Table Of ContentERROR ANALYSIS OF THE BERGMAN KERNEL
METHOD WITH SINGULAR BASIS FUNCTIONS
1
M. A.LYTRIDESAND N. S.STYLIANOPOULOS
1
0
2 Abstract. Let G be a bounded Jordan domain in the complex plane
n withpiecewiseanalyticboundary. Wepresenttheoreticalestimatesand
a numerical evidencefor certain phenomena,regarding theapplication of
J theBergmankernelmethodwithalgebraicandpolesingularbasisfunc-
3 tions, for approximating the conformal mapping of G onto the normal-
izeddisk. Inthisway,wecompletethetaskofprovidingfulltheoretical
] justification of thismethod.
A
N
.
h
t
a 1. Introduction
m
[ Let G be a bounded, simply-connected domain in the complex plane C
whose boundary Γ := ∂G is a Jordan curve and let Ω := C G denote the
1 \
complement of G with respect to the extended complex plane. Fix z G
v 0
∈
1 and let f denote the conformal map of G onto the disk D(0,r ) := z :
0 0
5 z < r , normalized by the conditions f (z ) = 0 and f′(z ) = 1. T{he
5 | | 0} 0 0 0 0
quantity r := r (G,z ) is called the conformal radius of G with respect to
0 0 0 0
1. z0.
For the inner product
0
1
1 f,g := f(z)g(z)dA(z), (1.1)
: h i
v ZG
Xi where dA denotes the differential of the area measure on C, we consider the
r Hilbert space
a
L2(G) := f : f analytic in G, f,f < , (1.2)
a h i ∞
(cid:8) 1 (cid:9)
with corresponding norm f L2(G) := f,f 2.
k k h i
Let K(,z ) denote the Bergman kernel function of G with respect to z .
0 0
·
This is the unique function of L2(G) satisfying the reproducing property
a
g,K(,z ) = g(z ), for all g L2(G). (1.3)
h · 0 i 0 ∈ a
Date: January 4, 2011.
1991 Mathematics Subject Classification. Primary 30C30; Secondary 30E10, 30C40,
65AE05.
Key words and phrases. Bergman orthogonal polynomials, Numerical conformal map-
ping, Bergman kernel method,Singular basis function.
1
2 M.LYTRIDESANDN.STYLIANOPOULOS
It follows from (1.3) that the kernel K(,z ) is related to the mapping func-
0
·
tion f by means of
0
K(z,z ) 1 z
f′(z) = 0 and f (z) = K(ζ,z )dζ, (1.4)
0 K(z ,z ) 0 K(z ,z ) 0
0 0 0 0 Zz0
see e.g. [3, p.33]. These yield the two relations,
1 1
K(z,z ) = f′(z) and r = . (1.5)
0 πr2 0 0 πK(z ,z )
0 0 0
Now, let {Pn}∞n=0 denote the sequence of thepBergman polynomials of G.
This is defined as the sequence of polynomials
P (z) = λ zn+ , λ > 0, n = 0,1,2,..., (1.6)
n n n
···
that are orthonormal with respect to the inner product (1.1), i.e.,
P (z)P (z)dA(z) = δ . (1.7)
m n m,n
ZG
The Bergman polynomials form a complete orthonormal system in L2(G).
a
Therefore, in view of the reproducing property (1.3),
∞
K(z,z ) = P (z )P (z), (1.8)
0 j 0 j
j=0
X
locally uniformly with respect to z G.
∈
The Bergman kernel method (BKM) is an orthonormalization method for
computing approximations to the conformal map f (z). It is based on the
0
factthatthekernelK(z,z )isgivenexplicitlyintermsoftheBergmanpoly-
0
nomials P (z) ∞ . Thus, the partial sums of the Fourier series expansion
{ n }n=0
of K(z,z ) are given by
0
n
K (z,z ) := P (z )P (z), n N. (1.9)
n 0 j 0 j
∈
j=0
X
The polynomials K (z,z ) ∞ are the so-called kernel polynomials of G,
{ n 0 }n=0
with respect to z . They provide the best L2(G)- approximation to K(,z )
0 0
out of the space P of complex polynomials of degree at most n. ·
n
In accordance with (1.4), the n-th BKM approximation to f is given by
0
1 z
π (z) := K (ζ,z )dζ, n N. (1.10)
n n−1 0
K (z ,z ) ∈
n−1 0 0 Zz0
This defines the sequence π ∞ of the Bieberbach polynomials of G, with
{ n}n=1
respect to z . Thepolynomial π solves the following minimal problem: Let
0 n
P∗ := p :p P , with p(z ) = 0 and p′(z ) = 1 .
n { ∈ n 0 0 }
Then, for each n N, the polynomial π minimizes uniquely the two norms
n
f′ p′ and∈ p′ over all p P∗; see e.g. [2, Kap. III, 1].
k 0− kL2(G) k kL2(G) ∈ n §
Regarding the convergence of the method, we note that in cases when f
0
has an analytic continuation across Γ, then this is a consequence of Walsh’s
ERROR ANALYSIS OF BKM/AB 3
theory of maximal convergence [22, 4.7, 5.3]. In order to be more specific,
§ §
let Φ denote the conformal map of Ω onto ∆ := w : w > 1 , normalized
{ | | }
so that near infinity,
γ γ
1 2
Φ(z) = γz+γ + + + , γ > 0. (1.11)
0 z z2 ···
Note that γ = 1/cap(Γ), where cap(Γ) denotes the (logarithmic) capacity
of Γ. Then,
1
f π = O , (1.12)
k 0− nkL∞(G) Rn
(cid:18) (cid:19)
holds for any 1 < R < Φ(z ), but for no R > Φ(z ), where z denotes
1 1 1
| | | |
the nearest singularity of f in Ω; see also [3, Ch. I]. (We use to
0 k·kL∞(G)
denote the sup-norm on G.)
In cases when Γ is piecewise analytic and f has singularities on Γ, then
0
Levin, Papamichael and Siderides were the first to observe in [11] that the
error (1.12) depends on the boundary singularities of the mapping func-
tion f on Γ, and also on the singularities of the extension of f across the
0 0
segments of Γ into Ω. Accordingly, in order to improve the numerical per-
formance of the BKM, they extended the method by orthonormalizing a
system of basis functions consisting from monomials, as in the BKM, and
also from functions that reflect the dominant singularities of f on Γ and in
0
Ω. This extension is known as BKM/AB (AB stands for augmented basis).
The BKM/AB was used subsequently in [15] and [16].
The most precise results regarding the convergence of the BKM are due
to D. Gaier [6]. In particular, under the assumption that Γ is piecewise
analytic without cusps, Gaier derived the estimate
1
f π = O(logn) , (1.13)
k 0− nkL∞(G) ns
wheres := λ/(2 λ) and λπ (0 < λ < 2) denotes the smallest exterior angle
−
where two analytic arcs of Γ meet. Regarding sharpness of the estimate
(1.13), it was shown in [5, Thm. 4] that there are cases where the exponent
s can not be replaced by a smaller number. However, the factor logn can
be replaced by √logn, see [1] and [12, Rem. 3.1]. A lower estimate of the
form
1
f π c , (1.14)
k 0− nkL∞(G) ≥ ns
provided that 1/(2 λ) is not a positive integer, where c is a constant that
−
does not dependon n, was established in [12, Thm. 3.2] by Maymeskul, Saff
and the second author.
The theoretical justification of the BKM/AB with basis function that
reflect the corner singularities of f was given in [12], by means of sharp
0
estimates for the associated BKM/AB errors. The purpose of the present
paper is to derive theoretical results that justify the use of basis functions
that reflect (a) pole singularities of f and (b) both corner and pole singu-
0
larities of f . More specifically, we derive upper and lower estimates for the
0
4 M.LYTRIDESANDN.STYLIANOPOULOS
BKM/AB errors in the case (a), and upper estimates for the BKM/AB er-
rors in the case (b). In doing so, we complete the task that was put forward
byYu.E.Khokhlov,revieweroftheintroductorypaper[11]oftheBKM/AB
in the Mathematical Reviews, who concluded that: “A proof of the conver-
gence of the numerical method given and an investigation of its convergence
rate are lacking, so the results obtained are of a heuristic nature”.
Thepaperis organized as follows: InSection 2wesetupthenotation and
recall the BKM/AB. Section 3 is devoted to the study of the various BKM
and BKM/AB errors, in cases when f has an analytic continuation across
0
Γ, hence only basis functions reflecting poles are used in the BKM/AB.
In Section 4, we consider the case when both corner and pole basis func-
tions are included in BKM/AB. Finally, in Section 5, we present numerical
computations that illustrate the theory of Sections 3 and 4.
2. The Bergman kernel method with singular basis functions
2.1. Corner singularities. Throughout this section we assume that the
boundary Γ of G consist of N analytic arcs that meet at corner points τ ,
k
k = 1,2,...,N, where they form interior angles α π, 0< α < 2. Then, we
k k
have the following asymptotic expansions for f , valid near τ :
0 k
(i) If α is irrational, then
k
f (z) = f (τ )+ B (z τ )p+q/αk, (2.1)
0 0 k p,q k
−
p,q
X
where p and q run over all integers p 0, q 1 and B = 0.
0,1
≥ ≥ 6
(ii) If α = a/b, with a and b relative prime numbers, then
k
f (z) = f (τ )+ B (z τ )p+q/αk(log(z τ ))m, (2.2)
0 0 k p,q,m k k
− −
p,q,m
X
wherep,q and m runover all integers p 0, 1 q a, 1 m p/b
≥ ≤ ≤ ≤ ≤
and B = 0.
0,1,0
6
(iii) If τ is formed by two straight-line segments, then
k
∞
f (z) = f (τ )+ B (z τ )l/αk, (2.3)
0 0 k l k
−
l=1
X
where B = 0. Furthermore, (2.1) holds in the case when τ is
1 k
6
formed by two circular arcs, or a straight-line and a circular arc.
In the above, (i) and (ii) are due to Lehman [9], while (iii) emerges easily
from the reflection principle; see also [6, 2.1] and [14, pp. 6–7].
§
It follows from (iii) that if G is a half-disk or a rectangle, then f has
0
a Taylor series expansion valid around each corner, and thus an analytic
continuation across Γ into Ω. In this case, the only singularities of f are
0
simple poles in Ω. This shows that the study of the BKM/AB, even with
only pole basis function is important in the applications.
ERROR ANALYSIS OF BKM/AB 5
For simplicity in the exposition, we shall assume throughout this paper
that no logarithmic terms occur in the asymptotic expansion of f near
0
the corner τ , k = 1,2,...,N. This, for example, will be the case in the
k
expansions (2.1) and (2.3) above. Nevertheless, our method of study can be
adjusted to cover logarithmic singularities as well.
LetM denotethenumberofcornersofΓforwhichα isnotofthespecial
k
form 1/m, m N. When we present results for corner singularities we shall
∈
assume that M 1. We index such corners by τ , k = 1,...,M. That is, if
k
≥
N > M, then the mapping function f has an analytic continuation in some
0
neighborhood of the corner τ .
N
For k = 1,...,M, we denote by γ(k) ∞ the increasing arrangement
{ j }j=1
of the possible powers p+q/α of (z τ ) that appear in the asymptotic
k k
−
expansion of f (z) near τ . In particular, if τ is formed by two straight-line
0 k k
(k)
segments, then γ = j/α , j = 1,2,.... Also, if α is irrational, or the
j k k
corner τ is formed by two circular arcs, then
k
(k)
γ = 1/α ;
1 k
(k)
γ = 1/α +min(1/α ,1);
2 k k
1/α +2, 0< α < 1/2,
k k
(k)
γ = 2/α , 1/2 < α < 1,
3 k k
1/α +1, 1< α < 2;
k k
.
.
.
Remark 2.1. Under the assumption regarding the no-appearance of loga-
rithmic terms, the asymptotic expansion near τ , k = 1,2,...,M, can be
k
written in the form
∞
f0(z) = a(jk)(z−τk)γj(k), (2.4)
j=0
X
(k) (k) (k)
where, γ := 0 and a = 0. Note that, we always have γ > 1/2, and
0 1 6 1
since τk is not a special corner, γ1(k) ∈/ N. Therefore (z − τk)γ1(k) has an
algebraic singularity at τ . However, when α is rational, it is possible that
k k
γj(k) ∈N, for indices j ≥ 2, so that (z−τk)γj(k) is analytic at τk.
2.2. Pole singularities. Since f (z ) = 0, z G, it follows from the
0 0 0
∈
reflection principle for analytic arcs that the extension of f across any
0
segment constituting Γ would have a pole or a pole-type singularity at the
reflected images of z . For example, if Γ consists explicitly from straight-
0
line segments and/or circular arcs, then f has a simple pole (due to the
0
univalency of f ) at every mirror image of z (with respect to the straight-
0 0
lines)andateverygeometric inverseofz (withrespecttothecirculararcs),
0
that lies in Ω. More generally, f may have at points z Ω a pole, or a
0 j
∈
poly-type, singularity of the form
(z z )−kj/mj, k ,m N. (2.5)
j j j
− ∈
6 M.LYTRIDESANDN.STYLIANOPOULOS
According to [17, 5.1], the following three special cases occur frequently in
§
the applications:
(i) k = m = 1. In this case, f has a simple pole at z .
j j 0 j
(ii) k = 2, m = 1. In this case, f has a double pole at z .
j j 0 j
(iii) k = 1, m = 2. In this case, f has a rational pole singularity at z .
j j 0 j
In order to describe the BKM/AB, we assume that the nearest singu-
larities of f in Ω are poles or rational poles, of the form (2.5) at points
0
z , j = 1,2,...κ, where Φ(z ) Φ(z ) ... Φ(z ) and that the
j 1 2 κ
| | ≤ | | ≤ ≤ | |
other singularities of f in Ω occur at points z ,z ,..., where Φ(z ) <
0 κ+1 κ+2 k
| |
Φ(z ) Φ(z ) ....
κ+1 κ+2
| | ≤ | | ≤
2.3. BKM/AB. Using the above notation, the BKM/AB with n mono-
mials, κ poles and p corner singularities at each (non-special) corner τ ,
k k
k = 1,2...,M, can be summarized as follows:
(i) Start with the augmented system η consisting of:
j
{ }
(1) the nearest poles or rational poles, i.e., for j = 1,2...κ,
1 kj/mj ′
η (z) = ; (2.6)
j
z z
(cid:20)(cid:18) − j(cid:19) (cid:21)
(2) the dominant r := M p algebraic singular functions, i.e.,
M k=1 k
for each non-special corner τ , k = 1,2,...,M,
k
P
ηκ+j(z) = [(z τk)γj(k)]′, j = 1,2,...pk; (2.7)
−
(3) the monomials
η (z) = (zj)′, j = 1,2,...,n. (2.8)
κ+rM+j
(As it was noted in Remark 2.1, it might be possible that γ(k) N.
j ∈
If this happens, we avoid redundancy in the basis by omitting such
(k)
γ .)
j
(ii) Orthonormalize η , bymeansoftheGram-Schmidtprocesstopro-
j
{ }
duce the orthonormal set P , where
j
{ }
j
P (z) = be η (z), b > 0. (2.9)
j j,i i j,j
i=1
X
(iii) Approximate Ke(z,z ) by its finite Fourier expansion with respect to
0
P :
j
{ }
κ+rM+n κ+rM+n
Ke (z,z ) := P (z )P (z) = d η (z). (2.10)
n 0 j 0 j n,j j
j=1 j=1
X X
e e e
(iv) Approximate f (z) by
0
1 z κ+rM+n
π (z) := K (ζ,z )dζ = c µ (z), (2.11)
n+1 n 0 n,j j
Kn(z0,z0)Zz0 j=1
X
e e
e
ERROR ANALYSIS OF BKM/AB 7
where
z
µ (z) := η (ζ)dζ. (2.12)
j j
Zz0
Wecallthefunctions P theaugmentedBergmanpolynomials ofG,with
j
{ }
respectto η , andthefunctions π theaugmented Bieberbach polynomi-
j n
als over th{e sy}stem µ .eClearly,{π }(z ) = 0 and π′(z ) = 1, n N. Note
{ j} n 0 n 0 ∈
that{Pj}∞j=1 formsacomplete orthoenormalsysteminL2a(G). Consequently,
e∞ e
e K(z,z ) = P (z )P (z), (2.13)
0 j 0 j
j=1
X
e e
locally uniformly with respect to z G, cf. (1.8).
∈
We conclude this section by presenting a result which shows that the two
errors f′ π′ and K(,z ) K (,z ) areof thesameorder.
k 0− n+1kL2(G) k · 0 − n · 0 kL2(G)
This fact will be used below in Sections 3 and 4.
Inwhatfoellows wedenoteby c, c1, c2e,..., constants thatare independent
of n. For quantities A > 0, B > 0, we use the notation A B (inequality
(cid:22)
with respect to the order) if A cB. The expression A B means that
≤ ≍
A B and B A simultaneously.
(cid:22) (cid:22)
Lemma 2.1. It holds that,
f′ π′ K(,z ) K (,z ) . (2.14)
k 0− n+1kL2(G) ≍ k · 0 − n · 0 kL2(G)
Proof. We set m := κ+r +n and note that (1.5), (2.10)–(2.13), imply:
e M e
∞ m m
f′(z) π′ (z) =πr2 P (z )P (z) P (z )2 −1 P (z )P (z)
0 − n+1 0 j 0 j −{ | j 0 | } j 0 j
j=1 j=1 j=1
X X X
e m e me e e e
= πr2 P (z )2 −1 P (z )P (z)
0 −{ | j 0 | } j 0 j
j=1 j=1
X(cid:2) X (cid:3)
∞ e e e
+πr2 P (z )P (z).
0 j 0 j
j=m+1
X
e e
Therefore, by using the orthonormality of P we see that,
j
m ∞
f′ π′ 2 = πr2 1/K (z ,z ) 2eP (z )2+(πr2)2 P (z )2.
k 0− n+1kL2(G) 0− n 0 0 | j 0 | 0 | j 0 |
j=1 j=m+1
X(cid:2) (cid:3) X
Now,eusing once more (1.5), weeobtain, afteresome trivial calculatione, that
−1
f′ π′ 2 = K(z ,z ) K (z ,z ) K(z ,z )K (z ,z ) .
k 0− n+1kL2(G) 0 0 − n 0 0 0 0 n 0 0
h ih i
This ande(1.3), with g(·) = K(·,z0)−eKn(·,z0), leads to e
K(,z ) K (,z ) 2
f′ π′ 2 = k e· 0 − n · 0 kL2(G) (2.15)
k 0− n+1kL2(G) K(z ,z )K (z ,z )
0 0 n 0 0
e
e
e
8 M.LYTRIDESANDN.STYLIANOPOULOS
and the result (2.14) follows from the set of the obvious inequalities,
P (z ) = K (z ,z ) K (z ,z ) K(z ,z ) = (πr2)−1,
| 1 0 | 1 0 0 ≤ n 0 0 ≤ 0 0 0
with constants depending on r and P (z ) only. (cid:3)
0 1 0
e e e | |
Remark 2.2. It is clear from the proof that the result of Lemma 2.1 holds
e
true for any complete orthonormal system. We note that for the system
P ∞ to be complete it suffices that Γ is a bounded Jordan curve. In
{ j}j=1
particular, (2.14) holds with π and K in the place of π and K .
n+1 n n+1 n
e
3. BKM/AB with pole singularitiese e
InthissectionwestudytheBKMandBKM/ABerrors f π and
k 0− nkL∞(G)
f π , under the assumption that f has an analytic continuation
k 0− nkL∞(G) 0
across Γ in Ω and its only singularities are poles, or rational poles, of the
type (e2.5). More precisely, we refine the classical result (1.12) for the BKM
error, and at the same time we obtain a lower estimate for it. Furthermore,
we establish upper and lower estimates for the BKM/AB error. The lower
estimates and the refinement are obtained by exploiting the assumption
regarding the singularities of f and by using certain important results of
0
E.B. Saff on polynomial interpolation of meromorphic functions [18]. Since
the results of [18] were established for domains with smooth boundaries, we
show in the next lemma that they hold true for domains with corners.
In order to do so, we use the Faber polynomials F ∞ of G. We recall
{ n}n=0
that F (z) is defined as the polynomial part of the Laurent series expansion
n
of Φn at infinity, i.e.,
1
F (z) Φn(z) = O , as z . (3.1)
n
− z → ∞
(cid:18) (cid:19)
This, in view of (1.11), gives F P and
n n
∈
F (z) = γnzn + . (3.2)
n
···
Let L (R 1), denote the level curve of index R of Φ, i.e.,
R
≥
L := z : Φ(z) = R , (3.3)
R
{ | | }
sothatL Γ. NotethatL ,forR > 1,isananalyticJordancurve. Weuse
1 R
≡
G to denote its interior, i.e., G := int(L ). The following result gives the
R R R
exact rate of convergence of the minimum uniform error in approximating
meromorphic functions by polynomials.
Lemma 3.1. Assume that the boundary Γ of G is piecewise Dini-smooth
and consider a function f which is analytic on G , for some ̺ > 1, apart
̺
from a finite number of poles on L . Let m denote the highest order of the
̺
poles of f on L . Then,
̺
nm−1
inf f p . (3.4)
p∈Pnk − kL∞(G) ≍ ̺n
ERROR ANALYSIS OF BKM/AB 9
AcurveΓispiecewiseDini-smoothifitconsistsofafinitenumberofDini-
smooth arcs. An arc z = z(s), where s [a,b] stands for the arclength, is
∈
called Dini-smooth if z′(s) is continuous on [a,b], and if z′(s) has a modulus
α
of continuity ω which satisfies [ω(t)/t] dt < , for some α > 0. We note,
0 ∞
in particular, that a piecewise Dini-smooth curve may have corners or cusps
R
and that a piecewise analytic Jordan curve is also piecewise Dini-smooth.
Proof. We recall the following two facts regarding Faber polynomials:
(i) For any r, R, with 1< r < R, it holds
rn
F (z) = Φn(z) 1+O , z L , (3.5)
n Rn ∈ R
(cid:26) (cid:18) (cid:19)(cid:27)
see e.g. [21, p. 43].
(ii) Under the assumption on Γ, the Faber polynomials are uniformly
bounded on G (see [7]), i.e.,
F c(Γ), n N, (3.6)
k nkL∞(G) ≤ ∈
where c(Γ) is a positive constant that depends on Γ only.
Observe that (3.5) implies that the sequence F (z) ∞ has no limit
{ n }n=1
point of zeros exterior to G. Also, from (3.5) and (3.6) we have for z G
∈
and t L that,
̺
∈
F (z) c (Γ)
| n | 1 , n N. (3.7)
F (t) ≤ ̺n ∈
n
| |
Now, following the proof of Theorem 2 of [18] and using the sequence of the
Faber polynomials F in the place of ω , we conclude that there exist
n n
{ } { }
polynomials p ∞ , such that
{ n}n=1
nm−1
f p c (Γ) , n N, (3.8)
k − nkL∞(G) ≤ 2 ̺n ∈
see also [19, p. 399]. This yields the upper bound in (3.4). The lower bound
follows at once from Theorem 10 of [18], by observing that Ω is simply-
connected and hence its Green function with pole at infinity has no critical
points. (cid:3)
The following result is the so-called Andrievskii’s lemma for polynomials
and rational polynomials. Its proof, for bounded Jordan domains such that
the inverse conformal map g : D G satisfies a Lipschitz condition on
D, can be found in [4]. This condit→ion is certainly satisfied by the type of
domains considered below.
Lemma 3.2. Assume that Γ is piecewise analytic without cusps. Then:
(i) For any P P , with P (z )= 0, it holds
n n n 0
∈
P logn P′ , n 2. (3.9)
k nkL∞(G) (cid:22) k nkL2(G) ≥
p
10 M.LYTRIDESANDN.STYLIANOPOULOS
(ii) For any P P , with P (z )= 0, and q a fixed polynomial with no
n n n 0
∈
zeros on G, it holds that
P /q logn (P /q)′ , n 2. (3.10)
k n kL∞(G) (cid:22) k n kL2(G) ≥
3.1. BKM. The next theorempcomplements the classical result (1.12) of
Walsh, in the sense that it provides a lower estimate and, in addition, uses
theprecise̺ = Φ(z ) inthedenominator,insteadofanyR,with1< R < ̺.
1
| |
This is done by utilizing extra information on the nature of the singularities
of f in Ω.
0
Theorem 3.1. Assume that Γ is piecewise analytic without cusps. Assume
further that the conformal map f has an analytic continuation across Γ,
0
such that f is analytic on G , for some ̺ > 1, apart from a finite number
0 ̺
of poles on L . Let m denote the highest order of the poles of f on L .
̺ 0 ̺
Then,
nm−1 nm√logn
f π , n 2. (3.11)
̺n (cid:22) k 0− nkL∞(G) (cid:22) ̺n ≥
Proof. We observe first that the kernel K(z,z ) shares the same analytic
0
properties with f on G , apart from an unit increase on the order of its
0 ̺
poles on L . Therefore, using Lemma 3.1 with f K(,z ), we conclude
̺ 0
≡ ·
that
nm
K(,z ) p , n N, (3.12)
k · 0 − nkL∞(G) (cid:22) ̺n ∈
for some sequence of polynomials p ∞ . Since the L2(G)-norm is domi-
{ n}n=1
nated by the L∞(G)-norm, (3.12) leads to the estimate
nm
K(,z ) p . (3.13)
k · 0 − nkL2(G) (cid:22) ̺n
Then, the minimum property of the kernel polynomials implies that
nm
K(,z ) K (,z ) , (3.14)
k · 0 − n · 0 kL2(G) (cid:22) ̺n
which, in conjunction with Remark 2.2, yields the estimate
nm
f′ π′ . (3.15)
k 0− nkL2(G) (cid:22) ̺n
Now, we use Andrievskii’s Lemma 3.2(i) and employ the method of An-
drievskii and Simonenko, see e.g. [5, 2.1]. This method enables the transi-
§
tion from an upper bound of the error f′ π′ to a similar bound for
k 0− nkL2(G)
theerror f π , withthe extracost of a√lognfactor, andleads to
k 0− nkL∞(G)
the upper estimate in (3.11). The lower estimate follows immediately from
Lemma 3.1. (cid:3)
The following pointwise estimate is useful in the study of the distribution
of the zeros of the Bergman polynomials; see e.g., [10], [13], [19] and [8].
