \alpha \leq 1, in this case, is
O((\mathrm{l}\mathrm{o}\mathrm{g} n) - \alpha /2). The result follows from the classical Mergelyan theorem (see [25], Chapter 1, \S 7,
Theorem 8).
Next, we formulate the main results of this section. Let G1 and G2 be open, bounded Jordan
domains such that
[ - 1, 0) \subset G1, G1 \subset
\bigl\{
z \in \BbbC : \Re (z) < 0
\bigr\}
\cup \{ 0\} ,
(0, 1] \subset G2, G2 \subset
\bigl\{
z \in \BbbC : \Re z > 0
\bigr\}
\cup \{ 0\} .
Following [7, p. 322], we say that a continuum
K := G1 \cup G2 (2.6)
consisting of two touching domains has the VP-property, if for f1, defined by (2.1), (2.2) holds.
From now on till the end of this section, we always assume that the continuum K is defined by
(2.6), usually without further mention. Let \phi : [0, 1] \rightarrow \BbbR + := [0,\infty ) be a continuous function such
that \phi (x)/x is increasing on (0, 1) and \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow 0+(\phi (x)/x) = 0. For 0 < \varepsilon \leq 1, let
S\varepsilon (\phi ) :=
\bigl\{
z \in \BbbC :
\bigm| \bigm| \Re (z)\bigm| \bigm| \leq \varepsilon , | \Im (z)| \leq \phi
\bigl( \bigm| \bigm| \Re (z)\bigm| \bigm| \bigr) \bigr\} .
According to [7, p. 331] (Theorem 1.4), if K has the VP-property, then there exist a sufficiently
small constant \varepsilon and a function \phi as above such that
K2\varepsilon := K \cap D(0, 2\varepsilon ) \subset S2\varepsilon (\phi ). (2.7)
Without loss of generality, we can assume that\bigm| \bigm| \bigm| \bigm| \Im (z)\Re (z)
\bigm| \bigm| \bigm| \bigm| < 0.01, z \in K2\varepsilon . (2.8)
For z \in \BbbC \setminus \BbbR , denote by \Delta z the closed equilateral triangle with one vertex at z and one side on \BbbR .
Consider the “envelope” of K\delta , 0 < \delta \leq 2\varepsilon , i.e., the set
E\delta = E\delta (K) :=
\bigcup
z\in \partial K\cap D(0,\delta )
\Delta z \supset K\delta ,
where \Delta 0 := \{ 0\} .
Let the nonnegative functions \psi \pm : [ - \varepsilon , \varepsilon ] \rightarrow \BbbR be defined by the condition\bigl\{
z \in E2\varepsilon : | \Re (z)| \leq \varepsilon
\bigr\}
=
\bigl\{
z \in \BbbC : | \Re (z)| \leq \varepsilon , - \psi - (\Re (z)) \leq \Im (z) \leq \psi +(\Re (z))
\bigr\}
. (2.9)
Theorem 3 [5]. The continuum K has the VP-property if and only if both conditions below
hold:
(i) there exist \varepsilon and \phi satisfying (2.7) and (2.8);
(ii) for \psi \pm defined by (2.9),
\varepsilon \int
- \varepsilon
\psi \pm (x)
x2
dx <\infty .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
154 V. ANDRIEVSKII
Since conditions (i) and (ii) have purely geometric nature, Theorem 3 provides a natural and
intrinsic characterization of continua with the VP-property. In particular, it provides the justification
of the Grothmann and Saff conjecture: take \phi (x) = \psi \pm (x) = x2.
Next, for \alpha > 0, denote by PA\alpha (K) the class of “piecewise analytic” functions
f(z) =
\left\{
g1(z) if z \in G1,
g2(z) if z \in G2,
0 if z = 0,
with the following two properties:
(i) gj , j = 1, 2, has analytic continuation to \BbbC \setminus
\bigl\{
x \in \BbbR : ( - 1)j - 1x \geq 0
\bigr\}
;
(ii) for z \in \BbbC \setminus \BbbR and j = 1, 2, we have | gj(z)| \leq | z| \alpha .
Theorem 4 [5]. Let K have the VP-property. Then, for \alpha > 0 and f \in PA\alpha (K),
En(f,K) = O(n - \alpha ) as n\rightarrow \infty .
Since by [7, p. 331] (Theorem 1.5), the continuum from Theorem 1 has the VP-property, The-
orem 4 implies Theorem 1 and extends it to the most general family of continua for which such
statements can hold (at least for \alpha = 1).
Theorem 5 [5]. Let K be as in Theorem 2. Then, for \alpha > 0 and f \in PA\alpha (K),
En(f,K) = O
\bigl(
(\mathrm{l}\mathrm{o}\mathrm{g} n) - \alpha
\bigr)
as n\rightarrow \infty . (2.10)
This statement proves the Gaier conjecture.
Theorem 6 [5]. Let K = D( - 1, 1) \cup D(1, 1), \alpha > 0, and
f\ast \alpha (z) =
\left\{ z
\alpha if z \in D(1, 1),
0 if z \in D( - 1, 1).
Then
En(f
\ast
\alpha ,K) \geq c(\mathrm{l}\mathrm{o}\mathrm{g} n) - \alpha , n \in \BbbN \setminus \{ 1\} ,
holds with a constant c = c(\alpha ) > 0.
Hence, the estimate (2.10), conjectured by Gaier, cannot, in general, be improved.
3. Approximation of functions by reciprocals of polynomials. The starting point of our
investigation is the following result by Levin and Saff [16, 17]. Let, as above, I := [ - 1, 1] and for
a continuous function f : I \rightarrow \BbbC , let
E0,n(f, I) := \mathrm{i}\mathrm{n}\mathrm{f}
p\in \BbbP n
\bigm\| \bigm\| \bigm\| \bigm\| f - 1
p
\bigm\| \bigm\| \bigm\| \bigm\|
I
. (3.1)
Theorem 7 [17]. We have
E0,n(f, I) = O
\bigl(
\omega f,I(n
- 1)
\bigr)
as n\rightarrow \infty , (3.2)
where \omega f,I denotes the modulus of continuity of f on I.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
APPLICATION OF DZYADYK’S POLYNOMIAL KERNELS IN THE CONSTRUCTIVE FUNCTION THEORY 155
The approximation by reciprocals of polynomials is a well-known research direction in con-
structive function theory. The results related to Theorem 7 and further references can be found in
[8, 15, 18, 21, 26].
It was observed in [16] that the use of complex polynomials in Theorem 7 is essential even in
the case of real functions (which have a sign change in I ). Since f and p in (3.1) and (3.2) are
complex-valued, it seems quite natural to extend Theorem 7 to the case where I is changed to an arc
L in the complex plane \BbbC . Such an extension is the main subject of this section.
Note that Theorem 7 is an analogue of the classical Jackson theorem on polynomial approxima-
tion. Our results can be related in the same way to the Mergelyan’s extension of the Jackson theorem
for the complex plane (see [20] or [25], Chapter I, \S 7) and to results in [2, 10, 13, 14, 19, 24] concer-
ning polynomial approximation on arcs. A complete bibliography may be found in [10] (Chapter IX)
and [6] (Chapter 5).
Let L \subset \BbbC be a bounded Jordan arc, i.e., its complement \Omega := \BbbC \setminus L is a simply connected
domain. Denote by L(z1, z2) a subarc of L between points z1 \in L and z2 \in L.
We assume that L is quasismooth (in the sense of Lavrentiev), which means that\bigm| \bigm| L(z1, z2)\bigm| \bigm| \leq c| z1 - z2| , z1, z2 \in L,
where c = c(L) \geq 1 and
\bigm| \bigm| L(z1, z2)\bigm| \bigm| is the length of L(z1, z2).
Let function \Phi map \Omega conformally and univalently onto \BbbD \ast with standard normalization at
infinity and let for z \in L and \tau > 0,
L\tau :=
\bigl\{
\zeta \in \Omega : | \Phi (\zeta )| = 1 + \tau
\bigr\}
, \mathrm{i}\mathrm{n}\mathrm{t}L\tau := \BbbC \setminus
\bigl\{
z : | \Phi (z)| \geq 1 + \tau
\bigr\}
,
\delta 1/\tau = \delta 1/\tau (L) := \mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in intL\tau
d(\zeta , L\tau ).
Let C(L) be the set of continuous functions f : L\rightarrow \BbbC .
For f \in C(L) and \tau > 0 set
\omega f,L(\tau ) := \mathrm{s}\mathrm{u}\mathrm{p}
z1,z2\in L
| z1 - z2| \leq \tau
\bigm| \bigm| f(z1) - f(z2)
\bigm| \bigm| .
It is known (see, for example, [3]) that, for f \in C(L),
En(f, L) = O
\bigl(
\omega f,L(\delta n)
\bigr)
as n\rightarrow \infty . (3.3)
We are interested in the extension (3.3) to the case of functions approximated by reciprocals of
polynomials on a quasismooth arc in which the quantity n - 1 in (3.2) is replaced by \delta n.
The main result of this section is the following statement. For f \in C(L) and n \in \BbbN , let
E0,n(f, L) := \mathrm{i}\mathrm{n}\mathrm{f}
p\in \BbbP n
\bigm\| \bigm\| \bigm\| \bigm\| f - 1
p
\bigm\| \bigm\| \bigm\| \bigm\|
L
.
Theorem 8 [4]. For f \in C(L),
E0,n(f, L) = O
\bigl(
\omega f,L(\delta n)
\bigr)
as n\rightarrow \infty . (3.4)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
156 V. ANDRIEVSKII
From the reasoning in [16, 17] one can make an intuitive conclusion that the functions vanishing
on I are the worst ones for the approximation by reciprocals of polynomials. Surprisingly, for arcs
the situation seems to be different. We demonstrate this idea by showing the sharpness of Theorem 8
for the linear functions l\zeta (z) := z - \zeta .
For positive functions a and b we use the order inequality a \preceq b if a \leq cb with a constant
c > 0. The expression a \asymp b means that a \preceq b and b \preceq a simultaneously. Note that since
\omega l\zeta ,L(\tau ) = \tau , \tau > 0, the estimate (3.4) implies
E0,n(l\zeta , L) \preceq \delta n, n \in \BbbN , \zeta \in \BbbC .
Theorem 9 [4]. For n \in \BbbN ,
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in intL1/n
E0,n(l\zeta , L) \succeq \delta n, (3.5)
\mathrm{s}\mathrm{u}\mathrm{p}
\zeta \in L
E0,n(l\zeta , L) \preceq dn, (3.6)
where dn = dn(L) := \mathrm{s}\mathrm{u}\mathrm{p}z\in L d(z, L1/n).
If L is Dini-smooth, then according to the well-known distortion properties of \Phi , which can be
found in [22, p. 52] (Theorem 3.9) or [7, p. 32 – 36], we have dn \asymp \delta n \asymp n - 1. Therefore, in the case
of sufficiently smooth L (3.4) looks exactly like (3.2).
It also follows from the properties of \Phi (see [22, p. 52] (Theorem 3.9) or [7, p. 32 – 36]) that
if L consists of a finite number of Dini-smooth arcs which meet under angles \beta k\pi , 0 < \beta k \leq 1,
k = 1, . . . ,m, then
dn \asymp n - 1, \delta n \asymp n - \beta , \beta := \mathrm{m}\mathrm{i}\mathrm{n}
1\leq k\leq m
\beta k.
Therefore, for the piecewise smooth arcs (3.4) is “worse” than (3.2). However, by (3.5) the esti-
mate (3.4) is sharp. Moreover, according to (3.6) the “slowly approximable” linear functions have
zeros outside of L.
4. Acknowledgements. The author would like to warmly thank M. Nesterenko for many useful
remarks.
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Received 22.07.18
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
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| spelling | umjimathkievua-article-14272019-12-05T08:54:43Z Application of Dzyadyk’s polynomial kernels in the constructive function theory Застосування полiномiальних ядер Дзядика в конструктивнiй теорiї функцiй Andrievskii, V. V. Андрієвський, В. В. This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzjadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings. Наведено огляд нових результатiв у конструктивнiй теорiї функцiй, що отриманi автором iз застосуванням теорiї ядер Дзядика в поєднаннi з методами та результатами сучасної геометричної теорiї функцiй i теорiї квазiконформних вiдображень. Institute of Mathematics, NAS of Ukraine 2019-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1427 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 2 (2019); 151-157 Український математичний журнал; Том 71 № 2 (2019); 151-157 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1427/411 Copyright (c) 2019 Andrievskii V. V. |
| spellingShingle | Andrievskii, V. V. Андрієвський, В. В. Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title | Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title_alt | Застосування полiномiальних ядер Дзядика
в конструктивнiй теорiї функцiй |
| title_full | Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title_fullStr | Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title_full_unstemmed | Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title_short | Application of Dzyadyk’s polynomial kernels in the constructive function theory |
| title_sort | application of dzyadyk’s polynomial kernels in the constructive function theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1427 |
| work_keys_str_mv | AT andrievskiivv applicationofdzyadykspolynomialkernelsintheconstructivefunctiontheory AT andríêvsʹkijvv applicationofdzyadykspolynomialkernelsintheconstructivefunctiontheory AT andrievskiivv zastosuvannâpolinomialʹnihâderdzâdikavkonstruktivnijteoriífunkcij AT andríêvsʹkijvv zastosuvannâpolinomialʹnihâderdzâdikavkonstruktivnijteoriífunkcij |