Finite-infinite range inequalities in the complex plane

Abstract

Let E⫅C be closed, ω be a suitable weight function on E, σ be a positive Borel measure on E. We discuss the conditions on ω and σ which ensure the existence of a fixed compact subset K of E with the following property. For any p, 0<P≤∞, there exist positive constants c1, c2 depending only on E, ω, σ and p such that for every integer n≥1 and every polynomial P of degree at most n, ∫EK|ωnP|pdσ≤c1exp(−c2n)∫K|ωnP|pdσ. In particular, we shall show that the support of a certain extremal measure is, in some sense, the smallest set K which works. The conditions on σ are formulated in terms of certain localized Christoffel functions related to σ.

Full text

Page 1

Internat.
VOL.
J. Math.
4 (1991) 625-638
& Math.
Sci.
14 NO.
FINITE-INFINITE-RANGEINEQUALITIES IN THE COMPLEX PLANE
625
H.N. MHASKAR
Department of Mathematics, California State University,
Los Angeles, California, 90032. U.S.A.
(Received April 23,
1990)
ABSTRACT. Let E C C be closed, w be a suitable weight function on E, a be a positive
Borel measure on E. We discuss the conditions on w and a which ensure the existence of
a fixed compact subset K of E with the following property. For any p, 0 < p _< oo, there
exist positive constants cl, c2 depending only on E, w, a and p such that for every integer
n > 1 and every polynomial P of degree at most n,
f
In particular, we shall show that the support of a certain extremal measure is, in some
[w"P[’da <clexp(-c2n)/K[w"P[’da"
sense, the smallest set K which works. The conditions on a are formulated in terms of
certain localized Christoffel functions related to a.
KEY WORDS AND PHRASES. Finite-infinite range inequalities, orthogonal polynomials,
weighted polynomials, Nikolskii inequalities.
1980 SUBJECT CLASSIFICATION CODE. 26C05, 31A15, 42C05, 41A17
1. INTRODUCTION.
Let a > 1 and wo,(x) := exp(-[x[), x E R. A special case of an inequality proved by
G. Freud [1] is the following. There exist positive constants cl, c, ca tepending only on a
such that for every integer n > 1 and polynomial P of degree at most n,
l=l>c,[w:(x)P(x)12dx <c2 exp(-c3n)l=l<c,[w:(x)P(x)ldx"
An inequality of the form (1.1), known as afinite-infiniterange inequality, is of critical
importance in the study of weighted polynomial approximation on It. In the past few
(1.1)
years, many mathematicians have investigated inequalities of this form in great detail (el.
[3] for example). In particular, Mhaskar and Saff [4] have obtained a highly generalized
and precise form of (1.1), using potential theoretic ideas. These ideas, in turn, have been
extended to the case when the underlying sets are subsets of the complex plane 12 rather
than the real line. In studying the finite-infinite range inequalities on subsets of 12, an
immediate problem which must be solved is to decide upon a natural choice of a measure
to replace the one-dimensional Lebesgue measure which is so natural for subsets of the

Page 2

626
H.N. MHASKAR
real line. In particular, one has to include the arc-length measure on sufficiently smooth
curves as well as the area measure on domains and their combinations.
In this paper, we define one such class of ’natural’ measures and then extend the
results of [4] to the complex domain.
Christoffel functions in this theory. In turn, we estimate these Christoffel functions when
Our results will demonstrate the importance of
the measure under consideration satisfies certain density conditions. We also describe
certain applications to the theory of extremal polynomials in the complex plane.
In Section 2, we develop the necessary technical background including some of the
definitions. The main results and applications will be described in Section 3. The proofs
of the new results will be given in Section 4.
2. PREPARATORY INFORMATION.
We say that a function w" C--,[0, c) is admiaible (or an admissible weight function) if
each of the following conditions (Wl), (W2), (W3) holds.
(W1) w is upper semi-continuous.
{W2) The set E0(w)"= {z "w(z) > 0} has nonzero capacity.
(W3) IfE0(w) is unbounded, then Izw(z)l--,0 asIzl
Here and throughout this paper, the term ’capacity’ means the inner logarithmic
capacity (cf. [13], p.55). For any set A C_ C, its capacity will be denoted by cap(A). A
property is said to hold quasi-everywhere (q.e.) on a set A if the subset of A where it
does not hold is of capacity zero. Let E be a closed subset of C. A function w is called
oo, z E Eo(w).
admiaible on E if the functionwE that coincides with w on E and is zero on C\E is
Typical examples of admissible weight functions are (i)exp(-Izl"), a > 0,
which are admissible on every closed subset of C having positive capacity and (ii) any
positive continuous function on any compact subset of C which has positive capacity. Let
2(E) denote the class of all unit positive Borel measures supported on E. If a E .4(E),
admissible.
then its potential is defined by
U(a,z):=/Elg(1/Iz
The(w-modified) energy of a is defined by
We define
X(w,) :=fru(, z)do(z).
V(w,E) :=inf{I(w,a)"a e .A4(E)},
zeC.
(2.1)
(2.2)
Q(z) := log(liT(z)),
z
c.
(2.4)
For integer n >_ 0, IIn will denote the class of all algebraic polynomials of degree at most
n. Finally, for any set A C C and a complex valued function g on A, we write
(2.3)

Page 3

FINITE-INFINITE RANGE INEQUALITIES
627
I111
sup I()1.
zA
(.)
Our starting point is the following theorem proved in [5] in the case when E C_ R.
The version stated below can be proved in exactly the same way as in [5] with obvious and
minor modifications.
THEOREM 2.1.
Let E C C be closed, w be an admissiblefunctionon E.
(a) The quantity V(w, E) definedin (.3) isfinite.
(b) There exists a unique element
hasfinite logarithmic energy.
(c) The set S := S(w, E) := supp(l) is a compact subsetof E0(w)qE, and cap(S) > 0.
Moreover, Q is boundedfromabove on S.
:= p(w,E) E .M(E) such that I(w,l)
Y(w,E).
This measure
F := F(w,E):= V(w,E)-/Qdt,
u(,, z) + Q(z) > F,
q.. o, E.
(2.6)
(2.7)
(e) We have
U(t,,z) + Q(z) < F,
forevery z C S.
(2.8)
In particular, the equation
U(I, z) + Q(z)
F
(2.9)
holds q. e. on S.
(f) For any positive integer n and P
II,if
Iw"(z)P(z)l <_ M
q.c. on S,
(2.10)
then
IP(z)l <_ Mexp(-nU(#, z) + nF),
In particular, (ILI O) implies that
zCC.
(2.11)
Iw"(z)P(z)l < M
q.e. on E.
(2.12)
()
S" :=S’(w,E) := {z C E" U(I,Z) + Q(z) < F},
(2.13)
then ,.9* D_ S is a compact set, Q is bounded on S* andfor any integer n > 0 and P
II,,
IIw"PIl
IIw"PIIs..
(2.14)
In fact, forany compact subset K C_ E\S*, there are positive constants cl,c2 depending
only on w, E,K such thatforany integer n > 1 and P
II,,,
IIw"PIIK< c, exp(-cn)llw"Pllso.
(2.15)

Page 4

628
H.N. MHASKAR
The classical notions of capacity, transfinite diameter, Fekete and Chebyshev polyno-
mials, Fekete points and Chebyshev constant can also be generalized. ([10]). For example,
thew-modifiedcapacity of a closed set E is defined by
cap(w, E)
exp(-V(w, E)).
(2.16)
The Fekete points are the points {z,,,
k
1,2,..-,n,
n
2,3,...} for which
}2/,,(,,-)
r,(w, E):=
max
H
]zi
zt’lw(zj)w(z’)
zt,...,zn_E
l<_j<k<_n
II
)"
_<<:_<n
(2.17)
Thew-modified transfinitediameter of the set E is defined by
r(w, E)
inf r,,(w E).
n>2
(2.18)
An argument similar to that in ([2l, P. 161) can be used (cf. [10]) to show not only that
cap(w, E)
r(w,E),
(2.19)
but that for any set of Fekete points {z,
k
1,2,.-.,n,
continuous, compactly supported function g" E-
C,
n
2,3,...} and any
lim-1g(,zk.)=/gag,
k=l
(2.20)
where p is the equilibrium measure defined in Theorem 2.1.
The notion of the Chebyshev constant can also be explored in this generality. We
shall elaborate on this in the context of the LP extremal polynomials. However, we note
here the following simple consequence of(2.15) and (2.20).
PROPOSITION 2.2.
([I0])
integer n and polynomial P E II,,
IfK C_ E is compact, andfor every sufficiently large
IIw"ell
IIw"PIIK,
(2.21)
then the support set Sdefinedin Theorem .1 is a subsetofK.
If A C_ C is a Borel set, a is a positive Borel measure on A, g
A
C is Borel
measurable, then we define
if 0 < p < oo,
(2.22)
if p= oo.
The spaces LP(a, A) are then defined as usual. We observe that our definition of Ilgll,,A

Page 5

FINITE-INFINITE RANGE INEQUALITIES
629
is not the usual one, but in the present context, the distinction would be inconsequential.
Also, the fact that (2.22) does not define a norm if 0 < p < 1 will be irrelevant for our
purposes. Thus, we will continue to call the expressions defined in (2.22) ’norms’ even in
this case.
3. MAIN RESULTS.
Let E C_ (3 be closed, w be an admissible weight function on E, and a be a fixed positive
Borel measure defined on E. We shall assume that w is continuous on E and [zw(z)]" E
L"(a, E) for every r, 0 < r < oz. We also continue the notation described in Section 2.
In the sequel, we assume the notation and conditions of Theorem 2.1. The set E and
the weight function w being fixed quantities, they will not be mentioned in the notations.
We also adopt the following convention concerning constants. The symbols c, 0,c2,."
will denote positive constants depending only on w,E and other explicitly prescribed
parameters. Their value may not be the same in different occurrences of the stone symbol,
even within the same formula. Constants denoted by capital letters will retain their values
after they are introduced.
DEFINITION 3.1. Let K C_ E be compact, 0 < p < oo. We say that the Lt’(a,E)
norm of(w-) weighted polynomials liven on K if the following property holds. For every
e > 0, there exists a bounded Borel set A C E with a(A) < e and positive constants c,c2
(depending only on w, E,p,e) such that for every integer n > 1 and polynomial P E 1-In,
.Iw"P]I,.,.E<_(1+c,exp(-cn))llw"PIl,,.,-va
(3.1)
Theorem 2.1 shows that the sup norm of weighted polynomials lives on 8". We shall
show that for suitable measures a, the L’(a,E) norm also lives on $*. As mentioned in
the introduction, a major issue here is the dcfinition of a ’natural’ measure in this case.
We shall show that the following definition will work.
DEFINITION 3.2. The measure a will be called natural if it satisfies each of the
following conditions.
(M1) a is a regular measure in the sense that for any Borel set A C_ E,
a(A)=inf{a(O)’AC_O,
Oopen
(3.2)
sup{a(Ix’)"gC_ A,
K compact
(M2) For any compact subset K of E, a(K) < oz and the restriction of a to K has finite
logarithmic energy.
(M3) There exists an integer N > 0 such that
Izl-Nda(z) < oz.
(3.3)
(M4) For
> 0 and z
(3, let